generalizing pagerank (pisa)
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DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
The Choice of a Damping Function forPropagating Page Importance
in Link-Based Ranking
Ricardo Baeza-Yates1 Paolo Boldi2 and Carlos Castillo3
1 Yahoo Research ndash Barcelona Spain2 Universita di Milano ndash Italy
3 Universita di Roma ldquoLa Sapienzardquo ndash Italy
February 6th 2005
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
1 Notation
2 Rewriting PageRank
3 Functional Rankings
4 Algorithms
5 Comparison
6 Conclusions
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Let PNtimesN be the normalized link matrix of a graph
Row-normalized
No ldquosinksrdquo
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (PageRank)
Stationary state of
αP +(1minus α)
N1NtimesN
Follow links with probability α
Random jump with probability 1minus α
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (PageRank)
Stationary state of
αP +(1minus α)
N1NtimesN
Follow links with probability α
Random jump with probability 1minus α
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Rewriting PageRank [Boldi et al 2005]
r(α) =(1minus α)
N
infinsumt=0
(αP)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (Branching contribution of a path)
Given a path p = 〈x1 x2 xt〉 of length t = |p|
branching(p) =1
d1d2 middot middot middot dtminus1
where di are the out-degrees of the members of the path
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Distribution of shortest paths
it (40M pages) uk (18M pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
euint (800K pages) Synthetic graph (100K pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with α=08damping(t) with α=07
Exponential damping = PageRank
damping(t) = α(1minus α)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with L=15damping(t) with L=10
Linear damping
damping(t) =
2(Lminust)L(L+1) t lt L
0 t ge L
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
1 Notation
2 Rewriting PageRank
3 Functional Rankings
4 Algorithms
5 Comparison
6 Conclusions
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Let PNtimesN be the normalized link matrix of a graph
Row-normalized
No ldquosinksrdquo
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (PageRank)
Stationary state of
αP +(1minus α)
N1NtimesN
Follow links with probability α
Random jump with probability 1minus α
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (PageRank)
Stationary state of
αP +(1minus α)
N1NtimesN
Follow links with probability α
Random jump with probability 1minus α
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Rewriting PageRank [Boldi et al 2005]
r(α) =(1minus α)
N
infinsumt=0
(αP)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (Branching contribution of a path)
Given a path p = 〈x1 x2 xt〉 of length t = |p|
branching(p) =1
d1d2 middot middot middot dtminus1
where di are the out-degrees of the members of the path
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Distribution of shortest paths
it (40M pages) uk (18M pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
euint (800K pages) Synthetic graph (100K pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with α=08damping(t) with α=07
Exponential damping = PageRank
damping(t) = α(1minus α)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with L=15damping(t) with L=10
Linear damping
damping(t) =
2(Lminust)L(L+1) t lt L
0 t ge L
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Let PNtimesN be the normalized link matrix of a graph
Row-normalized
No ldquosinksrdquo
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (PageRank)
Stationary state of
αP +(1minus α)
N1NtimesN
Follow links with probability α
Random jump with probability 1minus α
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (PageRank)
Stationary state of
αP +(1minus α)
N1NtimesN
Follow links with probability α
Random jump with probability 1minus α
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Rewriting PageRank [Boldi et al 2005]
r(α) =(1minus α)
N
infinsumt=0
(αP)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (Branching contribution of a path)
Given a path p = 〈x1 x2 xt〉 of length t = |p|
branching(p) =1
d1d2 middot middot middot dtminus1
where di are the out-degrees of the members of the path
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Distribution of shortest paths
it (40M pages) uk (18M pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
euint (800K pages) Synthetic graph (100K pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with α=08damping(t) with α=07
Exponential damping = PageRank
damping(t) = α(1minus α)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with L=15damping(t) with L=10
Linear damping
damping(t) =
2(Lminust)L(L+1) t lt L
0 t ge L
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (PageRank)
Stationary state of
αP +(1minus α)
N1NtimesN
Follow links with probability α
Random jump with probability 1minus α
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (PageRank)
Stationary state of
αP +(1minus α)
N1NtimesN
Follow links with probability α
Random jump with probability 1minus α
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Rewriting PageRank [Boldi et al 2005]
r(α) =(1minus α)
N
infinsumt=0
(αP)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (Branching contribution of a path)
Given a path p = 〈x1 x2 xt〉 of length t = |p|
branching(p) =1
d1d2 middot middot middot dtminus1
where di are the out-degrees of the members of the path
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Distribution of shortest paths
it (40M pages) uk (18M pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
euint (800K pages) Synthetic graph (100K pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with α=08damping(t) with α=07
Exponential damping = PageRank
damping(t) = α(1minus α)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with L=15damping(t) with L=10
Linear damping
damping(t) =
2(Lminust)L(L+1) t lt L
0 t ge L
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (PageRank)
Stationary state of
αP +(1minus α)
N1NtimesN
Follow links with probability α
Random jump with probability 1minus α
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Rewriting PageRank [Boldi et al 2005]
r(α) =(1minus α)
N
infinsumt=0
(αP)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (Branching contribution of a path)
Given a path p = 〈x1 x2 xt〉 of length t = |p|
branching(p) =1
d1d2 middot middot middot dtminus1
where di are the out-degrees of the members of the path
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Distribution of shortest paths
it (40M pages) uk (18M pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
euint (800K pages) Synthetic graph (100K pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with α=08damping(t) with α=07
Exponential damping = PageRank
damping(t) = α(1minus α)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with L=15damping(t) with L=10
Linear damping
damping(t) =
2(Lminust)L(L+1) t lt L
0 t ge L
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Rewriting PageRank [Boldi et al 2005]
r(α) =(1minus α)
N
infinsumt=0
(αP)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (Branching contribution of a path)
Given a path p = 〈x1 x2 xt〉 of length t = |p|
branching(p) =1
d1d2 middot middot middot dtminus1
where di are the out-degrees of the members of the path
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Distribution of shortest paths
it (40M pages) uk (18M pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
euint (800K pages) Synthetic graph (100K pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with α=08damping(t) with α=07
Exponential damping = PageRank
damping(t) = α(1minus α)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with L=15damping(t) with L=10
Linear damping
damping(t) =
2(Lminust)L(L+1) t lt L
0 t ge L
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Definition (Branching contribution of a path)
Given a path p = 〈x1 x2 xt〉 of length t = |p|
branching(p) =1
d1d2 middot middot middot dtminus1
where di are the out-degrees of the members of the path
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Distribution of shortest paths
it (40M pages) uk (18M pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
euint (800K pages) Synthetic graph (100K pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with α=08damping(t) with α=07
Exponential damping = PageRank
damping(t) = α(1minus α)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with L=15damping(t) with L=10
Linear damping
damping(t) =
2(Lminust)L(L+1) t lt L
0 t ge L
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Distribution of shortest paths
it (40M pages) uk (18M pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
euint (800K pages) Synthetic graph (100K pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with α=08damping(t) with α=07
Exponential damping = PageRank
damping(t) = α(1minus α)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with L=15damping(t) with L=10
Linear damping
damping(t) =
2(Lminust)L(L+1) t lt L
0 t ge L
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Distribution of shortest paths
it (40M pages) uk (18M pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
euint (800K pages) Synthetic graph (100K pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with α=08damping(t) with α=07
Exponential damping = PageRank
damping(t) = α(1minus α)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with L=15damping(t) with L=10
Linear damping
damping(t) =
2(Lminust)L(L+1) t lt L
0 t ge L
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Explicit formula for PageRank [Newman et al 2001]
ri (α) =sum
pisinPath(minusi)
(1minus α)α|p|
Nbranching(p)
Path(minus i) are incoming paths in node i
General functional ranking
ri (α) =sum
pisinPath(minusi)
damping(|p|)N
branching(p)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Distribution of shortest paths
it (40M pages) uk (18M pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
euint (800K pages) Synthetic graph (100K pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with α=08damping(t) with α=07
Exponential damping = PageRank
damping(t) = α(1minus α)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with L=15damping(t) with L=10
Linear damping
damping(t) =
2(Lminust)L(L+1) t lt L
0 t ge L
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Distribution of shortest paths
it (40M pages) uk (18M pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
euint (800K pages) Synthetic graph (100K pages)
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
00
01
02
03
5 10 15 20 25 30
Freq
uenc
y
Distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with α=08damping(t) with α=07
Exponential damping = PageRank
damping(t) = α(1minus α)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with L=15damping(t) with L=10
Linear damping
damping(t) =
2(Lminust)L(L+1) t lt L
0 t ge L
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with α=08damping(t) with α=07
Exponential damping = PageRank
damping(t) = α(1minus α)t
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with L=15damping(t) with L=10
Linear damping
damping(t) =
2(Lminust)L(L+1) t lt L
0 t ge L
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
000
010
020
030
1 2 3 4 5 6 7 8 9 10
Wei
ght
Length of the path (t)
damping(t) with L=15damping(t) with L=10
Linear damping
damping(t) =
2(Lminust)L(L+1) t lt L
0 t ge L
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
For calculating LinearRank we use
LinearRank =1
N
infinsumt=0
damping(t)Pt
=1
N
Lminus1sumt=0
2(Lminus t)
L(L + 1)Pt
However we cannot hold the temporary Pt in memory
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
We have to rewrite to be able to calculate
R(0) =2
L + 1
R(k+1) =(Lminus k minus 1)
(Lminus k)R(k)P
LinearRank =Lminus1sumk=0
R(k)
Now we can give the algorithm
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for
4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 0
6 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for
11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquorsaquo
1 for i 1 N do Initialization2 Score[i] larr R[i] larr 2
L+13 end for4 for k 1 Lminus 1 do Iteration step5 Aux larr 06 for i 1 N do Follow links in the graph7 for all j such that there is a link from i to j do8 Aux[j] larr Aux[j] + R[i]outdegree(i)9 end for
10 end for11 for i 1 N do Add to ranking value12 R[i] larr Aux[i] times (Lminuskminus1)
(Lminusk)
13 Score[i] larr Score[i] + R[i]14 end for15 end for16 return Score
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquorsaquo
Other functions studied in the paper
Hyperbolic damping
Empirical damping
02
03
04
05
06
07
1 2 3 4 5
Ave
rage
text
sim
ilari
ty
Link distance
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquorsaquo
How to approximate one functional ranking with another
Analysis (in the paper) match the first few levels of theirdamping functions
In practice the orderings can be very similar
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquorsaquo
Experimental comparison 18-million nodes in the UK WebGraph
Calculated PageRank with α = 01 02 09
Calculated LinearRank with L = 5 10 25
For certain combinations of parameters the rankings arealmost equal
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquorsaquo
Experimental Comparison in the UK Web Graph
05 06
07 08
09
5 10
15 20
25
080085090095100
τ
τ ge 095
αL
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquorsaquo
Prediction of Best Parameter Combinations (Analysis)
5
10
15
20
25
05 06 07 08 09
L th
at m
axim
izes
Ken
dallrsquo
s τ
Exponent α
Actual optimumPredicted optimum with length=5
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquorsaquo
What have we done
Separate the damping from the calculation
Show that different damping functions can provide thesame ranking
Analysis and experiments in the paper
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullrsaquo
What can we do with this
Fast approximation of PageRank using linear damping
Fast calculation of other link-based rankings (eg HITS)
Spam detection (eg cut the first levels of links)
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
bullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbullbull
Thank you
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
DampingFunctions forLink Ranking
R Baeza-YatesP Boldi andC Castillo
Notation
RewritingPageRank
FunctionalRankings
Algorithms
Comparison
Conclusions
Baeza-Yates R Boldi P and Castillo C (2005)The choice of a damping function for propagating importance in link-basedrankingTechnical report Dipartimento di Scienze dellrsquoInformazione Universit degliStudi di Milano
Boldi P Santini M and Vigna S (2005)Pagerank as a function of the damping factorIn Proceedings of the 14th international conference on World Wide Webpages 557ndash566 Chiba Japan ACM Press
Newman M E Strogatz S H and Watts D J (2001)Random graphs with arbitrary degree distributions and their applicationsPhys Rev E Stat Nonlin Soft Matter Phys 64(2 Pt 2)
- Notation
- Rewriting PageRank
- Functional Rankings
- Algorithms
- Comparison
- Conclusions
-
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