cmu scs finding patterns in large, real networks christos faloutsos cmu
TRANSCRIPT
CMU SCS
Finding patterns in large, real networks
Christos Faloutsos
CMU
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Thanks to
• Deepayan Chakrabarti (CMU/Yahoo)
• Michalis Faloutsos (UCR)
• Spiros Papadimitriou (CMU/IBM)
• George Siganos (UCR)
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Introduction
Internet Map [lumeta.com]
Food Web [Martinez ’91]
Protein Interactions [genomebiology.com]
Friendship Network [Moody ’01]
Graphs are everywhere!
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Outline
• Laws– static
– time-evolution laws
– why so many power laws?
• tools/results
• future steps
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Motivating questions
• Q1: What do real graphs look like?• Q2: How to generate realistic graphs?
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Virus propagation
• Q3: How do viruses/rumors propagate?
• Q4: Who is the best person/computer to immunize against a virus?
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Graph clustering & mining
• Q5: which edges/nodes are ‘abnormal’?
• Q6: split a graph in k ‘natural’ communities - but how to determine k?
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Outline
• Laws– static
– time-evolution laws
– why so many power laws?
• tools/results
• future steps
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Topology
Q1: How does the Internet look like? Any rules?
(Looks random – right?)
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Laws – degree distributions
• Q: avg degree is ~2 - what is the most probable degree?
degree
count ??
2
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Laws – degree distributions
• Q: avg degree is ~2 - what is the most probable degree?
degreedegree
count ??
2
count
2
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I.Power-law: outdegree O
The plot is linear in log-log scale [FFF’99]
freq = degree (-2.15)
O = -2.15
Exponent = slope
Outdegree
Frequency
Nov’97
-2.15
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II.Power-law: rank R
• The plot is a line in log-log scale
R
Rank: nodes in decreasing degree order
April’98
log(rank)
log(degree)
-0.82
att.com
ibm.com
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III.Power-law: eigen E
• Eigenvalues in decreasing order (first 20)• [Mihail+, 02]: R = 2 * E
E = -0.48
Exponent = slope
Eigenvalue
Rank of decreasing eigenvalue
Dec’98
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IV. The Node Neighborhood
• N(h) = # of pairs of nodes within h hops
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IV. The Node Neighborhood
• Q: average degree = 3 - how many neighbors should I expect within 1,2,… h hops?
• Potential answer:1 hop -> 3 neighbors
2 hops -> 3 * 3
…
h hops -> 3h
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IV. The Node Neighborhood
• Q: average degree = 3 - how many neighbors should I expect within 1,2,… h hops?
• Potential answer:1 hop -> 3 neighbors
2 hops -> 3 * 3
…
h hops -> 3h
WE HAVE DUPLICATES!
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IV. The Node Neighborhood
• Q: average degree = 3 - how many neighbors should I expect within 1,2,… h hops?
• Potential answer:1 hop -> 3 neighbors
2 hops -> 3 * 3
…
h hops -> 3h
‘avg’ degree: meaningless!
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IV. Power-law: hopplot H
Pairs of nodes as a function of hops N(h)= hH
H = 4.86
Dec 98Hops
# of Pairs # of Pairs
Hops Router level ’95
H = 2.83
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Observation• Q: Intuition behind ‘hop exponent’?• A: ‘intrinsic=fractal dimensionality’ of the
network
N(h) ~ h1 N(h) ~ h2
...
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Hop plots
• More on fractal/intrinsic dimensionalities: very soon
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Any other ‘laws’?
Yes!
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Any other ‘laws’?
Yes!
• Small diameter– six degrees of separation / ‘Kevin Bacon’
– small worlds [Watts and Strogatz]
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Any other ‘laws’?• Bow-tie, for the web [Kumar+ ‘99]
• IN, SCC, OUT, ‘tendrils’
• disconnected components
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Any other ‘laws’?• power-laws in communities (bi-partite cores)
[Kumar+, ‘99]
2:3 core(m:n core)
Log(m)
Log(count)
n:1
n:2n:3
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More Power laws
• Also hold for other web graphs [Barabasi+, ‘99], [Kumar+, ‘99]
• citation graphs (see later)
• and many more
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Outline
• Laws– static
– time-evolution laws
– why so many power laws?
• tools/results
• future steps
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How do graphs evolve?
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Time Evolution: rank R
The rank exponent has not changed! [Siganos+, ‘03]
Domainlevel
#days since Nov. ‘97
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How do graphs evolve?
• degree-exponent seems constant - anything else?
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Evolution of diameter?
• Prior analysis, on power-law-like graphs, hints that
diameter ~ O(log(N)) or
diameter ~ O( log(log(N)))
• i.e.., slowly increasing with network size
• Q: What is happening, in reality?
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Evolution of diameter?
• Prior analysis, on power-law-like graphs, hints that
diameter ~ O(log(N)) or
diameter ~ O( log(log(N)))
• i.e.., slowly increasing with network size
• Q: What is happening, in reality?
• A: It shrinks(!!), towards a constant value
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Shrinking diameter
[Leskovec+05a]• Citations among physics
papers • 11yrs; @ 2003:
– 29,555 papers– 352,807 citations
• For each month M, create a graph of all citations up to month M
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Shrinking diameter• Authors &
publications
• 1992– 318 nodes
– 272 edges
• 2002– 60,000 nodes
• 20,000 authors
• 38,000 papers
– 133,000 edges
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Shrinking diameter• Patents & citations
• 1975– 334,000 nodes
– 676,000 edges
• 1999– 2.9 million nodes
– 16.5 million edges
• Each year is a datapoint
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Shrinking diameter• Autonomous
systems
• 1997– 3,000 nodes
– 10,000 edges
• 2000– 6,000 nodes
– 26,000 edges
• One graph per day
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Temporal evolution of graphs
• N(t) nodes; E(t) edges at time t
• suppose that
N(t+1) = 2 * N(t)
• Q: what is your guess for
E(t+1) =? 2 * E(t)
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Temporal evolution of graphs
• N(t) nodes; E(t) edges at time t
• suppose that
N(t+1) = 2 * N(t)
• Q: what is your guess for
E(t+1) =? 2 * E(t)
• A: over-doubled!
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Temporal evolution of graphs
• A: over-doubled - but obeying:
E(t) ~ N(t)a for all t
where 1<a<2
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Temporal evolution of graphs
• A: over-doubled - but obeying:
E(t) ~ N(t)a for all t
• Identically:
log(E(t)) / log(N(t)) = constant for all t
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Densification Power Law
ArXiv: Physics papers
and their citations
1.69
N(t)
E(t)
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Densification Power Law
ArXiv: Physics papers
and their citations
1.69
N(t)
E(t)
‘tree’
1
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Densification Power Law
ArXiv: Physics papers
and their citations
1.69
N(t)
E(t)‘clique’
2
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Densification Power Law
U.S. Patents, citing each other
1.66
N(t)
E(t)
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Densification Power Law
Autonomous Systems
1.18
N(t)
E(t)
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Densification Power Law
ArXiv: authors & papers
1.15
N(t)
E(t)
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Summary of ‘laws’
• Power laws for degree distributions
• …………... for eigenvalues, bi-partite cores
• Small & shrinking diameter (‘6 degrees’)
• ‘Bow-tie’ for web; ‘jelly-fish’ for internet
• ``Densification Power Law’’, over time
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Outline
• Laws– static
– time-evolution laws
– why so many power laws?• other power laws
• fractals & self-similarity
• case study: Kronecker graphs
• tools/results
• future steps
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Power laws
• Many more power laws, in diverse settings:
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A famous power law: Zipf’s law
• Bible - rank vs frequency (log-log)
log(rank)
log(freq)
“a”
“the”
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Power laws, cont’ed
• length of file transfers [Bestavros+]
• web hit counts [Huberman]
• magnitude of earthquakes (Guttenberg-Richter law)
• sizes of lakes/islands (Korcak’s law)
• Income distribution (Pareto’s law)
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The Peer-to-Peer Topology
• Frequency versus degree
• Number of adjacent peers follows a power-law
[Jovanovic+]
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Web Site Traffic
log(freq)
log(count)Zipf
Click-stream data
u-id’s url’s
log(freq)
log(count)
‘yahoo’
‘super-surfer’
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Lotka’s law
(Lotka’s law of publication count); and citation counts: (citeseer.nj.nec.com 6/2001)
log(#citations)
log(count)
J. Ullman
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Nodes: people (Females; Males)Links: sexual relationships
Liljeros et al. Nature 2001
4781 Swedes; 18-74; 59% response rate.
Albert Laszlo Barabasihttp://www.nd.edu/~networks/Publication%20Categories/04%20Talks/2005-norway-3hours.ppt
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Nodes: people (Females; Males)Links: sexual relationships
Liljeros et al. Nature 2001
4781 Swedes; 18-74; 59% response rate.
Albert Laszlo Barabasihttp://www.nd.edu/~networks/Publication%20Categories/04%20Talks/2005-norway-3hours.ppt
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Power laws
• Q: Why so many power laws?
• A1: self-similarity
• A2: ‘rich-get-richer’
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Digression: intro to fractals
• Fractals: sets of points that are self similar
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A famous fractal
e.g., Sierpinski triangle:
...zero area;
infinite length!
dimensionality = ??
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A famous fractal
e.g., Sierpinski triangle:
...zero area;
infinite length!
dimensionality = log(3)/log(2) = 1.58
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A famous fractal
equivalent graph:
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Intrinsic (‘fractal’) dimension
How to estimate it?
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Intrinsic (‘fractal’) dimension
• Q: fractal dimension of a line?
• A: nn ( <= r ) ~ r^1(‘power law’: y=x^a)
• Q: fd of a plane?
• A: nn ( <= r ) ~ r^2
fd== slope of (log(nn) vs log(r) )
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Sierpinsky triangle
log( r )
log(#pairs within <=r )
1.58
== ‘correlation integral’
= CDF of pairwise distances
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Sierpinsky triangle
log( r )
log(#pairs within <=r )
1.58
== ‘correlation integral’
= CDF of pairwise distances
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Line
log( r )
log(#pairs within <=r )
1.58
== ‘correlation integral’
= CDF of pairwise distances
1...
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2-d (Plane)
log( r )
log(#pairs within <=r )
1.58
== ‘correlation integral’
= CDF of pairwise distances
2
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Recall: Hop Plot
• Internet routers: how many neighbors within h hops? (= correlation integral!)
Reachability function: number of neighbors within r hops, vs r (log-log).
Mbone routers, 1995log(hops)
log(#pairs)
2.8
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Fractals and power laws
They are related concepts:
• fractals <=>
• self-similarity <=>
• scale-free <=>
• power laws ( y= xa )– F = C r(-2)
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Fractals and power laws
• Power laws and fractals are closely related• And fractals appear in MANY cases
– coast-lines: 1.1-1.5– brain-surface: 2.6– rain-patches: 1.3– tree-bark: ~2.1– stock prices / random walks: 1.5– ... [see Mandelbrot; or Schroeder]
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Outline
• Laws– static
– time-evolution laws
– why so many power laws?• other power laws
• fractals & self-similarity
• case study: Kronecker graph generator
• tools/results
• future steps
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Self-similarity at work
• Q2: How to generate realistic graphs?
• (Q2’) and how to grow it over time?
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Wish list for a generator:
• Power-law-tail in- and out-degrees
• Power-law-tail scree plots
• shrinking/constant diameter
• Densification Power Law
• communities-within-communities
Q: how to achieve all of them?
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Wish list for a generator:
• Power-law-tail in- and out-degrees
• Power-law-tail scree plots
• shrinking/constant diameter
• Densification Power Law
• communities-within-communities
Q: how to achieve all of them?
A: Kronecker matrix product [Leskovec+05b]
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Kronecker product
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Kronecker product
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Kronecker product
N N*N N**4
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Properties of Kronecker graphs:
• Power-law-tail in- and out-degrees
• Power-law-tail scree plots
• constant diameter
• perfect Densification Power Law
• communities-within-communities
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Properties of Kronecker graphs:
• Power-law-tail in- and out-degrees
• Power-law-tail scree plots
• constant diameter
• perfect Densification Power Law
• communities-within-communities
and we can prove all of the above
(first and only generator that does that)
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Kronecker - ArXiv
real
(stochastic)Kronecker
Degree Scree Diameter D.P.L.
(det. Kronecker)
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Kronecker - patents
Degree Scree Diameter D.P.L.
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Kronecker - A.S.
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Outline
• Laws– static
– time-evolution laws
– why so many power laws?
• tools/results– virus propagation
– connection subgraphs
– cross-associations
• future steps
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Virus propagation
• Q3: How do viruses/rumors propagate?
• Q4: Who is the best person/computer to immunize against a virus?
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The model: SIS
• ‘Flu’ like: Susceptible-Infected-Susceptible
• Virus ‘strength’ s= /
Infected
Healthy
NN1
N3
N2
Prob.
Prob. β
Prob.
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Epidemic threshold
of a graph: the value of , such that
if strength s = / < an epidemic can not happen
Thus,
• given a graph
• compute its epidemic threshold
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Epidemic threshold
What should depend on?
• avg. degree? and/or highest degree?
• and/or variance of degree?
• and/or third moment of degree?
• and/or diameter?
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Epidemic threshold
• [Theorem] We have no epidemic, if
β/δ <τ = 1/ λ1,A
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Epidemic threshold
• [Theorem] We have no epidemic, if
β/δ <τ = 1/ λ1,A
largest eigenvalueof adj. matrix A
attack prob.
recovery prob.epidemic threshold
Proof: [Wang+03]
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Experiments (Oregon)
/ > τ (above threshold)
/ = τ (at the threshold)
/ < τ (below threshold)
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Our result:
• Holds for any graph
• includes older results as special cases
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Virus propagation
• Q3: How do viruses/rumors propagate?
• Q4: Who is the best person/computer to immunize against a virus?
• A4: the one that causes the highest decrease in 1
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Outline
• Laws– static
– time-evolution laws
– why so many power laws?
• tools/results– virus propagation
– connection subgraphs
– cross-associations
• future steps
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(A-A) Kidman-Diaz
• What are the best paths between ‘Kidman’ and ‘Diaz’?
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(A-A) Kidman-Diaz
Strong, direct link
• What are the best paths between ‘Kidman’ and ‘Diaz’? [KDD’04, w/ McCurley+Tomkins]
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Outline
• Laws– static
– time-evolution laws
– why so many power laws?
• tools/results– virus propagation
– connection subgraphs
– cross-associations
• future steps
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Graph clustering & mining
• Q5: which edges/nodes are ‘abnormal’?
• Q6: split a graph in k ‘natural’ communities - but how to determine k?
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Graph partitioning
• Documents x terms
• Customers x products
• Users x web-sites
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Graph partitioning
• Documents x terms
• Customers x products
• Users x web-sites
• Q: HOW MANY PIECES?
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Graph partitioning
• Documents x terms
• Customers x products
• Users x web-sites
• Q: HOW MANY PIECES?
• A: MDL/ compression
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Cross-associations
1x2 2x2
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Cross-associations
2x3 3x3 3x4
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Cross-associations
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Cross-associations
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Graph clustering & mining
• Q5: which edges/nodes are ‘abnormal’?
• Q6: split a graph in k ‘natural’ communities - but how to determine k?
• A6: choose the k that leads to best overall compression (= MDL = Minimum Description Language)
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Cross-associations
outlier edgemissing edge
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Outline
• Laws– static
– time-evolution laws
– why so many power laws?
• tools/results
• other related projects
• future steps
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Motivation
water distribution network
normal operation
May have hundreds of measurements, butit is unlikely they are completely unrelated!
Phase 1 Phase 2 Phase 3
: : : : : :
: : : : : :
chlo
rin
e c
on
cen
trati
on
s
sensorsnear leak
sensorsawayfrom leak
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Phase 1 Phase 2 Phase 3
: : : : : :
: : : : : :
Motivation
water distribution network
normal operation major leak
May have hundreds of measurements, butit is unlikely they are completely unrelated!
chlo
rin
e c
on
cen
trati
on
s
sensorsnear leak
sensorsawayfrom leak
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Motivation
actual measurements(n streams)
k hidden variable(s)
We would like to discover a few “hidden(latent) variables” that summarize the key trends
Phase 1
: : : : : :
: : : : : :
chlo
rin
e c
on
cen
trati
on
s
Phase 1
k = 1
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Motivation
We would like to discover a few “hidden(latent) variables” that summarize the key trends
chlo
rin
e c
on
cen
trati
on
s
Phase 1 Phase 1Phase 2 Phase 2
actual measurements(n streams)
k hidden variable(s)
k = 2
: : : : : :
: : : : : :
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Motivation
We would like to discover a few “hidden(latent) variables” that summarize the key trends
chlo
rin
e c
on
cen
trati
on
s
Phase 1 Phase 1Phase 2 Phase 2Phase 3 Phase 3
actual measurements(n streams)
k hidden variable(s)
k = 1
: : : : : :
: : : : : :
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Stream mining
• Solution: SPIRIT [VLDB’05]– incremental, on-line PCA
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Outline
• Laws
• tools/results
• other related projects
• future steps
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Future steps
• connection sub-graphs on multi-graphs [Tong]
• traffic matrix, evolving over time [Sun+]
• influence propagation on blogs [Leskovec+]
• automatic web site classification [Leskovec]
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Conclusions
• Power laws & self similarity– excellent tools, for real datasets,
– where Gaussian, uniform etc assumptions fail
• Additional tools– Kronecker, for graph generators
– eigenvalues for virus propagation
– MDL/compression for graph partitioning
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References
• [Aiello+, '00] William Aiello, Fan R. K. Chung, Linyuan Lu: A random graph model for massive graphs. STOC 2000: 171-180
• [Albert+] Reka Albert, Hawoong Jeong, and Albert-Laszlo Barabasi: Diameter of the World Wide Web, Nature 401 130-131 (1999)
• [Barabasi, '03] Albert-Laszlo Barabasi Linked: How Everything Is Connected to Everything Else and What It Means (Plume, 2003)
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References, cont’d
• [Barabasi+, '99] Albert-Laszlo Barabasi and Reka Albert. Emergence of scaling in random networks. Science, 286:509--512, 1999
• [Broder+, '00] Andrei Broder, Ravi Kumar, Farzin Maghoul, Prabhakar Raghavan, Sridhar Rajagopalan, Raymie Stata, Andrew Tomkins, and Janet Wiener. Graph structure in the web, WWW, 2000
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References, cont’d
• [Chakrabarti+, ‘04] RMAT: A recursive graph generator, D. Chakrabarti, Y. Zhan, C. Faloutsos, SIAM-DM 2004
• [Dill+, '01] Stephen Dill, Ravi Kumar, Kevin S. McCurley, Sridhar Rajagopalan, D. Sivakumar, Andrew Tomkins: Self-similarity in the Web. VLDB 2001: 69-78
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References, cont’d
• [Fabrikant+, '02] A. Fabrikant, E. Koutsoupias, and C.H. Papadimitriou. Heuristically Optimized Trade-offs: A New Paradigm for Power Laws in the Internet. ICALP, Malaga, Spain, July 2002
• [FFF, 99] M. Faloutsos, P. Faloutsos, and C. Faloutsos, "On power-law relationships of the Internet topology," in SIGCOMM, 1999.
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References, cont’d
• [Leskovec+05a] Jure Leskovec, Jon Kleinberg and Christos Faloutsos Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations KDD 2005, Chicago, IL. (Best research paper award)
• [Leskovec+05b] Jure Leskovec, Deepayan Chakrabarti, Jon Kleinberg and Christos Faloutsos, Realistic, Mathematically Tractable Graph Generation and Evolution, Using Kronecker Multiplication, ECML/PKDD 2005, Porto, Portugal.
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References, cont’d
• [Jovanovic+, '01] M. Jovanovic, F.S. Annexstein, and K.A. Berman. Modeling Peer-to-Peer Network Topologies through "Small-World" Models and Power Laws. In TELFOR, Belgrade, Yugoslavia, November, 2001
• [Kumar+ '99] Ravi Kumar, Prabhakar Raghavan, Sridhar Rajagopalan, Andrew Tomkins: Extracting Large-Scale Knowledge Bases from the Web. VLDB 1999: 639-650
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References, cont’d
• [Leland+, '94] W. E. Leland, M.S. Taqqu, W. Willinger, D.V. Wilson, On the Self-Similar Nature of Ethernet Traffic, IEEE Transactions on Networking, 2, 1, pp 1-15, Feb. 1994.
• [Mihail+, '02] Milena Mihail, Christos H. Papadimitriou: On the Eigenvalue Power Law. RANDOM 2002: 254-262
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References, cont’d
• [Milgram '67] Stanley Milgram: The Small World Problem, Psychology Today 1(1), 60-67 (1967)
• [Montgomery+, ‘01] Alan L. Montgomery, Christos Faloutsos: Identifying Web Browsing Trends and Patterns. IEEE Computer 34(7): 94-95 (2001)
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References, cont’d
• [Palmer+, ‘01] Chris Palmer, Georgos Siganos, Michalis Faloutsos, Christos Faloutsos and Phil Gibbons The connectivity and fault-tolerance of the Internet topology (NRDM 2001), Santa Barbara, CA, May 25, 2001
• [Pennock+, '02] David M. Pennock, Gary William Flake, Steve Lawrence, Eric J. Glover, C. Lee Giles: Winners don't take all: Characterizing the competition for links on the
web Proc. Natl. Acad. Sci. USA 99(8): 5207-5211 (2002)
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References, cont’d
• [Schroeder, ‘91] Manfred Schroeder Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise W H Freeman & Co., 1991 (excellent book on fractals)
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References, cont’d
• [Siganos+, '03] G. Siganos, M. Faloutsos, P. Faloutsos, C. Faloutsos Power-Laws and the AS-level Internet Topology, Transactions on Networking, August 2003.
• [Wang+03] Yang Wang, Deepayan Chakrabarti, Chenxi Wang and Christos Faloutsos: Epidemic Spreading in Real Networks: an Eigenvalue Viewpoint, SRDS 2003, Florence, Italy.
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Reference
• [Watts+ Strogatz, '98] D. J. Watts and S. H. Strogatz Collective dynamics of 'small-world' networks, Nature, 393:440-442 (1998)
• [Watts, '03] Duncan J. Watts Six Degrees: The Science of a Connected Age W.W. Norton & Company; (February 2003)
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Thank you!
www.cs.cmu.edu/~christos
www.db.cs.cmu.edu