link counts
DESCRIPTION
GOOGLE Page Rank engine needs speedup. Link Counts. Taher’s Home Page. Sep’s Home Page. CS361. DB Pub Server. CNN. Yahoo!. Linked by 2 Unimportant pages. Linked by 2 Important Pages. adapted from G. Golub et al. importance of page i. importance of page j. - PowerPoint PPT PresentationTRANSCRIPT
1
Link Counts
Linked by 2 Important Pages
Linked by 2 Unimportant
pages
Sep’s Home Page
Taher’s Home Page
Yahoo! CNNDB Pub Server CS361
GOOGLE Page Rank engine needs speedup
adapted from G. Golub et al
2
Definition of PageRank
The importance of a page is given by the importance of the pages that link to it.
jBj j
i xN
xi
1
importance of page i
pages j that link to page i
number of outlinks from page j
importance of page j
3
Definition of PageRank
1/2 1/2 1 1
0.1 0.10.1
0.05
Yahoo!CNNDB Pub Server
Taher Sep
0.25
4
PageRank Diagram
Initialize all nodes to rank
0.333
0.333
0.333
nxi
1)0(
5
PageRank Diagram
Propagate ranks across links(multiplying by link weights)
0.167
0.167
0.333
0.333
6
PageRank Diagram
0.333
0.5
0.167
)0()1( 1j
Bj ji x
Nx
i
7
PageRank Diagram
0.167
0.167
0.5
0.167
8
PageRank Diagram
0.5
0.333
0.167
)1()2( 1j
Bj ji x
Nx
i
9
PageRank Diagram
After a while…
0.4
0.4
0.2
jBj j
i xN
xi
1
10
Computing PageRank Initialize:
Repeat until convergence:
)()1( 1 kj
Bj j
ki x
Nx
i
nxi
1)0(
importance of page i
pages j that link to page i
number of outlinks from page j
importance of page j
11
Matrix Notation
jBj j
i xN
xi
1
0 .2 0 .3 0 0 .1 .4 0 .1=
.1
.3
.2
.3
.1
.1
.2
.1
.3
.2
.3
.1
.1TP
x
12
Matrix Notation
.1
.3
.2
.3
.1
.1
0 .2 0 .3 0 0 .1 .4 0 .1=
.1
.3
.2
.3
.1
.1
.2
xPx TFind x that satisfies:
13
Power Method Initialize:
Repeat until convergence:
(k)T1)(k xPx
T(0)x
nn
1...
1
14
PageRank doesn’t actually use PT. Instead, it uses A=cPT + (1-c)ET.
So the PageRank problem is really:
not:
A side note
AxxFind x that satisfies:
xPx TFind x that satisfies:
15
Power Method And the algorithm is really . . .
Initialize:
Repeat until convergence:
T(0)x
nn
1...
1
(k)1)(k Axx
16
Power Method
u1
1u2
2
u3
3
u4
4
u5
5
Express x(0) in terms of eigenvectors of A
17
Power Method
u1
1u2
22
u3
33
u4
44
u5
55
)(1x
18
Power Method)2(x
u1
1u2
222
u3
332
u4
442
u5
552
19
Power Method
u1
1u2
22k
u3
33k
u4
44k
u5
55k
)(kx
20
Power Method
u1
1u2
u3
u4
u5
)(x
21
Why does it work?
Imagine our n x n matrix A has n distinct eigenvectors ui.
ii uAu i
n0 uuux n ...221)(
u1
1u2
2
u3
3
u4
4
u5
5
Then, you can write any n-dimensional vector as a linear combination of the eigenvectors of A.
22
Why does it work? From the last slide:
To get the first iterate, multiply x(0) by A.
First eigenvalue is 1.
Therefore:
...;1 211
n0 uuux n ...221)(
n
n
(0)(1)
uuu
AuAuAu
Axx
nn
n
...
...
22211
221
n(1) uuux nn ...2221
All less than 1
23
Power Method
n0 uuux n ...221)(
u1
1u2
2
u3
3
u4
4
u5
5
u1
1u2
22
u3
33
u4
44
u5
55
n(1) uuux nn ...2221
n)( uuux 2
22221
2 ... nn u1
1u2
222
u3
332
u4
442
u5
552
24
The smaller 2, the faster the convergence of the Power Method.
Convergence
n)( uuux k
nnkk ...2221
u1
1u2
22k
u3
33k
u4
44k
u5
55k