Extrapolation Methods for Accelerating PageRank Computations

Sepandar D. Kamvar

Taher H. Haveliwala

Christopher D. Manning

Gene H. Golub

Stanford University

2

Results:

1. The Official Site of the San Francisco Giants

Search: Giants

Results:

1. The Official Site of the New York Giants

Motivation Problem:

Speed up PageRank

Motivation: Personalization “Freshness”

Note: PageRank Computations don’t get faster as computers do.

3

0.4

0.2

0.4

(k)1)(k Axx Repeat:

u1 u2 u3 u4 u5

u1 u2 u3 u4 u5

Outline Definition of PageRank

Computation of PageRank

Convergence Properties

Outline of Our Approach

Empirical Results

4

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

5

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

6

Definition of PageRank

1/2 1/2 1 1

0.1 0.10.1

0.05

Yahoo!CNNDB Pub Server

Taher Sep

0.25

7

PageRank Diagram

Initialize all nodes to rank

0.333

0.333

0.333

nxi

1)0(

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PageRank Diagram

Propagate ranks across links(multiplying by link weights)

0.167

0.167

0.333

0.333

9

PageRank Diagram

0.333

0.5

0.167

)0()1( 1j

Bj ji x

Nx

i

10

PageRank Diagram

0.167

0.167

0.5

0.167

11

PageRank Diagram

0.5

0.333

0.167

)1()2( 1j

Bj ji x

Nx

i

12

PageRank Diagram

After a while…

0.4

0.4

0.2

jBj j

i xN

xi

1

13

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

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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

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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:

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Power Method Initialize:

Repeat until convergence:

(k)T1)(k xPx

T(0)x

nn

1...

1

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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:

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Power Method And the algorithm is really . . .

Initialize:

Repeat until convergence:

T(0)x

nn

1...

1

(k)1)(k Axx

19

0.4

0.2

0.4

(k)1)(k Axx Repeat:

u1 u2 u3 u4 u5

u1 u2 u3 u4 u5

Outline Definition of PageRank

Computation of PageRank

Convergence Properties

Outline of Our Approach

Empirical Results

20

Power Method

u1

1u2

2

u3

3

u4

4

u5

5

Express x(0) in terms of eigenvectors of A

21

Power Method

u1

1u2

22

u3

33

u4

44

u5

55

)(1x

22

Power Method)2(x

u1

1u2

222

u3

332

u4

442

u5

552

23

Power Method

u1

1u2

22k

u3

33k

u4

44k

u5

55k

)(kx

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Power Method

u1

1u2

u3

u4

u5

)(x

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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.

26

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

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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

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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

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Our Approach

u1 u2 u3 u4 u5

Estimate components of current iterate in the directions of second two eigenvectors, and eliminate them.

30

Why this approach? For traditional problems:

A is smaller, often dense. 2 often close to , making the power method slow.

In our problem, A is huge and sparse More importantly, 2 is small1.

Therefore, Power method is actually much faster than other methods.

1(“The Second Eigenvalue of the Google Matrix” dbpubs.stanford.edu/pub/2003-20.)

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Using Successive Iterates

u1

x(0)

u1 u2 u3 u4 u5

32

Using Successive Iterates

u1

x(1)

x(0)

u1 u2 u3 u4 u5

33

Using Successive Iterates

u1

x(1)

x(0)

x(2)

u1 u2 u3 u4 u5

34

Using Successive Iterates

x(0)

u1

x(1)

x(2)

u1 u2 u3 u4 u5

35

Using Successive Iterates

x(0)

x’ = u1

x(1)

u1 u2 u3 u4 u5

36

How do we do this? Assume x(k) can be written as a linear

combination of the first three eigenvectors (u1, u2, u3) of A.

Compute approximation to {u2,u3}, and subtract it from x(k) to get x(k)’

37

Assume Assume the x(k) can be represented by

first 3 eigenvectors of A

33322211 uuuAxx )()( kk

n)( uuux 3221 k

32332

2221

2 uuux )( k

33332

3221

3 uuux )( k

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Linear Combination Let’s take some linear combination of

these 3 iterates.

)()()( xxx 33

22

11

kkk

)( 32332

22212 uuu

)( 33332

32213 uuu

)( 33322211 uuu

39

Rearranging Terms We can rearrange the terms to get:

)()()( xxx 33

22

11

kkk

1321 )( u

2323

222212 )( u

3333

232313 )( u

Goal: Find 1,2,3 so that coefficients of u2 and u3 are 0, and coefficient of u1 is 1.

40

Summary We make an assumption about the

current iterate. Solve for dominant eigenvector as a

linear combination of the next three iterates.

We use a few iterations of the Power Method to “clean it up”.

41

u1 u2 u3 u4 u5

u1 u2 u3 u4 u5

0.4

0.2

0.4

(k)1)(k Axx Repeat:

Outline Definition of PageRank

Computation of PageRank

Convergence Properties

Outline of Our Approach

Empirical Results

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ResultsQuadratic Extrapolation speeds up convergence. Extrapolation was only used 5 times!

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ResultsExtrapolation dramatically speeds up convergence, for high values of c (c=.99)

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Take-home message Speeds up PageRank by a fair amount,

but not by enough for true Personalized PageRank.

Ideas are useful for further speedup algorithms.

Quadratic Extrapolation can be used for a whole class of problems.

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The End Paper available at

http://dbpubs.stanford.edu/pub/2003-16