2007 csam outlinks and pr presentation2

8/14/2019 2007 Csam Outlinks and PR Presentation2

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Googles PageRankand the Choice of the Outlinks

Laure Ninove

Joint work with Cristobald de Kerchove and Paul Van Dooren

CESAMEUniversit catholique de Louvain, Belgium

CESAME SeminarFebruary 27, 2007

Laure Ninove (CESAME) Outlinks and PR 1 / 27
http://goforward/http://find/http://goback/


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Googles power

Googles search engine

guides websurfers in their

visits.

A good ranking is vital for a

webpage to be read.

How to improve yourGoogle rank?



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Googles power

Googles search engine

guides websurfers in their

visits.

A good ranking is vital for a

webpage to be read.

How to improve yourGoogle rank?



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Outline

1

Preliminaries: What is under Googles PageRank?A brief history

A story of links

PageRank equations

2 How to improve your PageRank?Add inlinks

Choose outlinks

3

Optimal outlink structureFor a single node

For a set of nodes



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Outline

1


A story of links

PageRank equations


Choose outlinks

3

Optimal outlink structureFor a single node

For a set of nodes



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A brief history of the Web search engine Google

1996: a research project, by L. Page and S. Brin

1998: Google Inc. company, 25 million webpages indexed

2005: 8 billion webpages indexed

2006: "to google" added to the Oxford English Dictionary

The primary goal is to provide high quality search results

over a rapidly growing World Wide Web. Google employs a

number of techniques to improve search quality including

page rank, anchor text, and proximity information.

Brin & Page, 1998

The anatomy of a large-scale hypertextual web search engine



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Googles PageRank: a story of links

An hyperlink from i to j

is vote of confidence in j.

A page j has a high PageRank j if it is pointed to by many pages with

a high PageRank,

few outlinks.



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Votes of confidence

Example

1

2

4

3

2/11

2/11

?

1/11

1/11

2/11

1 =1

2 2 + 1 4 =

3

11



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PageRank equationsVote of confidence

j = cij

idi

+ (1 c)zj

j

j = 1

sum of parents weighted scores

normalization of the PageRanks

damping with personalization score

T = cTD1A + (1 c)zT

T

e = 1

A {0, 1}n: webgraphs adjacency matrix(zero diagonal, no zero row)

D = diag(Ae): outdegrees matrix

c ]0, 1[: damping factor

z > 0, zTe = 1: personalization vector



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j = cij

idi

+ (1 c)zj

j

j = 1




T = cTD1A + (1 c)zT

T

e = 1







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j = cij

idi

+ (1 c)zj

j

j = 1




T = cTD1A + (1 c)zT

Te=

1







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PageRank equationsRandom walk

Google matrix:

G = c D1

A + (1 c) ezT

Irreducible, stochastic matrix transition probability matrix

Random walk on the webgraph:

P(i j) = Gij, with P(follow hyperlinks) = c

P(zap according z) = 1 c

PageRank vector : stationary distribution of this Markov chainTG = T

Te = 1



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Damping with a personalization score

Example

1

2

4

3

0.19?

c*0.095

c*0.19

z

(1c)*0.25

0.19

1 = c

1

22 + 4

+ (1 c) z1



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Outline

1


A story of links

PageRank equations


Choose outlinks

3 Optimal outlink structure

For a single node

For a set of nodes



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How to improve your PageRank?


H i P R k?


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How to improve your PageRank?Add inlinks

Add inlinks?

j = c

i

j

idi

+ (1 c)zj

Always your PR

Ipsen & Wills, 2006

Mathematical properties and analysis of Googles PageRank


H i P R k?


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Add inlinks?

j = ci

j

idi

+ (1 c)zj

Always your PR

Ipsen & Wills, 2006



H t i P R k?


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Example

1 1

1 = 0.196 < (inlink)1 = 0.245


H t i P R k?


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Add inlinks?

j = cij

idi

+ (1 c)zj

Always your PR

But no control

on your inlinks

Ipsen & Wills, 2006



Ho to impro e o r PageRank?


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How to improve your PageRank?Choose outlinks

Choose outlinks?

You control them

Constraints:

at least one outlink

no loopImpact not obvious:

adding outlinks can

or your PR

Sydow, 2005

Can one out-link change your PageRank?




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Choose outlinks?

You control them

Constraints:

at least one outlink

no loopImpact not obvious:

adding outlinks can

or your PR

Sydow, 2005

Can one out-link change your PageRank?




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Example

1 1 1

(outlink a)

1 = 0.182 < 1 = 0.196 < (outlink b)

1 = 0.211


Outline


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Outline

1 Preliminaries: What is under Googles PageRank?

A brief history

A story of links

PageRank equations


Choose outlinks

3 Optimal outlink structure

For a single node

For a set of nodes


Notation


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Notation

LetI be the considered set of nodes.

Up to a permutation of the indices,

A =

AI Aout(I)

Ain(I) AI

.


Optimal outlink structure for a single node


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SupposeI = {1}.

We want to maximize 1(Aout({1})).

With Aout({1}) = eTL, where L = {children of 1} = .

Proposition

1(eTL) is maximal = L L

= arg mini

eTi (I GI)1e.

Proof.1(e

TL) =

1

c

iL

eTi (I GI)1e

|L|+ constant

.




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SupposeI = {1}.



Proposition

1(eTL) is maximal = L L = arg mini

eTi (I GI)1e.

Proposition

Suppose that 1 has some parents. Then

1(eTL) is maximal = L {parents of 1}.




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SupposeI = {1}.



Proposition

1(eTL) is maximal = L L = arg mini

eTi (I GI)1e.

Proposition

Suppose that 1 has some parents. Then

1(eTL) is maximal = L {parents of 1}.

But




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Optimal outlink structure for a single nodeExample

Example

1

2

3*

*

*

In order to maximize its PageRank,

Node 1 should linkto some node(s) (parents).

But it is better for 1 to link

to node 3 (grand-parent)

rather than to node 2 (parent).


Optimal outlink structure for a set of nodes


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Consider now a set I of nodes.Internal link structure AI given, with AI has no zero row.

External outlink structure Aout(I) to be determined.Goal: to maximize the sum of PageRanks:

maxAout(I)

iI

i (Aout(I)).




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Proposition

Under the assumption that I has at least m external outlinks,

iI

i (Aout(I)) is maximal = I has exactly m external outlinks.


Optimal outlink structure for a set of node


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Proof.

Removing a link i j from the graph perturbation:

G(i,j) = c(D1A + ei(i,j)T) + (1 c) ezT.

Difference between new and old PageRank sums:

sI

(i,j)s

sI

s = ci(i,j)T(I cD1A)1eI

1 c(i,j)T(I cD1A)1ei.

For every link i j, c(i,j)T

(I c D1

A)1

ei < 1.There exists an external outlink k with k I, / I, such that

(k,)T(I cD1A)1eI > 0.




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p

Example

Sometimes, removing an outlink for Imay decrease the PageRank sum for I.

2

3 4 5

1 2

3 4 5

1 2

3 4 5

1

iI

i(I 3)

2007 csam outlinks and pr presentation2

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