Lectures 6 & 7 Centrality Measures February 2, 2009

Lectures 6 & 7Lectures 6 & 7Centrality MeasuresCentrality MeasuresFebruary 2, 2009

Monojit Choudhurymonojitc@microsoft.com

A brief Intro toA brief Intro to

MyselfYourselfThe courseThe classes

◦Please ask questions◦Don’t disturb otherwise◦Please go back and read

I shall assume that you I shall assume that you knowknowBasic graph theory

◦Adjacency matrix representation◦Degree, in-degree, out-degree◦Connected component, shortest paths

Basic linear algebra◦Symmetric matrix, transpose◦Vectors, multiplication of vectors with

vectors and matrices, orthogonality◦Eigenvectors and Eigenvalues

Lecture 5Lecture 5Centrality MeasuresCentrality MeasuresFebruary 2, 2009

Monojit Choudhurymonojitc@microsoft.com

Question 1: Information Question 1: Information percolationpercolation

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In this friendship network of 8 persons, suppose that someone comes to know about an interesting news. Who are most likely to receive this news fast?

Question 2: Searching the Question 2: Searching the WebWeb

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In this hyperlinked network of webpages, which pages are most likely to contain authoritative information ?

Question 3: Spreading of Question 3: Spreading of STDs STDs

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In this hypothetical sexual interaction network, who are most likely to be affected by STDs such as AIDS?

A common answer to all the A common answer to all the questionsquestions

Nodes which are most “CENTRAL” to the network

Centrality of a node measures its◦Power, Prestige, Prominence &

imPortance◦The 4 “P”s

Degree CentralityDegree CentralityHow many friends do you have?

Measure of centralization of the network◦Star network – most centralized◦Line graph – least centralized

Thus, the variance of degree centrality is the measure of (de)centralization of a network

How much is this network How much is this network centralized?centralized?

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When is centralization When is centralization good/bad?good/bad?Fault tolerance

◦Centralized: bad◦Decentralized: good

However, for random attacks◦Centralized: good

What happens in a scale-free network?

Closeness CentralityCloseness CentralityReciprocal of the sum of

shortest paths to all the nodesCompute closeness centrality

for nodes 3 and 6

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Closeness CentralityCloseness CentralityWhat does variance of closeness

centrality indicate?

What would this variance be for◦A Clique◦A Tree◦A Ring

Spreading of STDs Spreading of STDs

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Who should be removed from this network to make this community less susceptible to spreading of STDs?

Betweenness CentralityBetweenness Centrality

Joydeep

Subrata

Rich (in what?)

Joydeep has the opportunity to play a information broker – but Subrata

doesn’t

Mathematical DefinitionMathematical Definition

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t

v

Can be extende

d to edges

Which networks haveWhich networks haveNodes with very small

betweenness centralityNode(s) with very high

betweenness centrality

What is the betweenness centrality of the nodes in a complete bipartite network?

Question 2: Searching the Question 2: Searching the WebWeb

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In this hyperlinked network of webpages, which pages are most popular?

The basic idea The basic idea I am popular if my friends are

popular

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p6 = p2 + p5 + p7 + p8

Computing PopularityComputing Popularity

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Computing PopularityComputing Popularity

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Oops! Popularity

grows unboundedly

!!

A better approachA better approach

1/8

1/8

1/8

1/8

1/81/8

1/8

1/8

4/8

2/8

2/8

3/8

1/84/8

3/8

3/8

4/22

2/22

2/22

3/22

1/224/22

3/22

3/22

Computing popularityComputing popularity

4/22

2/22

2/22

3/22

1/224/22

3/22

3/22

13/22

6/22

6/22

10/22

4/229/22

10/22

10/22

13/68

6/68

6/68

10/68

4/689/68

10/68

10/68

Computing popularityComputing popularity

13/68

6/68

6/68

10/68

4/689/68

10/68

10/68

39/68

15/68

15/68

33/68

9/6829/68

33/68

33/68

39/206

15/206

15/206

33/206

9/20629/206

33/206

33/206

Is it converging?Is it converging?

39/206

15/206

15/206

33/206

9/20629/206

33/206

33/2061 1/8 2/22 6/68 15/206

1 .125 .091 .088 .073

2 1/8 4/22 9/68 29/206

2 .125 .182 .132 .141

5 1/8 3/22 10/68 33/206

5 .125 .136 .147 .160

6 1/8 4/22 13/68 39/206

6 .125 .182 .191 .189

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ObservationsObservationsThe popularity values eventually

convergeNodes which are isomorphic have the

same popularity

What happens when we start from a different initialization?

Does it converge for every graph?What happens for a disconnected

graph?

An alternative view to An alternative view to popularitypopularityRandom surfer model:

◦The surfer lands up on a random page

◦With probability w it stays in the same page, but with probability (1-w) it visits any other random link from the page

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What’s the probability that What’s the probability that the surfer is at node the surfer is at node i i ??

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p6 = wp6 + (1-w) [p2/4+ p5 + p7/3 + p8]

What’s the probability that What’s the probability that the surfer is at node the surfer is at node i i ??

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pi = wpi + (1-w)jajipj/dj

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1 0 0 0 0 0 0 0 0

2 1 0 1 1 0 1 0 0

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Therefore, popularity isTherefore, popularity isEigenvector CentralityIntroduced by Bonacich (1972)

A slightly different variant is used as “PageRank”

pi = (1-w)+ wjajipj/dj

Does all networks have Does all networks have == 1 1Yes!Actually, all stochastic matrices

(aka Markov Matrices) have the largest Eigenvalue 1 = 1

Perron-Frobenius Theorem◦If A is a positive matrix, so is its largest

Eigenvalue 1 > all other | i |. Every component of the corresponding Eigenvector is also positive.