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Lectures 6 & 7Lectures 6 & 7Centrality MeasuresCentrality MeasuresFebruary 2, 2009
Monojit [email protected]
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A brief Intro toA brief Intro to
MyselfYourselfThe courseThe classes
◦Please ask questions◦Don’t disturb otherwise◦Please go back and read
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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
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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?
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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 ?
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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?
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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
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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
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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?
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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
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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?
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Betweenness CentralityBetweenness Centrality
Joydeep
Subrata
Rich (in what?)
Joydeep has the opportunity to play a information broker – but Subrata
doesn’t
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Mathematical DefinitionMathematical Definition
s
t
v
Can be extende
d to edges
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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?
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Question 2: Searching the Question 2: Searching the WebWeb
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In this hyperlinked network of webpages, which pages are most popular?
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The basic idea The basic idea I am popular if my friends are
popular
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p6 = p2 + p5 + p7 + p8
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Computing PopularityComputing Popularity
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Computing PopularityComputing Popularity
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Oops! Popularity
grows unboundedly
!!
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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
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Computing popularityComputing popularity
4/22
2/22
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3/22
1/224/22
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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
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Computing popularityComputing popularity
13/68
6/68
6/68
10/68
4/689/68
10/68
10/68
39/68
15/68
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33/68
9/6829/68
33/68
33/68
39/206
15/206
15/206
33/206
9/20629/206
33/206
33/206
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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?
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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]
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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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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
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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.