Lectures 6 & 7 Centrality Measures Lectures 6 & 7 Centrality Measures February 2, 2009 Monojit Choudhury

A brief Intro to Myself Yourself The course The classes ◦ Please ask questions ◦ Don’t disturb otherwise ◦ Please go back and read

I shall assume that you know Basic 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 5 Centrality Measures Lecture 5 Centrality Measures February 2, 2009 Monojit Choudhury

Question 1: Information percolation 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 Web In this hyperlinked network of webpages, which pages are most likely to contain authoritative information ?

Question 3: Spreading of STDs In this hypothetical sexual interaction network, who are most likely to be affected by STDs such as AIDS?

A common answer to all the questions Nodes which are most “CENTRAL” to the network Centrality of a node measures its ◦ Power, Prestige, Prominence & imPortance ◦ The 4 “P”s

Degree Centrality How 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 centralized?

When is centralization good/bad? Fault tolerance ◦ Centralized: bad ◦ Decentralized: good However, for random attacks ◦ Centralized: good What happens in a scale-free network?

Closeness Centrality Reciprocal of the sum of shortest paths to all the nodes Compute closeness centrality for nodes 3 and

Closeness Centrality What does variance of closeness centrality indicate? What would this variance be for ◦ A Clique ◦ A Tree ◦ A Ring

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

Betweenness Centrality Joydeep Subrata Rich (in what?) Joydeep has the opportunity to play a information broker – but Subrata doesn’t

Mathematical Definition s t v Can be extended to edges

Which networks have Nodes with very small betweenness centrality Node(s) with very high betweenness centrality What is the betweenness centrality of the nodes in a complete bipartite network?

Question 2: Searching the Web In this hyperlinked network of webpages, which pages are most popular?

The basic idea I am popular if my friends are popular p 6 = p 2 + p 5 + p 7 + p 8

Computing Popularity

Oops! Popularity grows unboundedly!!

A better approach 1/8 4/8 2/8 3/8 1/8 4/8 3/8 4/22 2/22 3/22 1/22 4/22 3/22

Computing popularity 4/22 2/22 3/22 1/22 4/22 3/22 13/22 6/22 10/22 4/22 9/22 10/22 13/68 6/68 10/68 4/68 9/68 10/68

Computing popularity 13/68 6/68 10/68 4/68 9/68 10/68 39/68 15/68 33/68 9/68 29/68 33/68 39/206 15/206 33/206 9/206 29/206 33/206

Is it converging? 39/206 15/206 33/206 9/206 29/206 33/206 11/82/226/6815/ /84/229/6829/ /83/2210/6833/ /84/2213/6839/

Observations The popularity values eventually converge Nodes 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 popularity Random 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

What’s the probability that the surfer is at node i ? p 6 = wp 6 + (1-w) [p 2 /4+ p 5 + p 7 /3 + p 8 ]

What’s the probability that the surfer is at node i ? p i = wp i + (1-w)  j a ji p j /d j

Therefore, popularity is Eigenvector Centrality Introduced by Bonacich (1972) A slightly different variant is used as “PageRank” p i = (1-w)+ w  j a ji p j /d j

Does all networks have = 1 Yes! 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.