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Understanding Centrality Measures in Networks: Concepts and Applications

This lecture, presented by Monojit Choudhury, covers fundamental concepts of centrality measures in networks, including degree, closeness, and betweenness centralities. It explores how these measures identify central nodes in various scenarios like information diffusion, web authority, and disease spread, showcasing the significance of node prominence. Participants are encouraged to engage and question the relativeness of centralization in network structures. Applications discussed include the popularity of web pages and the identification of potential targets in sexual interaction networks.

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Understanding Centrality Measures in Networks: Concepts and Applications

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  1. Lectures 6 & 7Centrality MeasuresFebruary 2, 2009 Monojit Choudhury monojitc@microsoft.com

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

  3. 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

  4. Lecture 5Centrality MeasuresFebruary 2, 2009 Monojit Choudhury monojitc@microsoft.com

  5. Question 1: Information percolation 1 3 2 5 6 4 7 8 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?

  6. Question 2: Searching the Web 1 3 2 5 6 4 7 8 In this hyperlinked network of webpages, which pages are most likely to contain authoritative information ?

  7. Question 3: Spreading of STDs 1 3 2 5 6 4 7 8 In this hypothetical sexual interaction network, who are most likely to be affected by STDs such as AIDS?

  8. 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

  9. 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

  10. How much is this network centralized? 1 3 2 5 6 4 7 8

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

  12. Closeness Centrality • Reciprocal of the sum of shortest paths to all the nodes • Compute closeness centrality for nodes 3 and 6 1 3 2 5 6 4 7 8

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

  14. Spreading of STDs 1 3 2 5 6 4 7 8 Who should be removed from this network to make this community less susceptible to spreading of STDs?

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

  16. Mathematical Definition Can be extended to edges v s t

  17. 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?

  18. Question 2: Searching the Web 1 3 2 5 6 4 7 8 In this hyperlinked network of webpages, which pages are most popular?

  19. The basic idea • I am popular if my friends are popular 1 3 2 5 6 4 7 8 p6 = p2 + p5 + p7 + p8

  20. Computing Popularity 1 2 1 1 1 4 3 1 1 4 1 2 1 3 1 3

  21. Computing Popularity Oops! Popularity grows unboundedly!! 2 6 1 4 4 9 10 3 13 4 2 6 3 10 10 3

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

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

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

  25. Is it converging? 15/206 1 9/206 3 29/206 2 5 33/206 39/206 6 15/206 4 33/206 8 33/206 7

  26. 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?

  27. 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 1 3 2 5 6 4 7 8

  28. What’s the probability that the surfer is at node i? 1 3 2 5 6 4 7 8 p6 = wp6 + (1-w) [p2/4+ p5 + p7/3 + p8]

  29. What’s the probability that the surfer is at node i? 1 3 2 5 6 4 7 8 pi = wpi+ (1-w)jajipj/dj

  30. Therefore, popularity is • Eigenvector Centrality • Introduced by Bonacich (1972) • A slightly different variant is used as “PageRank” pi = (1-w)+ wjajipj/dj

  31. 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.

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