Complex Social System, Elections. Introduction to Network Analysis 1

Transcription

1 Complex Social System, Elections Introduction to Network Analysis 1

2 Complex Social System, Network I person A voted for B A is more central than B if more people voted for A In-degree centrality index = number of in-edges Introduction to Network Analysis 2

3 Complex Social System, Network II person A convinced B Central because it mediates information between groups - Betweeness centrality (between groups) Another example: friendship network, degree-based centrality Introduction to Network Analysis 3

4 Measures and Metrics Q: What are the most important nodes and edges? Degree centrality: Closeness centrality: Edge removal centrality: There are many different ways to quantify the importance Can you come up with your interesting new measure of importance? Is it important for different types of networks? Can you compute it efficiently? Introduction to Network Analysis 4

5 Eccentricity from U. Brandes Network Analysis Introduction to Network Analysis 5

6 Eigenvector Centrality degree centrality Repeating last step t times Rewrite this linear combination with eigenvectors of A d(i)=2 Eigenvector centrality is defined by values Introduction to Network Analysis 6

7 Eigenvector Centrality Introduction to Network Analysis 7

8 PageRank Markov Chain: probabilities of visiting the pages after k steps is P k x. Problem: dangling nodes (d + (i)=0) Solution: damping factor a (usually small) transition matrix regular random walk teleportation Note: P is still a Markov matrix with no 0s Introduction to Network Analysis 8

9 Eigenproblem-based Centralities: Computational Problems These problems are equivalent to solving eigenproblems, namely, P v = v. Gaussian elimination is very expensive. Iterative computation of P v is less expensive because P is usually sparse. Principal eigenvector of P is a PageRank. It can be solved by Power Method (iterative) and by Algebraic Multigrid. Parallel computation Sublinear computation Updating PageRank vector, evolving networks Further reading: Berkhin ``A Survey on PageRank computing Langville, Meyer ``Deeper Inside PageRank Introduction to Network Analysis 9

10 Hubs and Authorities Observation: In some networks a vertex may be important if it points to others with high centrality. Examples: X, Y, Z Cool Survey 9/11 contacts X sends an information to many leaders but they don t know X. X is influential. Scientific paper citations Raddicchi et al. Diffusion of scientific credits and the ranking of scientists Introduction to Network Analysis 10

11 Hubs and Authorities Authorities are nodes that contain useful information on a topic of interest. Hubs are nodes that point to best authorities. HITS Algorithm computes for each node i its authority and hub centralities x i and y i, respectively same leading (see eigenvector centrality) eigenvalue in both cases! All eigenvalues of AA T and A T A are the same (check and prove!). Multiplying both sides of the first equation by A T gives i.e., once we have a vector of authorities, the hub centrality can be calculated faster Note: Authority centrality of A = Eigenvector centrality of co-citation matrix AA T Current applications: Teoma.com and Ask.com Further reading: Kleinberg, Authoritative sources in a hyper-linked environment Introduction to Network Analysis 11

12 Closeness Centrality Problems: i-j path is infinity (for example different connected components) Solution: Harmonic Mean Distance Centrality In practice C spans relatively small range of values Geodesic distances are integers, often small because distance increases logarithmically with the size of the network Example: Internet Movie Database highest centrality = , lowest centrality = , ratio 3.6 for 500K actors Introduction to Network Analysis 12

13 Shortest Paths-based Centrality is a number of s-t shortest paths containing i is a number of all s-t shortest-paths Observation: In practice, communication or transport of goods in networks follow different kinds of paths that tend to be shortest. Question: How much work can be done by a node? Disadvantage? Stress Centrality for nodes Relation between stress centralities Betweenness Centrality for edges Interpretation: BC is a quantification of communication control which has a vertex over all pairs of nodes. Introduction to Network Analysis 13