Node Centrality and Ranking on Networks Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Social Network Analysis MAGoLEGO course Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 1 / 27
Lecture outline 1 Centrality and prestige 2 Centrality measures Degree centrality Closeness centrality Betweenness centrality Eigenvector centrality 3 Ranking on directed graphs Pagerank HITS Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 2 / 27
Centrality Measures Determine the most important or prominent actors in the network based on actor location. Marriage alliances among leading Florentine families 15th century. Padgett, 1993 Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 3 / 27
Centrality measures Undirected graphs: - Actor centrality - involvement (connections) with other actors Directed graphs: - Actor centrality - source of the ties (outgoing edges) - Actor prestige - recipient of many ties (incoming edges) Linton Freeman, 1979 Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 4 / 27
Three graphs Star graph Circle graph Line Graph Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 5 / 27
Degree centrality Degree centrality: number of nearest neighbors C D (i) = k(i) = j A ij = j A ji Normalized degree centrality C D (i) = 1 n 1 C D(i) = k(i) n 1 High centrality degree -direct contact with many other actors Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 6 / 27
Closeness centrality Closeness centrality: how close an actor to all the other actors in network Normalized closeness centrality C C (i) = j 1 d(i, j) CC (i) = (n 1)C C (i) = n 1 d(i, j) High closeness centrality - short communication path to others, minimal number of steps to reach others j Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 7 / 27
Betweenness centrality Betweenness centrality: number of shortest paths going through the actor σ st (i) C B (i) = σ st (i) σ st Normalized betweenness centrality s t i C B (i) = 2 (n 1)(n 2) C B(i) = 2 (n 1)(n 2) s t i Hight betweenness centrality - vertex lies on many shortest paths Probability that a communication from s to t will go through i σ st (i) σ st Linton Freeman, 1977 Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 8 / 27
Eigenvector centrality Importance of a node depends on the importance of its neighbors (recursive definition) v i j A ij v j v i = 1 A ij v j λ j Av = λv Select an eigenvector associated with largest eigenvalue λ = λ 1, v = v 1 Phillip Bonacich, 1972. Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 9 / 27
Centrality examples Closeness centrality igraph:closeness() from www.activenetworks.net Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 10 / 27
Centrality examples Betweenness centrality igraph:betweenness() from www.activenetworks.net Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 11 / 27
Centrality examples Eigenvector centrality igraph:evcent() from www.activenetworks.net Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 12 / 27
Centrality examples A) Degree centrality B) Closeness centrality C) Betweenness centrality D) Eigenvector centrality from Claudio Rocchini Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 13 / 27
Centralization Centralization (network measure) - how central the most central node in the network in relation to all other nodes. C x = N i [C x (p ) C x (p i )] max N i [C x (p ) C x (p i )] C x - one of the centrality measures p - node with the largest centrality value max - is taken over all graphs with the same number of nodes (for degree, closeness and betweenness the most centralized structure is the star graph) igraph: centralization.degree(), centralization.closeness(), centralization.betweenness(), centralization.evcent() Linton Freeman, 1979 Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 14 / 27
Directional relations Directed graph: distinguish between choices made (outgoing edges) and choices received (incoming edges) sending - receiving export - import cite papers - being cited Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 15 / 27
Centrality measures All based on outgoing edges Degree centrality (normalized): Closeness centrality (normalized): C D (i) = kout (i) n 1 CC (i) = n 1 d(i, j) **Betweenness centrality (normalized): C B (i) = 1 (n 1)(n 2) j s t i σ st (i) σ st Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 16 / 27
Status or Rank Prestige Leo Katz, 1953. Node prestige depends on prestige of directly connected actors iterate p i j N(i) p j = j A ji p j p t+1 = A T p t, p t=0 = p 0 Difficulties: - Absorbing nodes - Source nodes - Cycles ** Solution to p = A T p might not exist. Nontrivial solution only if det(i A T ) = 0. Need to constraint matrix Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 17 / 27
Web as a graph Hyperlinks - implicit endorsements Web graph - graph of endorsements (sometimes reciprocal) Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 18 / 27
PageRank PageRank can be thought of as a model of user behavior. We assume there is a random surfer who is given a web page at random and keeps clicking on links, never hitting back but eventually gets bored and starts on another random page. The probability that the random surfer visits a page is its PageRank. Sergey Brin and Larry Page, 1998 Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 19 / 27
Random walk Random walk on graph p t+1 i = j N(i) p t j d out j = j A ji dj out p j P = D 1 A, D ii = diag{d out i } p t+1 = P T p t with teleportation p t+1 = αp T p t + (1 α) e n Perron-Frobenius Theorem guarantees existence and uniqueness of the solution to p = αp T p + (1 α) e n Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 20 / 27
PageRank igraph: page.rank() Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 21 / 27
PageRank beyond the Web Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 22 / 27
Hubs and Authorities (HITS) Citation networks. Reviews vs original research (authoritative) papers authorities, contain useful information, a i hubs, contains links to authorities, h i Mutual recursion Good authorities referred by good hubs a i j A ji h j Good hubs point to good authorities h i j A ij a j Jon Kleinberg, 1999 Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 23 / 27
HITS System of linear equations a = αa T h h = βaa Symmetric eigenvalue problem (A T A)a = λa (AA T )h = λh where eigenvalue λ = (αβ) 1 Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 24 / 27
Hubs and Authorities Hubs Authorities igraph: hub.score(), authority.score() Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 25 / 27
Florentines families Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 26 / 27
References Centrality in Social Networks. Conceptual Clarification, Linton C. Freeman, Social Networks, 1, 215-239, 1979 Power and Centrality: A Family of Measures, Phillip Bonacich, The American Journal of Sociology, Vol. 92, No. 5, 1170-1182, 1987 A new status index derived from sociometric analysis, L. Katz, Psychometrika, 19, 39-43, 1953. Eigenvector-like measures of centrality for asymmetric relations, Phillip Bonacich, Paulette Lloyd, Social Networks 23, 191?201, 2001 The PageRank Citation Ranknig: Bringing Order to the Web. S. Brin, L. Page, R. Motwany, T. Winograd, Stanford Digital Library Technologies Project, 1998 Authoritative Sources in a Hyperlinked Environment. Jon M. Kleinberg, Proc. 9th ACM-SIAM Symposium on Discrete Algorithms A Survey of Eigenvector Methods of Web Information Retrieval. Amy N. Langville and Carl D. Meyer, 2004 PageRank beyond the Web. David F. Gleich, arxiv:1407.5107, 2014 Leonid E. Zhukov (HSE) Lecture 4 24.04.2015 27 / 27