Centrality Measures Leonid E. Zhukov School of Data Analysis and Artificial Intelligence Department of Computer Science National Research University Higher School of Economics Structural Analysis and Visualization of Networks Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 1 / 22
Graph-theoretic measures Which vertices are important? M.Grandjean, 2014 Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 2 / 22
Graph-theoretic measures The eccentricity ɛ(v) of a vertex v is the maximum distance between v and any other vertex u of the graph ɛ(v) = max u V d(u, v) Graph diameter is the maximum eccentricity d = max v V ɛ(v) Graph radius is the minimum eccentricity r = min v V ɛ(v). A point v is a central point of a graph if the eccentricity of the point equals the graph radius ɛ(v) = r from Eric Weisstein MathWorld Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 3 / 22
Graph-theoretic measures Graph center is a set of of vertices with graph eccentricity equal to the graph radius ɛ(v) = r - set of central points Graph periphery is a set of vertices that have graph eccentricities equal to the graph diameter ɛ(v) = d from Eric Weisstein MathWorld Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 4 / 22
Centrality Measures Sociology. Most "important"actors: actor location in the social network Actor centrality - involvement with other actors, many ties, source or recipient. Undirected network. Actor prestige - recipient (object) of many ties, ties directed to an actor. Directed network. In this lecture: undirected graphs, symmetric matrix A ij = A ji, A = A T Linton Freeman, 1979 Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 5 / 22
Three graphs Star graph Circle graph Line Graph Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 6 / 22
Degree centrality Degree centrality: number of nearest neighbours C D (i) = k(i) = j A ij = j A ji Normalized degree centrality C D (i) = 1 n 1 C D(i) High centrality degree -direct contact with many other actors Low degree - not active, peripheral actor Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 7 / 22
Closeness centrality Closeness centrality: how close an actor to all the other actors in network C C (i) = j 1 d(i, j) Normalized closeness centrality C C (i) = (n 1)C C (i) Actor in the center can quickly interact with all others, short communication path to others, minimal number of steps to reach others [*** Harmonic centrality = j 1 d(i,j) ***] Alex Bavelas, 1948 Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 8 / 22
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) Probability that a communication from s to t will go through i (geodesics) Linton Freeman, 1977 Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 9 / 22
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 5 10.02.2015 10 / 22
Centrality examples Closeness centrality from www.activenetworks.net Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 11 / 22
Centrality examples Betweenness centrality from www.activenetworks.net Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 12 / 22
Centrality examples Eigenvector centrality from www.activenetworks.net Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 13 / 22
Katz status index Weighted count of all paths coming to the node: the weight of path of length n is counted with attenuation factor β n, β < 1 λ 1 k i = β j A ij + β 2 j A 2 ij + β 3 j A 3 ij +... k = (βa + β 2 A 2 + β 3 A 3 +...)e = (β n A n )e = ( (βa) n I)e n=1 n=0 (βa) n = (I βa) 1 n=0 k = ((I βa) 1 I)e (I βa)k = βae Leo Katz, 1953 k = β(i βa) 1 Ae Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 14 / 22
Bonacich centrality Two-parametric centrality measure c(α, β) β - degree to which an individual status is a function of the statuses of those to whom he is connected (can be positive if connected to powerful and negative, if connected to powerless ) α - normalization parameter c i (α, β) = j (α + βc j )A ij c = αae + βac (I βa)c = αae c = α(i βa) 1 Ae α - found from normalization c 2 = c 2 i = 1 Phillip Bonacich, 1987 Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 15 / 22
Modified versions Katz centrality (Newman): x i = α j A ij x j + β i Alpha-centrality (Bonacich): x = αax + β x = (I αa) 1 β x i = α j A ij x j + 1 x = αax + e x = (I αa) 1 e Bonacich, 2001,Newman, 2010 Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 16 / 22
Centrality examples A) Degree centrality B) Closeness centrality C) Betweenness centrality D) Eigenvector centrality E) Katz centrality F) Alpha centrality from Claudio Rocchini Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 17 / 22
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) Linton Freeman, 1979 Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 18 / 22
Prestige Prestige - measure of node importance in directed graphs Degree prestige k in (i) Proximity prestige (closeness) Status or Rank prestige (Katz, Bonacich) Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 19 / 22
Metrics comparison Pearson correlation coefficient r = n i=1 (X i X )(Y i Ȳ ) n i=1 (X n i X ) 2 i=1 (Y i Ȳ ) 2 Shows linear dependence between variables, 1 r 1 (perfect when related by linear function) Spearman rank correlation coefficient (Sperman s rho): Convert raw scores to ranks - sort by score: X i x i, Y i y i ρ = 1 6 n i=1 (x i y i ) 2 n(n 2 1) Shows strength of monotonic association (perfect for monotone increasing/decreasing relationship) Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 20 / 22
Ranking comparison The Kendall tau rank distance is a metric that counts the number of pairwise disagreements between two ranking lists Kendall rank correlation coefficient, commonly referred to as Kendall s tau coefficient τ = n c n d n(n 1)/2 n c - number of concordant pairs, n d - number of discordant pairs 1 τ 1, perfect agreement τ = 1, reversed τ = 1 Example Rank 1 A B C D E Rank 2 C D A B E τ = 6 4 5(5 1)/2 = 0.2 Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 21 / 22
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 Leonid E. Zhukov (HSE) Lecture 5 10.02.2015 22 / 22