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Communicability Betweenness CentralityDefinition
Communicability betweenness measure makes use of the number of walks connecting every pair of nodes as the basis of a betweenness centrality measure.
Let G=(V,E) be a simple undirected graph with n nodes and m edges, and A denote the adjacency matrix of G. Let G(r)=(V,E(r)) be the graph resulting from removing all edges connected to node r but not the node itself. The adjacency matrix for G(r) is A+E(r), where E(r) has nonzeros only in row and column r. The communicability betweenness of a node r is: where G_{prq}=(e^{A})_{pq}−(e^{A+E(r)})_{pq} is the number of walks involving node r, G_{pq}=(e^{A})_{pq} is the number of closed walks starting at node p and ending at node q, and C=(n−1)^{2}−(n−1) is a normalization factor equal to the number of terms in the sum. The resulting ω_{r} takes values between zero and one. The lower bound cannot be attained for a connected graph, and the upper bound is attained in the star graph. SoftwareReferences
