The communicability betweenness of a node r is: $$\omega_{r} = \frac{1}{C} \sum_{p}\sum_{q}\frac{G_{prq}}{G_{pq}}, p\neq q,p\neq r, q\neq r$$ where 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.
communibet(graph, vids = V(graph), normalized = FALSE)The input graph as igraph object
Vertex sequence, the vertices for which the centrality values are returned. Default is all vertices.
Logical scalar, whether to calculate the normalized score.
A numeric vector contaning the centrality scores for the selected vertices.
Communicability betweenness measure makes use of the number of walks connecting every pair of nodes as the basis of a betweenness centrality measure. The resulting \(\omega_{r}\) 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. More detail at Communicability Betweenness Centrality
Estrada, Ernesto, Desmond J. Higham, and Naomichi Hatano. "Communicability betweenness in complex networks." Physica A: Statistical Mechanics and its Applications 388.5 (2009): 764-774.
Hagberg, Aric, Pieter Swart, and Daniel S Chult. Exploring network structure, dynamics, and function using NetworkX. No. LA-UR-08-05495; LA-UR-08-5495. Los Alamos National Laboratory (LANL), 2008.
# NOT RUN {
g <- graph(c(1,2,2,3,2,6,6,5,3,5,3,4,5,4,4,7), directed=FALSE)
communibet(g)
# }
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