# Communicability Betweenness Centrality

#### Definition

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 Gprq=(eA)pq−(eA+E(r))pq is the number of walks involving node r, Gpq=(eA)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 ωr 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.

#### References

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