# 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 G

The resulting ω

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.#### Software

#### References

- A HAGBERG, D. S., P SWART. Exploring Network Structure, Dynamics, and Function using NetworkX. In: G VAROQUAUX, T. V., J MILLMAN, ed. Proceedings of the 7th Python in Science conference (SciPy 2008), 2008. 11-15.
- ESTRADA, E., HIGHAM, D. J. & HATANO, N. 2009. Communicability betweenness in complex networks. Physica A: Statistical Mechanics and its Applications, 388, 764-774. DOI: 10.1016/j.physa.2008.11.011