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_{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.
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. 1115.
 ESTRADA, E., HIGHAM, D. J. & HATANO, N. 2009. Communicability betweenness in complex networks. Physica A: Statistical Mechanics and its Applications, 388, 764774. DOI: 10.1016/j.physa.2008.11.011
Comments
I know your explanation for the mathematical definition of this measure starts with "Let G=(V,E) be a simple undirected graph...", nevertheless I would like to ask if this centrality measure, along with your algorithm ("communibet" in package centiserve) can be used in directed weighted networks. Many thanks! 

Add Replay  written July 29, 2021, 11:57 am by Anonymous User 