InformationCentrality returns the information centrality for a specified vertex in the given graph G, or if no vertex is specified, returns a list of the information centralities for each vertex in G.

Let G be a graph with vertex set V of size n, and let L be the Laplacian matrix of G.

Define B to be the matrix inverse of L+J, where J represents the n by n matrix in which every entry is 1.

Then the information centrality IC(v) of a vertex v in G is defined as

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$G≔\mathrm{Graph}\left(6\,\left\{\left\{1\,3\right\}\,\left\{1\,6\right\}\,\left\{2\,4\right\}\,\left\{2\,6\right\}\,\left\{3\,6\right\}\,\left\{4\,5\right\}\,\left\{4\,6\right\}\,\left\{5\,6\right\}\right\}\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 6\; vertices\; and\; 8\; edge(s)}$

$\mathrm{DrawGraph}\left(G\right)$

$\mathrm{InformationCentrality}\left(G\right)$

$\left[\frac{72}{61}\,\frac{36}{29}\,\frac{72}{61}\,\frac{72}{49}\,\frac{36}{29}\,\frac{72}{37}\right]$

K. Stephenson and M. Zelen. "Rethinking centrality: Methods and examples." Social Networks, 11(1):1–37, 1989. doi:10.1016/0378-8733(89)90016-6

L. Shan, Y. Yi, Z. Zhang, "Improving Information Centrality of a Node in Complex Networks by Adding Edges", Proc IJCAI 2018. doi:10.24963/ijcai.2018/491

The GraphTheory[InformationCentrality] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.