InformationCentrality - Maple Help

GraphTheory

 InformationCentrality
 compute information centrality

 Calling Sequence InformationCentrality(G) InformationCentrality(G, v)

Parameters

 G - graph v - (optional) a vertex of G

Description

 • 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.

Definition

 • 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

$\mathrm{IC}\left(v\right)=\frac{n}{{\sum }_{u\in V}\left({B}_{u,u}-{B}_{v,v}-2{B}_{u,v}\right)}$

Examples

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

Compute the information centrality for a specified graph.

 > $G≔\mathrm{Graph}\left(6,\left\{\left\{1,3\right\},\left\{1,6\right\},\left\{2,4\right\},\left\{2,6\right\},\left\{3,6\right\},\left\{4,5\right\},\left\{4,6\right\},\left\{5,6\right\}\right\}\right)$
 ${G}{≔}{\mathrm{Graph 1: an undirected unweighted graph with 6 vertices and 8 edge\left(s\right)}}$ (1)
 > $\mathrm{DrawGraph}\left(G\right)$
 > $\mathrm{InformationCentrality}\left(G\right)$
 $\left[\frac{{72}}{{61}}{,}\frac{{36}}{{29}}{,}\frac{{72}}{{61}}{,}\frac{{72}}{{49}}{,}\frac{{36}}{{29}}{,}\frac{{72}}{{37}}\right]$ (2)

References

 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

Compatibility

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