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

The betweenness centrality of a vertex v is a measure of the proportion of shortest paths through the graph which pass through v. It is the sum over all pairs of vertices a and b (both different from v) of the number of least-weight paths from a to b which pass through v divided by the total number of least-weight paths from a to b.

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

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

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

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

$\mathrm{BetweennessCentrality}\left(G\,6\right)$

$\frac{13}{2}$

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

$\left[0\,0\,0\,\frac{1}{2}\,0\,\frac{13}{2}\right]$

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

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