BUILT-IN MATHEMATICA SYMBOL

ClosenessCentrality

ClosenessCentrality[g]
gives a list of closeness centralities for the vertices in the graph g.

ClosenessCentrality for a graph g is given as where is the average distance from vertex to all other vertices connected to .

If is the distance matrix then the average distance from vertex to all connected vertices is given by , where the sum is taken over all finite and is the number of vertices connected to .

Find the closeness centrality for each vertex in a connected graph:

Closeness is computed separately for each connected component:

Find the closeness centrality for each vertex in a connected graph:

Out[1]=

Closeness is computed separately for each connected component:

Out[1]=

ClosenessCentrality for undirected graphs:

Highlight the vertex with highest centrality:

Directed graphs:

Highlight the vertex with highest centrality:

Work with large graphs:

Highlight the closeness centrality for CycleGraph:

An unbalanced tree:

Create a social network:

Find the people with more direct influence on others:

ClosenessCentrality is the inverse of average finite distances to other vertices:

An undirected graph:

A directed graph:

Closeness centrality for isolated vertices is taken to be zero:

Closeness centralities for an undirected graph are equivalent to centralities for each component:

Computing the centralities for each component yields the same result: