This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.
 GRAPH UTILITIES PACKAGE SYMBOL

# ClosenessCentrality

 ClosenessCentrality[g] finds the closeness centrality.
• The closeness centrality of a vertex u is defined as the inverse of the sum of the distance from u to all other vertices. The closeness centrality of a vertex in a disconnected graph is based on the closeness centrality of the component where this vertex belongs.
• The following options can be given:
 Weighted True whether edge weight is to be used in calculating distance Normalize False whether to normalize the output
This defines a small graph:
This shows that vertex 2, being at the center of this graph, has a higher closeness centrality:
This defines a disconnected graph and finds the closeness centrality:
Needs["GraphUtilities`"]
This defines a small graph:
 Out[3]=
This shows that vertex 2, being at the center of this graph, has a higher closeness centrality:
 Out[4]=

Needs["GraphUtilities`"]
This defines a disconnected graph and finds the closeness centrality:
 Out[3]=
 Out[4]=
 Scope   (1)
The centrality of a vertex that cannot reach all other vertices in its component is zero:
The centrality of a disconnected graph is calculated by treating each component separately:
 Options   (1)
This defines a graph with edge weights:
By default, edge weights are taken into account:
This gives the closeness centrality if edge weights are assumed to be 1.:
 Applications   (1)
A plot of a grid graph with vertices of high centrality in red: