# Wolfram Language & System 10.0 (2014)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
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# EccentricityCentrality

gives a list of eccentricity centralities for the vertices in the graph g.

## DetailsDetails

• EccentricityCentrality will give high centralities to vertices that are at short maximum distances to every other reachable vertex.
• EccentricityCentrality for a graph g is given by , where is the maximum distance from vertex to all other vertices connected to .
• The eccentricity centrality for isolated vertices is taken to be zero.
• EccentricityCentrality works with undirected graphs, directed graphs, weighted graphs, multigraphs, and mixed graphs.

## BackgroundBackground

• EccentricityCentrality returns a list of non-negative machine numbers ("eccentricity centralities") that approximate particular centrality measures of the vertices of a graph. Eccentricity centrality is a measure of the centrality of a node in a network based on having a small maximum distance from a node to every other reachable node (i.e. the graph eccentricities). This measure has found applications in social networks, transportation, biology, and the social sciences.
• If is the maximum distance from vertex to all other vertices connected to , then the eccentricity centralities are given by . The eccentricity centrality for isolated vertices is taken to be zero. Eccentricity centralities lie between 0 and 1 inclusive.
• The eccentricity centrality of a vertex is the reciprocal of its VertexEccentricity. The full distance matrix of a graph can be computed using GraphDistanceMatrix.

## ExamplesExamplesopen allclose all

### Basic Examples  (2)Basic Examples  (2)

Compute eccentricity centralities:

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Highlight:

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Rank vertices. Highest-ranked vertices are at short distances to every other reachable vertex:

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