# EccentricityCentrality

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

EccentricityCentrality[{vw,}]

uses rules vw to specify the graph g.

# Details • 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.

# Background & Context

• 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.

# Examples

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## Basic Examples(2)

Compute eccentricity centralities:

Highlight:

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

## Scope(7)

EccentricityCentrality works with undirected graphs:

Directed graphs:

Weighted graphs:

Multigraphs:

Mixed graphs:

Use rules to specify the graph:

EccentricityCentrality works with large graphs:

## Applications(5)

Rank vertices by their eccentricity:

Highlight the eccentricity centrality for CycleGraph:

Based on the infrastructure network of the Marshall Islands in eastern Micronesia, find a strategically located island:

It is different with the median of the graph:

A road network linking Chicago suburbs. Find the best location for hospitals and fire departments, to minimize the distance traveled by emergency vehicles:

For graphs with vertices, the largest sum in differences in eccentricity centrality between the most central vertex and all other vertices is the inverse of :

Measure how central the most central vertex is with respect to other vertices:

Centralization of social networks:

## Properties & Relations(6)

EccentricityCentrality is the inverse of maximum distances to other reachable vertices:

The inverse of maximum distances to other reachable vertices:

Eccentricity centrality of a vertex is the reciprocal of the VertexEccentricity:

Eccentricity centrality is between 0 and 1:

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

Computing the centralities for each component yields the same result:

Use GraphCenter to find vertices with the highest eccentricity centrality:

Use VertexIndex to obtain the centrality for a specific vertex: