# Wolfram Language & System 11.0 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# EigenvectorCentrality

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

EigenvectorCentrality[g,"In"]
gives a list of in-centralities for a directed graph g.

EigenvectorCentrality[g,"Out"]
gives a list of out-centralities for a directed graph g.

EigenvectorCentrality[{vw,},]
uses rules vw to specify the graph g.

## Details and OptionsDetails and Options

• EigenvectorCentrality will give high centralities to vertices that are connected to many other well-connected vertices.
• EigenvectorCentrality gives a list of centralities that can be expressed as a weighted sum of centralities of its neighbors.
• With being the largest eigenvalue of the adjacency matrix for the graph g, you have:
•  EigenvectorCentrality[g] EigenvectorCentrality[g,"In"] , left eigenvector EigenvectorCentrality[g,"Out"] , right eigenvector
• Eigenvector centralities are normalized.
• For a directed graph g, is equivalent to EigenvectorCentrality[g,"In"].
• The option can be used to control precision used in internal computations.
• EigenvectorCentrality works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

## Background & ContextBackground & Context

• EigenvectorCentrality returns a list of non-negative numbers ("eigenvector centralities", also known as Gould indices) that are particular centrality measures of the vertices of a graph. The returned centralities are always normalized so that they sum to 1. Eigenvector centrality is a measure of the centrality of a node in a network, based on the weighted sum of centralities of its neighbors. It therefore identifies nodes in the network that are connected to many other well-connected nodes. This measure has found applications in social networks, transportation, biology, and social sciences.
• For a connected undirected graph, the vector of eigenvector centralities satisfies the eigenvector equation , where is the largest eigenvalue of the graph's adjacency matrix . In other words, for a connected undirected graph, the vector of eigenvector centralities is given by the (suitably normalized) eigenvector of corresponding to its largest eigenvalue. For a disconnected undirected graph, the vector of eigenvector centralities is given by a (suitably normalized) weighted sum of connected component eigenvector centralities.
• For a connected directed graph, the in-centrality vector satisfies the equation and the out-centrality satisfies . An additional "In" or "Out" argument may be specified to obtain a list of in-centralities or out-centralities, respectively, for a directed graph.
• EigenvectorCentrality returns machine numbers by default but supports a WorkingPrecision argument to allow high-precision or exact (by specifying Infinity as the precision) values to be computed. EigenvectorCentrality is a normalized special case of KatzCentrality with and . A related centrality is PageRankCentrality. Eigenvectors, Eigenvalues, and Eigensystem can be used to compute eigenproperties of a given square matrix, and AdjacencyMatrix to obtain the adjacency matrix of a given graph.

## ExamplesExamplesopen allclose all

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

Compute eigenvector centralities:

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Highlight:

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Rank the vertices. Highest-ranked vertices are connected to many well-connected vertices:

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