BUILTIN WOLFRAM LANGUAGE SYMBOL
EigenvectorCentrality[g]
gives a list of eigenvector centralities for the vertices in the graph g.
EigenvectorCentrality[g,"In"]
gives a list of incentralities for a directed graph g.
EigenvectorCentrality[g,"Out"]
gives a list of outcentralities for a directed graph g.
Details and OptionsDetails and Options
 EigenvectorCentrality will give high centralities to vertices that are connected to many other wellconnected 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, EigenvectorCentrality[g] is equivalent to EigenvectorCentrality[g,"In"].
 The option WorkingPrecision>p 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 nonnegative 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 wellconnected 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 incentrality vector satisfies the equation and the outcentrality satisfies . An additional or argument may be specified to obtain a list of incentralities or outcentralities, respectively, for a directed graph.
 EigenvectorCentrality returns machine numbers by default but supports a WorkingPrecision argument to allow highprecision 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.
Introduced in 2010
(8.0)
 Updated in 2014 (10.0)
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