EigenvectorCentrality

EigenvectorCentrality[g]
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.

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]c=TemplateBox[{{{1, /, {lambda, _, 1}}, a}}, Transpose].c
    EigenvectorCentrality[g,"In"]c=TemplateBox[{{{1, /, {lambda, _, 1}}, a}}, Transpose].c, 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
Background

  • 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 or 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.
Introduced in 2010
(8.0)
| Updated in 2014
(10.0)