The eigenvector centrality of a graph vertex is a centrality measure in which a vertex receives a large score when it is adjacent to
vertices that themselves have large scores. For a graph with adjacency
matrix 
,
the vector of eigenvector centralities is therefore an eigenvector
of 
,
usually chosen for the largest eigenvalue and normalized.
Historically, this dominant-eigenvector measure is also known as Gould's index of accessibility, especially in geographic and transportation-network contexts (Gould 1967, de Paula 2015).
Eigenvector-type centralities are used when influence or status is taken to come not just from having many neighbors, but from having important neighbors (Bonacich 1987). They are common in prestige and influence scoring, including settings where endorsements from already prominent vertices are weighted more heavily. Thus eigenvector centrality is recursive in a way that degree centrality is not. Katz centrality is a closely related attenuated-walk centrality, while page rank centrality and HITS centrality are eigenvector-style recursive scores for directed link networks.
Eigenvector centrality is implemented in the Wolfram Language as EigenvectorCentrality[g], and precomputed values for many named graphs can be obtained using GraphData[graph, "EigenvectorCentralities"].
