gives a list of page-rank centralities for the vertices in the graph g and weight α.


gives a list of page-rank centralities, using weight α and initial centralities β.


uses rules vw to specify the graph g.

Details and Options

  • Page-rank centralities represent the likelihood that a person randomly following links arrives at any particular page on the web graph.
  • PageRankCentrality gives a list of centralities that are solutions to c=alpha TemplateBox[{a}, Transpose].d.c+beta, where is the adjacency matrix of g and is the diagonal matrix consisting of , where is the out-degree of the ^(th) vertex. »
  • If β is a scalar, it is taken to mean {β,β,}.
  • PageRankCentrality[g,α] is equivalent to PageRankCentrality[g,α,1/VertexCount[g]].
  • Page-rank centralities are normalized.
  • The option WorkingPrecision->p can be used to control the precision used in internal computations.
  • PageRankCentrality works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

Background & Context

  • PageRankCentrality returns a list of normalized positive real numbers (page ranks) that are particular centrality measures of the vertices of a graph. The page rank centralities of the directed graph corresponding to a network of hyperlinked web pages give the probability that randomly clicking on hyperlinks in the network will arrive at a particular page. Page ranks can therefore be used to find the web page that is most likely to be visited after a large number of clicks. Other applications include finding a species whose extinction would lead to ecosystem collapse in a given food chain and finding proteins in a metabolic cellular network that play a major (or marginal) role in the system. Here, each such system is again represented as a directed graph.
  • Page ranks can be used to measure the importance of web pages or vertices in a network, based on the likelihood to reach each vertex. They are used by Google's search engine to rank the order in which websites are returned when performing searches. PageRank was named after Larry Page, one of the founders of Google.
  • PageRankCentrality requires specification of a weight (or damping factor) satisfying and may optionally also take a list of initial centralities of length VertexCount[g] whose values sum to 1.
  • A similar measure, HITSCentrality, rates web pages using hyperlink-induced topic search.


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Basic Examples  (2)

Compute page-rank centralities:

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Find the probability that a person randomly clicking on hyperlinks will arrive at a particular page:

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Rank web pages, with the most visible pages first:

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Scope  (7)

Options  (3)

Applications  (6)

Properties & Relations  (3)

Introduced in 2010
Updated in 2015