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Return the PageRank of the nodes in the graph.

PageRank computes a ranking of the nodes in the graph G based on the structure of the incoming links. It was originally designed as an algorithm to rank web pages.

Parameters :

G : graph

A NetworkX graph. Undirected graphs will be converted to a directed graph with two directed edges for each undirected edge.

alpha : float, optional

Damping parameter for PageRank, default=0.85.

personalization: dict, optional

The “personalization vector” consisting of a dictionary with a key for every graph node and nonzero personalization value for each node. By default, a uniform distribution is used.

max_iter : integer, optional

Maximum number of iterations in power method eigenvalue solver.

tol : float, optional

Error tolerance used to check convergence in power method solver.

nstart : dictionary, optional

Starting value of PageRank iteration for each node.

weight : key, optional

Edge data key to use as weight. If None weights are set to 1.

dangling: dict, optional

The outedges to be assigned to any “dangling” nodes, i.e., nodes without any outedges. The dict key is the node the outedge points to and the dict value is the weight of that outedge. By default, dangling nodes are given outedges according to the personalization vector (uniform if not specified). This must be selected to result in an irreducible transition matrix (see notes under google_matrix). It may be common to have the dangling dict to be the same as the personalization dict.

Returns :

pagerank : dictionary

Dictionary of nodes with PageRank as value


The eigenvector calculation is done by the power iteration method and has no guarantee of convergence. The iteration will stop after max_iter iterations or an error tolerance of number_of_nodes(G)*tol has been reached.

The PageRank algorithm was designed for directed graphs but this algorithm does not check if the input graph is directed and will execute on undirected graphs by converting each edge in the directed graph to two edges.


[R255]A. Langville and C. Meyer, “A survey of eigenvector methods of web information retrieval.” http://citeseer.ist.psu.edu/713792.html
[R256]Page, Lawrence; Brin, Sergey; Motwani, Rajeev and Winograd, Terry, The PageRank citation ranking: Bringing order to the Web. 1999 http://dbpubs.stanford.edu:8090/pub/showDoc.Fulltext?lang=en&doc=1999-66&format=pdf


>>> G = nx.DiGraph(nx.path_graph(4))
>>> pr = nx.pagerank(G, alpha=0.9)