Returns the Google matrix of the graph.


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


The damping factor.

personalization: dict, optional

The “personalization vector” consisting of a dictionary with a key some subset of graph nodes and personalization value each of those. At least one personalization value must be non-zero. If not specified, a nodes personalization value will be zero. By default, a uniform distribution is used.

nodelistlist, optional

The rows and columns are ordered according to the nodes in nodelist. If nodelist is None, then the ordering is produced by G.nodes().

weightkey, 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 below). It may be common to have the dangling dict to be the same as the personalization dict.

A2D NumPy ndarray

Google matrix of the graph

See also



The array returned represents the transition matrix that describes the Markov chain used in PageRank. For PageRank to converge to a unique solution (i.e., a unique stationary distribution in a Markov chain), the transition matrix must be irreducible. In other words, it must be that there exists a path between every pair of nodes in the graph, or else there is the potential of “rank sinks.”

This implementation works with Multi(Di)Graphs. For multigraphs the weight between two nodes is set to be the sum of all edge weights between those nodes.

Additional backends implement this function

graphblas : OpenMP-enabled sparse linear algebra backend.