7.5.2.2. networkx.algorithms.centrality.eigenvector_centrality_numpy¶

networkx.algorithms.centrality.eigenvector_centrality_numpy(G, weight='weight', max_iter=50, tol=0)[source]¶

Compute the eigenvector centrality for the graph G.

Eigenvector centrality computes the centrality for a node based on the centrality of its neighbors. The eigenvector centrality for node i is

\[\mathbf{Ax} = \lambda \mathbf{x}\]

where A is the adjacency matrix of the graph G with eigenvalue lambda. By virtue of the Perron–Frobenius theorem, there is a unique and positive solution if lambda is the largest eigenvalue associated with the eigenvector of the adjacency matrix A ([R535]).

Parameters:	G : graph A networkx graph weight : None or string, optional The name of the edge attribute used as weight. If None, all edge weights are considered equal. max_iter : integer, optional Maximum number of iterations in power method. tol : float, optional Relative accuracy for eigenvalues (stopping criterion). The default value of 0 implies machine precision.
Returns:	nodes : dictionary Dictionary of nodes with eigenvector centrality as the value.
Raises:	NetworkXPointlessConcept If the graph G is the null graph.

See also

eigenvector_centrality, pagerank, hits

Notes

The measure was introduced by [R534].

This algorithm uses the SciPy sparse eigenvalue solver (ARPACK) to find the largest eigenvalue/eigenvector pair.

For directed graphs this is “left” eigenvector centrality which corresponds to the in-edges in the graph. For out-edges eigenvector centrality first reverse the graph with G.reverse().

References

[R534]

(1, 2) Phillip Bonacich: Power and Centrality: A Family of Measures. American Journal of Sociology 92(5):1170–1182, 1986 http://www.leonidzhukov.net/hse/2014/socialnetworks/papers/Bonacich-Centrality.pdf

[R535]

(1, 2) Mark E. J. Newman: Networks: An Introduction. Oxford University Press, USA, 2010, pp. 169.

Examples

>>> G = nx.path_graph(4)
>>> centrality = nx.eigenvector_centrality_numpy(G)
>>> print(['%s %0.2f'%(node,centrality[node]) for node in centrality])
['0 0.37', '1 0.60', '2 0.60', '3 0.37']