7.5.2.1. networkx.algorithms.centrality.eigenvector_centrality¶

networkx.algorithms.centrality.eigenvector_centrality(G, max_iter=100, tol=1e-06, nstart=None, weight='weight')[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 ([R533]).

Parameters:	G : graph A networkx graph max_iter : integer, optional Maximum number of iterations in power method. tol : float, optional Error tolerance used to check convergence in power method iteration. nstart : dictionary, optional Starting value of eigenvector iteration for each node. weight : None or string, optional If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight.
Returns:	nodes : dictionary Dictionary of nodes with eigenvector centrality as the value.
Raises:	NetworkXPointlessConcept If the graph G is the null graph. NetworkXError If each value in nstart is zero.

See also

eigenvector_centrality_numpy, pagerank, hits

Notes

The measure was introduced by [R532] and is discussed in [R533].

The power iteration method is used to compute the eigenvector and convergence is not guaranteed. Our method stops after max_iter iterations or when the change in the computed vector between two iterations is smaller than an error tolerance of G.number_of_nodes() * tol. This implementation uses (A + I) rather than the adjacency matrix A because it shifts the spectrum to enable discerning the correct eigenvector even for networks with multiple dominant eigenvalues.

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

[R532]

(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>

[R533]

(1, 2, 3) Mark E. J. Newman. Networks: An Introduction. Oxford University Press, USA, 2010, pp. 169.

Examples

>>> G = nx.path_graph(4)
>>> centrality = nx.eigenvector_centrality(G)
>>> sorted((v, '{:0.2f}'.format(c)) for v, c in centrality.items())
[(0, '0.37'), (1, '0.60'), (2, '0.60'), (3, '0.37')]