eigenvector_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 \(A\) ([2]).

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 of nodes with eigenvector centrality as the value.
Return type:	dictionary

Examples

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

See also

eigenvector_centrality_numpy(), pagerank(), hits()

Notes

The measure was introduced by [1].

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.

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

[1]	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

[2]	Mark E. J. Newman: Networks: An Introduction. Oxford University Press, USA, 2010, pp. 169.