eigenvector_centrality¶

eigenvector_centrality
(G, max_iter=100, tol=1e06, 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 ofnumber_of_nodes(G)*tol
has been reached.For directed graphs this is “left” eigenvector centrality which corresponds to the inedges in the graph. For outedges 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/BonacichCentrality.pdf [2] Mark E. J. Newman: Networks: An Introduction. Oxford University Press, USA, 2010, pp. 169.