eigenvector_centrality_numpy#
- eigenvector_centrality_numpy(G, weight=None, 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
\[Ax = \lambda x\]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:
- Ggraph
A networkx graph
- weightNone or string, optional (default=None)
The name of the edge attribute used as weight. If None, all edge weights are considered equal. In this measure the weight is interpreted as the connection strength.
- max_iterinteger, optional (default=100)
Maximum number of iterations in power method.
- tolfloat, optional (default=1.0e-6)
Relative accuracy for eigenvalues (stopping criterion). The default value of 0 implies machine precision.
- Returns:
- nodesdictionary
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 [1].
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
[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.
Examples
>>> G = nx.path_graph(4) >>> centrality = nx.eigenvector_centrality_numpy(G) >>> print([f"{node} {centrality[node]:0.2f}" for node in centrality]) ['0 0.37', '1 0.60', '2 0.60', '3 0.37']