eigenvector_centrality#

eigenvector_centrality(G, max_iter=100, tol=1e-06, nstart=None, weight=None)[source]#

Compute the eigenvector centrality for the graph G.

Eigenvector centrality computes the centrality for a node by adding the centrality of its predecessors. The centrality for node $$i$$ is the $$i$$-th element of a left eigenvector associated with the eigenvalue $$\lambda$$ of maximum modulus that is positive. Such an eigenvector $$x$$ is defined up to a multiplicative constant by the equation

$\lambda x^T = x^T A,$

where $$A$$ is the adjacency matrix of the graph G. By definition of row-column product, the equation above is equivalent to

$\lambda x_i = \sum_{j\to i}x_j.$

That is, adding the eigenvector centralities of the predecessors of $$i$$ one obtains the eigenvector centrality of $$i$$ multiplied by $$\lambda$$. In the case of undirected graphs, $$x$$ also solves the familiar right-eigenvector equation $$Ax = \lambda x$$.

By virtue of the Perron–Frobenius theorem [1], if G is strongly connected there is a unique eigenvector $$x$$, and all its entries are strictly positive.

If G is not strongly connected there might be several left eigenvectors associated with $$\lambda$$, and some of their elements might be zero.

Parameters:
Ggraph

A networkx graph.

max_iterinteger, optional (default=100)

Maximum number of power iterations.

tolfloat, optional (default=1.0e-6)

Error tolerance (in Euclidean norm) used to check convergence in power iteration.

nstartdictionary, optional (default=None)

Starting value of power iteration for each node. Must have a nonzero projection on the desired eigenvector for the power method to converge. If None, this implementation uses an all-ones vector, which is a safe choice.

weightNone or string, optional (default=None)

If None, all edge weights are considered equal. Otherwise holds the name of the edge attribute used as weight. In this measure the weight is interpreted as the connection strength.

Returns:
nodesdictionary

Dictionary of nodes with eigenvector centrality as the value. The associated vector has unit Euclidean norm and the values are nonegative.

Raises:
NetworkXPointlessConcept

If the graph G is the null graph.

NetworkXError

If each value in nstart is zero.

PowerIterationFailedConvergence

If the algorithm fails to converge to the specified tolerance within the specified number of iterations of the power iteration method.

Notes

Eigenvector centrality was introduced by Landau [2] for chess tournaments. It was later rediscovered by Wei [3] and then popularized by Kendall [4] in the context of sport ranking. Berge introduced a general definition for graphs based on social connections [5]. Bonacich [6] reintroduced again eigenvector centrality and made it popular in link analysis.

This function computes the left dominant eigenvector, which corresponds to adding the centrality of predecessors: this is the usual approach. To add the centrality of successors first reverse the graph with G.reverse().

The implementation uses power iteration [7] to compute a dominant eigenvector starting from the provided vector nstart. Convergence is guaranteed as long as nstart has a nonzero projection on a dominant eigenvector, which certainly happens using the default value.

The method stops when the change in the computed vector between two iterations is smaller than an error tolerance of G.number_of_nodes() * tol or after max_iter iterations, but in the second case it raises an exception.

This implementation uses $$(A + I)$$ rather than the adjacency matrix $$A$$ because the change preserves eigenvectors, but it shifts the spectrum, thus guaranteeing convergence even for networks with negative eigenvalues of maximum modulus.

References

[1]

Abraham Berman and Robert J. Plemmons. “Nonnegative Matrices in the Mathematical Sciences.” Classics in Applied Mathematics. SIAM, 1994.

[2]

Edmund Landau. “Zur relativen Wertbemessung der Turnierresultate.” Deutsches Wochenschach, 11:366–369, 1895.

[3]

Teh-Hsing Wei. “The Algebraic Foundations of Ranking Theory.” PhD thesis, University of Cambridge, 1952.

[4]

Maurice G. Kendall. “Further contributions to the theory of paired comparisons.” Biometrics, 11(1):43–62, 1955. https://www.jstor.org/stable/3001479

[5]

Claude Berge “Théorie des graphes et ses applications.” Dunod, Paris, France, 1958.

[6]

Phillip Bonacich. “Technique for analyzing overlapping memberships.” Sociological Methodology, 4:176–185, 1972. https://www.jstor.org/stable/270732

Examples

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


nstart parameter is not used, but it is checked for validity.