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 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.

However, if G is not (strongly) connected, there might be several left eigenvectors associated with \(\lambda\), and some of their elements might be zero. Depending on the method used to choose eigenvectors, round-off error can affect which of the infinitely many eigenvectors is reported. This can lead to inconsistent results for the same graph, which the underlying implementation is not robust to. For this reason, only (strongly) connected graphs are accepted.

Parameters:

Ggraph: A connected NetworkX graph.
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.
max_iterinteger, optional (default=50): Maximum number of Arnoldi update iterations allowed.
tolfloat, optional (default=0): Relative accuracy for eigenvalues (stopping criterion). The default value of 0 implies machine precision.

Returns:

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

Raises:

NetworkXPointlessConcept: If the graph G is the null graph.
ArpackNoConvergence: When the requested convergence is not obtained. The currently converged eigenvalues and eigenvectors can be found as eigenvalues and eigenvectors attributes of the exception object.
AmbiguousSolution: If G is not connected.

See also

scipy.sparse.linalg.eigs()
eigenvector_centrality
pagerank()
hits()

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().

This implementation uses the SciPy sparse eigenvalue solver (ARPACK) to find the largest eigenvalue/eigenvector pair using Arnoldi iterations [7].

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

[7]

Arnoldi, W. E. (1951). “The principle of minimized iterations in the solution of the matrix eigenvalue problem”. Quarterly of Applied Mathematics. 9 (1): 17–29. https://doi.org/10.1090/qam/42792

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']