katz_centrality_numpy

katz_centrality_numpy(G, alpha=0.1, beta=1.0, normalized=True, weight=None)

Compute the Katz centrality for the graph G.

Katz centrality computes the centrality for a node based on the centrality of its neighbors. It is a generalization of the eigenvector centrality. The Katz centrality for node \(i\) is

\[x_i = \alpha \sum_{j} A_{ij} x_j + \beta,\]

where \(A\) is the adjacency matrix of graph G with eigenvalues \(\lambda\).

The parameter \(\beta\) controls the initial centrality and

\[\alpha < \frac{1}{\lambda_{\max}}.\]

Katz centrality computes the relative influence of a node within a network by measuring the number of the immediate neighbors (first degree nodes) and also all other nodes in the network that connect to the node under consideration through these immediate neighbors.

Extra weight can be provided to immediate neighbors through the parameter \(\beta\). Connections made with distant neighbors are, however, penalized by an attenuation factor \(\alpha\) which should be strictly less than the inverse largest eigenvalue of the adjacency matrix in order for the Katz centrality to be computed correctly. More information is provided in [1].

Parameters:

Ggraph

A NetworkX graph

alphafloat

Attenuation factor

betascalar or dictionary, optional (default=1.0)

Weight attributed to the immediate neighborhood. If not a scalar the dictionary must have an value for every node.

normalizedbool

If True normalize the resulting values.

weightNone or string, optional

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 Katz centrality as the value.

Raises:

NetworkXError

If the parameter beta is not a scalar but lacks a value for at least one node

See also

katz_centrality

eigenvector_centrality_numpy

eigenvector_centrality

pagerank()

hits()

Notes

Katz centrality was introduced by [2].

This algorithm uses a direct linear solver to solve the above equation. The parameter alpha should be strictly less than the inverse of largest eigenvalue of the adjacency matrix for there to be a solution. You can use max(nx.adjacency_spectrum(G)) to get \(\lambda_{\max}\) the largest eigenvalue of the adjacency matrix.

For strongly connected graphs, as \(\alpha \to 1/\lambda_{\max}\), and \(\beta > 0\), Katz centrality approaches the results for eigenvector centrality.

For directed graphs this finds “left” eigenvectors which corresponds to the in-edges in the graph. For out-edges Katz centrality, first reverse the graph with G.reverse().

References

[1]

Mark E. J. Newman: Networks: An Introduction. Oxford University Press, USA, 2010, p. 173.

[2]

Leo Katz: A New Status Index Derived from Sociometric Index. Psychometrika 18(1):39–43, 1953 https://link.springer.com/content/pdf/10.1007/BF02289026.pdf

Examples

>>> import math
>>> G = nx.path_graph(4)
>>> phi = (1 + math.sqrt(5)) / 2.0  # largest eigenvalue of adj matrix
>>> centrality = nx.katz_centrality_numpy(G, 1 / phi)
>>> for n, c in sorted(centrality.items()):
...     print(f"{n} {c:.2f}")
0 0.37
1 0.60
2 0.60
3 0.37