second_order_centrality

second_order_centrality(G, weight='weight')

Compute the second order centrality for nodes of G.

The second order centrality of a given node is the standard deviation of the return times to that node of a perpetual random walk on G:

Parameters:

Ggraph

A NetworkX connected and undirected graph.

weightstring or None, optional (default=”weight”)

The name of an edge attribute that holds the numerical value used as a weight. If None then each edge has weight 1.

Returns:

nodesdictionary

Dictionary keyed by node with second order centrality as the value.

Raises:

NetworkXException

If the graph G is empty, non connected or has negative weights.

See also

betweenness_centrality

Notes

Lower values of second order centrality indicate higher centrality.

The algorithm is from Kermarrec, Le Merrer, Sericola and Trédan [1].

This code implements the analytical version of the algorithm, i.e., there is no simulation of a random walk process involved. The random walk is here unbiased (corresponding to eq 6 of the paper [1]), thus the centrality values are the standard deviations for random walk return times on the transformed input graph G (equal in-degree at each nodes by adding self-loops).

Complexity of this implementation, made to run locally on a single machine, is O(n^3), with n the size of G, which makes it viable only for small graphs.

References

[1] (1,2)

Anne-Marie Kermarrec, Erwan Le Merrer, Bruno Sericola, Gilles Trédan “Second order centrality: Distributed assessment of nodes criticity in complex networks”, Elsevier Computer Communications 34(5):619-628, 2011.

Examples

>>> G = nx.star_graph(10)
>>> soc = nx.second_order_centrality(G)
>>> print(sorted(soc.items(), key=lambda x: x[1])[0][0])  # pick first id
0