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:


G (graph) – A NetworkX connected and undirected graph.


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

Return type



>>> 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

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


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.



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.