networkx.algorithms.centrality.second_order_centrality¶

second_order_centrality
(G)[source]¶ 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: G (graph) – A NetworkX connected and undirected graph. Returns: nodes – Dictionary keyed by node with second order centrality as the value. Return type: dictionary 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
Raises: NetworkXException
– If the graph G is empty, non connected or has negative weights.See also
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 indegree at each nodes by adding selfloops).
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) AnneMarie Kermarrec, Erwan Le Merrer, Bruno Sericola, Gilles Trédan “Second order centrality: Distributed assessment of nodes criticity in complex networks”, Elsevier Computer Communications 34(5):619628, 2011.