Source code for networkx.algorithms.centrality.second_order

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import networkx as nx
from networkx.utils import not_implemented_for

# Authors: Erwan Le Merrer (erwan.lemerrer@technicolor.com)

__all__ = ["second_order_centrality"]


[docs] @not_implemented_for("directed") @nx._dispatch(edge_attrs="weight") def 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 ---------- G : graph A NetworkX connected and undirected graph. weight : string 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 ------- nodes : dictionary Dictionary keyed by node with second order centrality as the value. 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 -------- 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] 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. """ import numpy as np n = len(G) if n == 0: raise nx.NetworkXException("Empty graph.") if not nx.is_connected(G): raise nx.NetworkXException("Non connected graph.") if any(d.get(weight, 0) < 0 for u, v, d in G.edges(data=True)): raise nx.NetworkXException("Graph has negative edge weights.") # balancing G for Metropolis-Hastings random walks G = nx.DiGraph(G) in_deg = dict(G.in_degree(weight=weight)) d_max = max(in_deg.values()) for i, deg in in_deg.items(): if deg < d_max: G.add_edge(i, i, weight=d_max - deg) P = nx.to_numpy_array(G) P /= P.sum(axis=1)[:, np.newaxis] # to transition probability matrix def _Qj(P, j): P = P.copy() P[:, j] = 0 return P M = np.empty([n, n]) for i in range(n): M[:, i] = np.linalg.solve( np.identity(n) - _Qj(P, i), np.ones([n, 1])[:, 0] ) # eq 3 return dict( zip(G.nodes, [np.sqrt(2 * np.sum(M[:, i]) - n * (n + 1)) for i in range(n)]) ) # eq 6