# Source code for networkx.algorithms.centrality.second_order

```
"""Copyright (c) 2015 – Thomson Licensing, SAS
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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._dispatchable(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,
(float(np.sqrt(2 * np.sum(M[:, i]) - n * (n + 1))) for i in range(n)),
)
) # eq 6
```