text: # -*- coding: utf-8 -*-
  '''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)
  ''' Second order centrality measure.'''
  
  __all__ = ['second_order_centrality']
  
  
  @not_implemented_for('directed')
  def second_order_centrality(G):
      """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
         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.
      """
  
      try:
          import numpy as np
      except ImportError:
          raise ImportError('Requires NumPy: http://scipy.org/')
  
      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_matrix(G)
      P = P / P.sum(axis=1)  # 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
  
  
  # fixture for pytest
  def setup_module(module):
      import pytest
      numpy = pytest.importorskip('numpy')
      scipy = pytest.importorskip('scipy')
  