Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/centrality/second_order.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """Copyright (c) 2015 – Thomson Licensing, SAS |
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2 |
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3 Redistribution and use in source and binary forms, with or without |
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4 modification, are permitted (subject to the limitations in the |
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5 disclaimer below) provided that the following conditions are met: |
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6 |
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7 * Redistributions of source code must retain the above copyright |
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8 notice, this list of conditions and the following disclaimer. |
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9 |
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10 * Redistributions in binary form must reproduce the above copyright |
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11 notice, this list of conditions and the following disclaimer in the |
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12 documentation and/or other materials provided with the distribution. |
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13 |
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14 * Neither the name of Thomson Licensing, or Technicolor, nor the names |
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15 of its contributors may be used to endorse or promote products derived |
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16 from this software without specific prior written permission. |
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17 |
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18 NO EXPRESS OR IMPLIED LICENSES TO ANY PARTY'S PATENT RIGHTS ARE |
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19 GRANTED BY THIS LICENSE. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT |
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20 HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED |
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21 WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF |
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22 MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE |
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23 DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE |
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24 LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR |
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25 CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF |
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26 SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR |
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27 BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, |
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28 WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE |
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29 OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN |
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30 IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. |
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31 """ |
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32 |
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33 import networkx as nx |
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34 from networkx.utils import not_implemented_for |
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35 |
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36 # Authors: Erwan Le Merrer (erwan.lemerrer@technicolor.com) |
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37 """ Second order centrality measure.""" |
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38 |
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39 __all__ = ["second_order_centrality"] |
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40 |
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41 |
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42 @not_implemented_for("directed") |
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43 def second_order_centrality(G): |
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44 """Compute the second order centrality for nodes of G. |
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45 |
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46 The second order centrality of a given node is the standard deviation of |
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47 the return times to that node of a perpetual random walk on G: |
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48 |
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49 Parameters |
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50 ---------- |
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51 G : graph |
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52 A NetworkX connected and undirected graph. |
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53 |
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54 Returns |
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55 ------- |
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56 nodes : dictionary |
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57 Dictionary keyed by node with second order centrality as the value. |
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58 |
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59 Examples |
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60 -------- |
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61 >>> G = nx.star_graph(10) |
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62 >>> soc = nx.second_order_centrality(G) |
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63 >>> print(sorted(soc.items(), key=lambda x: x[1])[0][0]) # pick first id |
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64 0 |
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65 |
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66 Raises |
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67 ------ |
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68 NetworkXException |
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69 If the graph G is empty, non connected or has negative weights. |
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70 |
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71 See Also |
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72 -------- |
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73 betweenness_centrality |
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74 |
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75 Notes |
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76 ----- |
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77 Lower values of second order centrality indicate higher centrality. |
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78 |
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79 The algorithm is from Kermarrec, Le Merrer, Sericola and Trédan [1]_. |
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80 |
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81 This code implements the analytical version of the algorithm, i.e., |
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82 there is no simulation of a random walk process involved. The random walk |
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83 is here unbiased (corresponding to eq 6 of the paper [1]_), thus the |
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84 centrality values are the standard deviations for random walk return times |
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85 on the transformed input graph G (equal in-degree at each nodes by adding |
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86 self-loops). |
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87 |
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88 Complexity of this implementation, made to run locally on a single machine, |
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89 is O(n^3), with n the size of G, which makes it viable only for small |
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90 graphs. |
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91 |
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92 References |
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93 ---------- |
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94 .. [1] Anne-Marie Kermarrec, Erwan Le Merrer, Bruno Sericola, Gilles Trédan |
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95 "Second order centrality: Distributed assessment of nodes criticity in |
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96 complex networks", Elsevier Computer Communications 34(5):619-628, 2011. |
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97 """ |
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98 |
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99 try: |
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100 import numpy as np |
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101 except ImportError as e: |
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102 raise ImportError("Requires NumPy: http://numpy.org/") from e |
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103 |
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104 n = len(G) |
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105 |
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106 if n == 0: |
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107 raise nx.NetworkXException("Empty graph.") |
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108 if not nx.is_connected(G): |
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109 raise nx.NetworkXException("Non connected graph.") |
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110 if any(d.get("weight", 0) < 0 for u, v, d in G.edges(data=True)): |
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111 raise nx.NetworkXException("Graph has negative edge weights.") |
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112 |
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113 # balancing G for Metropolis-Hastings random walks |
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114 G = nx.DiGraph(G) |
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115 in_deg = dict(G.in_degree(weight="weight")) |
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116 d_max = max(in_deg.values()) |
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117 for i, deg in in_deg.items(): |
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118 if deg < d_max: |
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119 G.add_edge(i, i, weight=d_max - deg) |
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120 |
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121 P = nx.to_numpy_array(G) |
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122 P /= P.sum(axis=1)[:, np.newaxis] # to transition probability matrix |
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123 |
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124 def _Qj(P, j): |
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125 P = P.copy() |
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126 P[:, j] = 0 |
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127 return P |
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128 |
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129 M = np.empty([n, n]) |
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130 |
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131 for i in range(n): |
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132 M[:, i] = np.linalg.solve( |
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133 np.identity(n) - _Qj(P, i), np.ones([n, 1])[:, 0] |
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134 ) # eq 3 |
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135 |
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136 return dict( |
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137 zip(G.nodes, [np.sqrt(2 * np.sum(M[:, i]) - n * (n + 1)) for i in range(n)]) |
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138 ) # eq 6 |