Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/centrality/subgraph_alg.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 """ |
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2 Subraph centrality and communicability betweenness. |
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3 """ |
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4 import networkx as nx |
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5 from networkx.utils import not_implemented_for |
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6 |
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7 __all__ = [ |
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8 "subgraph_centrality_exp", |
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9 "subgraph_centrality", |
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10 "communicability_betweenness_centrality", |
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11 "estrada_index", |
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12 ] |
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13 |
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14 |
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15 @not_implemented_for("directed") |
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16 @not_implemented_for("multigraph") |
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17 def subgraph_centrality_exp(G): |
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18 r"""Returns the subgraph centrality for each node of G. |
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19 |
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20 Subgraph centrality of a node `n` is the sum of weighted closed |
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21 walks of all lengths starting and ending at node `n`. The weights |
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22 decrease with path length. Each closed walk is associated with a |
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23 connected subgraph ([1]_). |
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24 |
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25 Parameters |
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26 ---------- |
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27 G: graph |
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28 |
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29 Returns |
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30 ------- |
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31 nodes:dictionary |
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32 Dictionary of nodes with subgraph centrality as the value. |
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33 |
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34 Raises |
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35 ------ |
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36 NetworkXError |
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37 If the graph is not undirected and simple. |
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38 |
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39 See Also |
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40 -------- |
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41 subgraph_centrality: |
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42 Alternative algorithm of the subgraph centrality for each node of G. |
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43 |
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44 Notes |
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45 ----- |
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46 This version of the algorithm exponentiates the adjacency matrix. |
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47 |
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48 The subgraph centrality of a node `u` in G can be found using |
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49 the matrix exponential of the adjacency matrix of G [1]_, |
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50 |
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51 .. math:: |
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52 |
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53 SC(u)=(e^A)_{uu} . |
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54 |
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55 References |
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56 ---------- |
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57 .. [1] Ernesto Estrada, Juan A. Rodriguez-Velazquez, |
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58 "Subgraph centrality in complex networks", |
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59 Physical Review E 71, 056103 (2005). |
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60 https://arxiv.org/abs/cond-mat/0504730 |
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61 |
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62 Examples |
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63 -------- |
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64 (Example from [1]_) |
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65 >>> G = nx.Graph( |
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66 ... [ |
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67 ... (1, 2), |
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68 ... (1, 5), |
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69 ... (1, 8), |
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70 ... (2, 3), |
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71 ... (2, 8), |
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72 ... (3, 4), |
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73 ... (3, 6), |
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74 ... (4, 5), |
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75 ... (4, 7), |
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76 ... (5, 6), |
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77 ... (6, 7), |
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78 ... (7, 8), |
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79 ... ] |
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80 ... ) |
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81 >>> sc = nx.subgraph_centrality_exp(G) |
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82 >>> print([f"{node} {sc[node]:0.2f}" for node in sorted(sc)]) |
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83 ['1 3.90', '2 3.90', '3 3.64', '4 3.71', '5 3.64', '6 3.71', '7 3.64', '8 3.90'] |
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84 """ |
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85 # alternative implementation that calculates the matrix exponential |
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86 import scipy.linalg |
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87 |
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88 nodelist = list(G) # ordering of nodes in matrix |
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89 A = nx.to_numpy_array(G, nodelist) |
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90 # convert to 0-1 matrix |
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91 A[A != 0.0] = 1 |
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92 expA = scipy.linalg.expm(A) |
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93 # convert diagonal to dictionary keyed by node |
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94 sc = dict(zip(nodelist, map(float, expA.diagonal()))) |
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95 return sc |
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96 |
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97 |
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98 @not_implemented_for("directed") |
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99 @not_implemented_for("multigraph") |
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100 def subgraph_centrality(G): |
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101 r"""Returns subgraph centrality for each node in G. |
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102 |
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103 Subgraph centrality of a node `n` is the sum of weighted closed |
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104 walks of all lengths starting and ending at node `n`. The weights |
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105 decrease with path length. Each closed walk is associated with a |
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106 connected subgraph ([1]_). |
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107 |
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108 Parameters |
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109 ---------- |
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110 G: graph |
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111 |
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112 Returns |
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113 ------- |
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114 nodes : dictionary |
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115 Dictionary of nodes with subgraph centrality as the value. |
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116 |
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117 Raises |
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118 ------ |
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119 NetworkXError |
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120 If the graph is not undirected and simple. |
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121 |
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122 See Also |
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123 -------- |
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124 subgraph_centrality_exp: |
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125 Alternative algorithm of the subgraph centrality for each node of G. |
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126 |
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127 Notes |
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128 ----- |
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129 This version of the algorithm computes eigenvalues and eigenvectors |
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130 of the adjacency matrix. |
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131 |
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132 Subgraph centrality of a node `u` in G can be found using |
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133 a spectral decomposition of the adjacency matrix [1]_, |
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134 |
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135 .. math:: |
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136 |
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137 SC(u)=\sum_{j=1}^{N}(v_{j}^{u})^2 e^{\lambda_{j}}, |
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138 |
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139 where `v_j` is an eigenvector of the adjacency matrix `A` of G |
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140 corresponding corresponding to the eigenvalue `\lambda_j`. |
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141 |
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142 Examples |
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143 -------- |
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144 (Example from [1]_) |
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145 >>> G = nx.Graph( |
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146 ... [ |
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147 ... (1, 2), |
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148 ... (1, 5), |
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149 ... (1, 8), |
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150 ... (2, 3), |
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151 ... (2, 8), |
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152 ... (3, 4), |
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153 ... (3, 6), |
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154 ... (4, 5), |
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155 ... (4, 7), |
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156 ... (5, 6), |
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157 ... (6, 7), |
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158 ... (7, 8), |
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159 ... ] |
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160 ... ) |
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161 >>> sc = nx.subgraph_centrality(G) |
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162 >>> print([f"{node} {sc[node]:0.2f}" for node in sorted(sc)]) |
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163 ['1 3.90', '2 3.90', '3 3.64', '4 3.71', '5 3.64', '6 3.71', '7 3.64', '8 3.90'] |
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164 |
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165 References |
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166 ---------- |
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167 .. [1] Ernesto Estrada, Juan A. Rodriguez-Velazquez, |
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168 "Subgraph centrality in complex networks", |
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169 Physical Review E 71, 056103 (2005). |
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170 https://arxiv.org/abs/cond-mat/0504730 |
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171 |
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172 """ |
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173 import numpy as np |
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174 import numpy.linalg |
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175 |
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176 nodelist = list(G) # ordering of nodes in matrix |
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177 A = nx.to_numpy_array(G, nodelist) |
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178 # convert to 0-1 matrix |
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179 A[np.nonzero(A)] = 1 |
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180 w, v = numpy.linalg.eigh(A) |
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181 vsquare = np.array(v) ** 2 |
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182 expw = np.exp(w) |
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183 xg = np.dot(vsquare, expw) |
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184 # convert vector dictionary keyed by node |
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185 sc = dict(zip(nodelist, map(float, xg))) |
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186 return sc |
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187 |
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188 |
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189 @not_implemented_for("directed") |
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190 @not_implemented_for("multigraph") |
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191 def communicability_betweenness_centrality(G, normalized=True): |
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192 r"""Returns subgraph communicability for all pairs of nodes in G. |
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193 |
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194 Communicability betweenness measure makes use of the number of walks |
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195 connecting every pair of nodes as the basis of a betweenness centrality |
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196 measure. |
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197 |
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198 Parameters |
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199 ---------- |
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200 G: graph |
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201 |
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202 Returns |
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203 ------- |
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204 nodes : dictionary |
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205 Dictionary of nodes with communicability betweenness as the value. |
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206 |
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207 Raises |
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208 ------ |
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209 NetworkXError |
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210 If the graph is not undirected and simple. |
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211 |
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212 Notes |
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213 ----- |
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214 Let `G=(V,E)` be a simple undirected graph with `n` nodes and `m` edges, |
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215 and `A` denote the adjacency matrix of `G`. |
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216 |
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217 Let `G(r)=(V,E(r))` be the graph resulting from |
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218 removing all edges connected to node `r` but not the node itself. |
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219 |
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220 The adjacency matrix for `G(r)` is `A+E(r)`, where `E(r)` has nonzeros |
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221 only in row and column `r`. |
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222 |
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223 The subraph betweenness of a node `r` is [1]_ |
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224 |
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225 .. math:: |
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226 |
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227 \omega_{r} = \frac{1}{C}\sum_{p}\sum_{q}\frac{G_{prq}}{G_{pq}}, |
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228 p\neq q, q\neq r, |
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229 |
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230 where |
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231 `G_{prq}=(e^{A}_{pq} - (e^{A+E(r)})_{pq}` is the number of walks |
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232 involving node r, |
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233 `G_{pq}=(e^{A})_{pq}` is the number of closed walks starting |
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234 at node `p` and ending at node `q`, |
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235 and `C=(n-1)^{2}-(n-1)` is a normalization factor equal to the |
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236 number of terms in the sum. |
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237 |
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238 The resulting `\omega_{r}` takes values between zero and one. |
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239 The lower bound cannot be attained for a connected |
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240 graph, and the upper bound is attained in the star graph. |
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241 |
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242 References |
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243 ---------- |
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244 .. [1] Ernesto Estrada, Desmond J. Higham, Naomichi Hatano, |
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245 "Communicability Betweenness in Complex Networks" |
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246 Physica A 388 (2009) 764-774. |
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247 https://arxiv.org/abs/0905.4102 |
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248 |
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249 Examples |
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250 -------- |
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251 >>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)]) |
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252 >>> cbc = nx.communicability_betweenness_centrality(G) |
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253 >>> print([f"{node} {cbc[node]:0.2f}" for node in sorted(cbc)]) |
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254 ['0 0.03', '1 0.45', '2 0.51', '3 0.45', '4 0.40', '5 0.19', '6 0.03'] |
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255 """ |
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256 import numpy as np |
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257 import scipy.linalg |
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258 |
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259 nodelist = list(G) # ordering of nodes in matrix |
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260 n = len(nodelist) |
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261 A = nx.to_numpy_array(G, nodelist) |
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262 # convert to 0-1 matrix |
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263 A[np.nonzero(A)] = 1 |
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264 expA = scipy.linalg.expm(A) |
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265 mapping = dict(zip(nodelist, range(n))) |
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266 cbc = {} |
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267 for v in G: |
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268 # remove row and col of node v |
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269 i = mapping[v] |
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270 row = A[i, :].copy() |
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271 col = A[:, i].copy() |
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272 A[i, :] = 0 |
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273 A[:, i] = 0 |
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274 B = (expA - scipy.linalg.expm(A)) / expA |
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275 # sum with row/col of node v and diag set to zero |
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276 B[i, :] = 0 |
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277 B[:, i] = 0 |
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278 B -= np.diag(np.diag(B)) |
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279 cbc[v] = float(B.sum()) |
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280 # put row and col back |
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281 A[i, :] = row |
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282 A[:, i] = col |
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283 # rescaling |
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284 cbc = _rescale(cbc, normalized=normalized) |
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285 return cbc |
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286 |
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287 |
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288 def _rescale(cbc, normalized): |
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289 # helper to rescale betweenness centrality |
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290 if normalized is True: |
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291 order = len(cbc) |
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292 if order <= 2: |
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293 scale = None |
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294 else: |
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295 scale = 1.0 / ((order - 1.0) ** 2 - (order - 1.0)) |
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296 if scale is not None: |
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297 for v in cbc: |
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298 cbc[v] *= scale |
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299 return cbc |
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300 |
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301 |
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302 def estrada_index(G): |
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303 r"""Returns the Estrada index of a the graph G. |
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304 |
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305 The Estrada Index is a topological index of folding or 3D "compactness" ([1]_). |
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306 |
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307 Parameters |
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308 ---------- |
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309 G: graph |
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310 |
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311 Returns |
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312 ------- |
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313 estrada index: float |
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314 |
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315 Raises |
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316 ------ |
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317 NetworkXError |
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318 If the graph is not undirected and simple. |
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319 |
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320 Notes |
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321 ----- |
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322 Let `G=(V,E)` be a simple undirected graph with `n` nodes and let |
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323 `\lambda_{1}\leq\lambda_{2}\leq\cdots\lambda_{n}` |
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324 be a non-increasing ordering of the eigenvalues of its adjacency |
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325 matrix `A`. The Estrada index is ([1]_, [2]_) |
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326 |
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327 .. math:: |
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328 EE(G)=\sum_{j=1}^n e^{\lambda _j}. |
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329 |
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330 References |
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331 ---------- |
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332 .. [1] E. Estrada, "Characterization of 3D molecular structure", |
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333 Chem. Phys. Lett. 319, 713 (2000). |
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334 https://doi.org/10.1016/S0009-2614(00)00158-5 |
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335 .. [2] José Antonio de la Peñaa, Ivan Gutman, Juan Rada, |
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336 "Estimating the Estrada index", |
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337 Linear Algebra and its Applications. 427, 1 (2007). |
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338 https://doi.org/10.1016/j.laa.2007.06.020 |
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339 |
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340 Examples |
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341 -------- |
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342 >>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)]) |
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343 >>> ei = nx.estrada_index(G) |
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344 >>> print(f"{ei:0.5}") |
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345 20.55 |
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346 """ |
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347 return sum(subgraph_centrality(G).values()) |