Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/communicability_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 Communicability. |
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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__ = ["communicability", "communicability_exp"] |
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8 |
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9 |
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10 @not_implemented_for("directed") |
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11 @not_implemented_for("multigraph") |
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12 def communicability(G): |
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13 r"""Returns communicability between all pairs of nodes in G. |
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14 |
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15 The communicability between pairs of nodes in G is the sum of |
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16 walks of different lengths starting at node u and ending at node v. |
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17 |
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18 Parameters |
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19 ---------- |
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20 G: graph |
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21 |
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22 Returns |
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23 ------- |
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24 comm: dictionary of dictionaries |
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25 Dictionary of dictionaries keyed by nodes with communicability |
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26 as the value. |
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27 |
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28 Raises |
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29 ------ |
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30 NetworkXError |
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31 If the graph is not undirected and simple. |
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32 |
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33 See Also |
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34 -------- |
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35 communicability_exp: |
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36 Communicability between all pairs of nodes in G using spectral |
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37 decomposition. |
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38 communicability_betweenness_centrality: |
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39 Communicability betweeness centrality for each node in G. |
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40 |
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41 Notes |
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42 ----- |
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43 This algorithm uses a spectral decomposition of the adjacency matrix. |
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44 Let G=(V,E) be a simple undirected graph. Using the connection between |
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45 the powers of the adjacency matrix and the number of walks in the graph, |
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46 the communicability between nodes `u` and `v` based on the graph spectrum |
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47 is [1]_ |
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48 |
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49 .. math:: |
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50 C(u,v)=\sum_{j=1}^{n}\phi_{j}(u)\phi_{j}(v)e^{\lambda_{j}}, |
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51 |
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52 where `\phi_{j}(u)` is the `u\rm{th}` element of the `j\rm{th}` orthonormal |
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53 eigenvector of the adjacency matrix associated with the eigenvalue |
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54 `\lambda_{j}`. |
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55 |
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56 References |
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57 ---------- |
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58 .. [1] Ernesto Estrada, Naomichi Hatano, |
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59 "Communicability in complex networks", |
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60 Phys. Rev. E 77, 036111 (2008). |
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61 https://arxiv.org/abs/0707.0756 |
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62 |
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63 Examples |
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64 -------- |
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65 >>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)]) |
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66 >>> c = nx.communicability(G) |
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67 """ |
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68 import numpy |
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69 |
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70 nodelist = list(G) # ordering of nodes in matrix |
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71 A = nx.to_numpy_array(G, nodelist) |
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72 # convert to 0-1 matrix |
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73 A[A != 0.0] = 1 |
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74 w, vec = numpy.linalg.eigh(A) |
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75 expw = numpy.exp(w) |
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76 mapping = dict(zip(nodelist, range(len(nodelist)))) |
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77 c = {} |
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78 # computing communicabilities |
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79 for u in G: |
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80 c[u] = {} |
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81 for v in G: |
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82 s = 0 |
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83 p = mapping[u] |
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84 q = mapping[v] |
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85 for j in range(len(nodelist)): |
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86 s += vec[:, j][p] * vec[:, j][q] * expw[j] |
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87 c[u][v] = float(s) |
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88 return c |
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89 |
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90 |
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91 @not_implemented_for("directed") |
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92 @not_implemented_for("multigraph") |
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93 def communicability_exp(G): |
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94 r"""Returns communicability between all pairs of nodes in G. |
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95 |
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96 Communicability between pair of node (u,v) of node in G is the sum of |
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97 walks of different lengths starting at node u and ending at node v. |
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98 |
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99 Parameters |
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100 ---------- |
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101 G: graph |
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102 |
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103 Returns |
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104 ------- |
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105 comm: dictionary of dictionaries |
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106 Dictionary of dictionaries keyed by nodes with communicability |
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107 as the value. |
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108 |
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109 Raises |
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110 ------ |
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111 NetworkXError |
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112 If the graph is not undirected and simple. |
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113 |
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114 See Also |
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115 -------- |
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116 communicability: |
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117 Communicability between pairs of nodes in G. |
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118 communicability_betweenness_centrality: |
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119 Communicability betweeness centrality for each node in G. |
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120 |
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121 Notes |
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122 ----- |
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123 This algorithm uses matrix exponentiation of the adjacency matrix. |
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124 |
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125 Let G=(V,E) be a simple undirected graph. Using the connection between |
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126 the powers of the adjacency matrix and the number of walks in the graph, |
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127 the communicability between nodes u and v is [1]_, |
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128 |
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129 .. math:: |
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130 C(u,v) = (e^A)_{uv}, |
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131 |
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132 where `A` is the adjacency matrix of G. |
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133 |
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134 References |
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135 ---------- |
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136 .. [1] Ernesto Estrada, Naomichi Hatano, |
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137 "Communicability in complex networks", |
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138 Phys. Rev. E 77, 036111 (2008). |
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139 https://arxiv.org/abs/0707.0756 |
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140 |
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141 Examples |
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142 -------- |
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143 >>> G = nx.Graph([(0, 1), (1, 2), (1, 5), (5, 4), (2, 4), (2, 3), (4, 3), (3, 6)]) |
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144 >>> c = nx.communicability_exp(G) |
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145 """ |
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146 import scipy.linalg |
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147 |
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148 nodelist = list(G) # ordering of nodes in matrix |
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149 A = nx.to_numpy_array(G, nodelist) |
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150 # convert to 0-1 matrix |
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151 A[A != 0.0] = 1 |
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152 # communicability matrix |
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153 expA = scipy.linalg.expm(A) |
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154 mapping = dict(zip(nodelist, range(len(nodelist)))) |
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155 c = {} |
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156 for u in G: |
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157 c[u] = {} |
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158 for v in G: |
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159 c[u][v] = float(expA[mapping[u], mapping[v]]) |
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160 return c |