Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/linalg/modularitymatrix.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 """Modularity matrix of graphs. |
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2 """ |
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3 import networkx as nx |
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4 from networkx.utils import not_implemented_for |
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5 |
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6 __all__ = ["modularity_matrix", "directed_modularity_matrix"] |
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7 |
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8 |
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9 @not_implemented_for("directed") |
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10 @not_implemented_for("multigraph") |
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11 def modularity_matrix(G, nodelist=None, weight=None): |
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12 r"""Returns the modularity matrix of G. |
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13 |
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14 The modularity matrix is the matrix B = A - <A>, where A is the adjacency |
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15 matrix and <A> is the average adjacency matrix, assuming that the graph |
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16 is described by the configuration model. |
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17 |
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18 More specifically, the element B_ij of B is defined as |
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19 |
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20 .. math:: |
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21 A_{ij} - {k_i k_j \over 2 m} |
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22 |
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23 where k_i is the degree of node i, and where m is the number of edges |
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24 in the graph. When weight is set to a name of an attribute edge, Aij, k_i, |
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25 k_j and m are computed using its value. |
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26 |
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27 Parameters |
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28 ---------- |
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29 G : Graph |
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30 A NetworkX graph |
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31 |
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32 nodelist : list, optional |
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33 The rows and columns are ordered according to the nodes in nodelist. |
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34 If nodelist is None, then the ordering is produced by G.nodes(). |
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35 |
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36 weight : string or None, optional (default=None) |
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37 The edge attribute that holds the numerical value used for |
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38 the edge weight. If None then all edge weights are 1. |
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39 |
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40 Returns |
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41 ------- |
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42 B : Numpy matrix |
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43 The modularity matrix of G. |
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44 |
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45 Examples |
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46 -------- |
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47 >>> k = [3, 2, 2, 1, 0] |
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48 >>> G = nx.havel_hakimi_graph(k) |
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49 >>> B = nx.modularity_matrix(G) |
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50 |
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51 |
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52 See Also |
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53 -------- |
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54 to_numpy_array |
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55 modularity_spectrum |
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56 adjacency_matrix |
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57 directed_modularity_matrix |
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58 |
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59 References |
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60 ---------- |
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61 .. [1] M. E. J. Newman, "Modularity and community structure in networks", |
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62 Proc. Natl. Acad. Sci. USA, vol. 103, pp. 8577-8582, 2006. |
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63 """ |
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64 if nodelist is None: |
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65 nodelist = list(G) |
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66 A = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, weight=weight, format="csr") |
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67 k = A.sum(axis=1) |
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68 m = k.sum() * 0.5 |
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69 # Expected adjacency matrix |
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70 X = k * k.transpose() / (2 * m) |
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71 return A - X |
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72 |
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73 |
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74 @not_implemented_for("undirected") |
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75 @not_implemented_for("multigraph") |
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76 def directed_modularity_matrix(G, nodelist=None, weight=None): |
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77 """Returns the directed modularity matrix of G. |
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78 |
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79 The modularity matrix is the matrix B = A - <A>, where A is the adjacency |
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80 matrix and <A> is the expected adjacency matrix, assuming that the graph |
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81 is described by the configuration model. |
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82 |
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83 More specifically, the element B_ij of B is defined as |
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84 |
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85 .. math:: |
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86 B_{ij} = A_{ij} - k_i^{out} k_j^{in} / m |
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87 |
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88 where :math:`k_i^{in}` is the in degree of node i, and :math:`k_j^{out}` is the out degree |
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89 of node j, with m the number of edges in the graph. When weight is set |
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90 to a name of an attribute edge, Aij, k_i, k_j and m are computed using |
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91 its value. |
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92 |
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93 Parameters |
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94 ---------- |
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95 G : DiGraph |
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96 A NetworkX DiGraph |
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97 |
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98 nodelist : list, optional |
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99 The rows and columns are ordered according to the nodes in nodelist. |
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100 If nodelist is None, then the ordering is produced by G.nodes(). |
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101 |
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102 weight : string or None, optional (default=None) |
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103 The edge attribute that holds the numerical value used for |
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104 the edge weight. If None then all edge weights are 1. |
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105 |
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106 Returns |
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107 ------- |
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108 B : Numpy matrix |
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109 The modularity matrix of G. |
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110 |
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111 Examples |
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112 -------- |
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113 >>> G = nx.DiGraph() |
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114 >>> G.add_edges_from( |
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115 ... ( |
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116 ... (1, 2), |
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117 ... (1, 3), |
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118 ... (3, 1), |
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119 ... (3, 2), |
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120 ... (3, 5), |
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121 ... (4, 5), |
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122 ... (4, 6), |
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123 ... (5, 4), |
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124 ... (5, 6), |
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125 ... (6, 4), |
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126 ... ) |
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127 ... ) |
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128 >>> B = nx.directed_modularity_matrix(G) |
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129 |
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130 |
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131 Notes |
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132 ----- |
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133 NetworkX defines the element A_ij of the adjacency matrix as 1 if there |
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134 is a link going from node i to node j. Leicht and Newman use the opposite |
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135 definition. This explains the different expression for B_ij. |
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136 |
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137 See Also |
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138 -------- |
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139 to_numpy_array |
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140 modularity_spectrum |
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141 adjacency_matrix |
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142 modularity_matrix |
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143 |
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144 References |
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145 ---------- |
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146 .. [1] E. A. Leicht, M. E. J. Newman, |
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147 "Community structure in directed networks", |
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148 Phys. Rev Lett., vol. 100, no. 11, p. 118703, 2008. |
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149 """ |
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150 if nodelist is None: |
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151 nodelist = list(G) |
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152 A = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, weight=weight, format="csr") |
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153 k_in = A.sum(axis=0) |
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154 k_out = A.sum(axis=1) |
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155 m = k_in.sum() |
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156 # Expected adjacency matrix |
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157 X = k_out * k_in / m |
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158 return A - X |