Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/generators/spectral_graph_forge.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 """Generates graphs with a given eigenvector structure""" |
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2 |
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3 |
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4 import networkx as nx |
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5 from networkx.utils import np_random_state |
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6 |
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7 __all__ = ["spectral_graph_forge"] |
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8 |
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9 |
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10 def _mat_spect_approx(A, level, sorteigs=True, reverse=False, absolute=True): |
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11 """ Returns the low-rank approximation of the given matrix A |
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12 |
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13 Parameters |
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14 ---------- |
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15 A : 2D numpy array |
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16 level : integer |
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17 It represents the fixed rank for the output approximation matrix |
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18 sorteigs : boolean |
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19 Whether eigenvectors should be sorted according to their associated |
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20 eigenvalues before removing the firsts of them |
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21 reverse : boolean |
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22 Whether eigenvectors list should be reversed before removing the firsts |
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23 of them |
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24 absolute : boolean |
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25 Whether eigenvectors should be sorted considering the absolute values |
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26 of the corresponding eigenvalues |
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27 |
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28 Returns |
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29 ------- |
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30 B : 2D numpy array |
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31 low-rank approximation of A |
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32 |
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33 Notes |
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34 ----- |
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35 Low-rank matrix approximation is about finding a fixed rank matrix close |
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36 enough to the input one with respect to a given norm (distance). |
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37 In the case of real symmetric input matrix and euclidean distance, the best |
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38 low-rank approximation is given by the sum of first eigenvector matrices. |
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39 |
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40 References |
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41 ---------- |
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42 .. [1] G. Eckart and G. Young, The approximation of one matrix by another |
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43 of lower rank |
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44 .. [2] L. Mirsky, Symmetric gauge functions and unitarily invariant norms |
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45 |
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46 """ |
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47 |
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48 import numpy as np |
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49 |
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50 d, V = np.linalg.eigh(A) |
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51 d = np.ravel(d) |
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52 n = len(d) |
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53 if sorteigs: |
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54 if absolute: |
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55 k = np.argsort(np.abs(d)) |
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56 else: |
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57 k = np.argsort(d) |
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58 # ordered from the lowest to the highest |
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59 else: |
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60 k = range(n) |
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61 if not reverse: |
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62 k = np.flipud(k) |
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63 |
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64 z = np.zeros(n) |
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65 for i in range(level, n): |
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66 V[:, k[i]] = z |
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67 |
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68 B = V @ np.diag(d) @ V.T |
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69 return B |
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70 |
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71 |
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72 @np_random_state(3) |
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73 def spectral_graph_forge(G, alpha, transformation="identity", seed=None): |
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74 """Returns a random simple graph with spectrum resembling that of `G` |
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75 |
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76 This algorithm, called Spectral Graph Forge (SGF), computes the |
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77 eigenvectors of a given graph adjacency matrix, filters them and |
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78 builds a random graph with a similar eigenstructure. |
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79 SGF has been proved to be particularly useful for synthesizing |
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80 realistic social networks and it can also be used to anonymize |
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81 graph sensitive data. |
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82 |
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83 Parameters |
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84 ---------- |
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85 G : Graph |
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86 alpha : float |
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87 Ratio representing the percentage of eigenvectors of G to consider, |
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88 values in [0,1]. |
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89 transformation : string, optional |
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90 Represents the intended matrix linear transformation, possible values |
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91 are 'identity' and 'modularity' |
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92 seed : integer, random_state, or None (default) |
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93 Indicator of numpy random number generation state. |
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94 See :ref:`Randomness<randomness>`. |
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95 |
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96 Returns |
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97 ------- |
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98 H : Graph |
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99 A graph with a similar eigenvector structure of the input one. |
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100 |
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101 Raises |
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102 ------ |
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103 NetworkXError |
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104 If transformation has a value different from 'identity' or 'modularity' |
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105 |
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106 Notes |
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107 ----- |
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108 Spectral Graph Forge (SGF) generates a random simple graph resembling the |
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109 global properties of the given one. |
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110 It leverages the low-rank approximation of the associated adjacency matrix |
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111 driven by the *alpha* precision parameter. |
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112 SGF preserves the number of nodes of the input graph and their ordering. |
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113 This way, nodes of output graphs resemble the properties of the input one |
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114 and attributes can be directly mapped. |
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115 |
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116 It considers the graph adjacency matrices which can optionally be |
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117 transformed to other symmetric real matrices (currently transformation |
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118 options include *identity* and *modularity*). |
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119 The *modularity* transformation, in the sense of Newman's modularity matrix |
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120 allows the focusing on community structure related properties of the graph. |
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121 |
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122 SGF applies a low-rank approximation whose fixed rank is computed from the |
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123 ratio *alpha* of the input graph adjacency matrix dimension. |
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124 This step performs a filtering on the input eigenvectors similar to the low |
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125 pass filtering common in telecommunications. |
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126 |
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127 The filtered values (after truncation) are used as input to a Bernoulli |
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128 sampling for constructing a random adjacency matrix. |
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129 |
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130 References |
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131 ---------- |
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132 .. [1] L. Baldesi, C. T. Butts, A. Markopoulou, "Spectral Graph Forge: |
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133 Graph Generation Targeting Modularity", IEEE Infocom, '18. |
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134 https://arxiv.org/abs/1801.01715 |
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135 .. [2] M. Newman, "Networks: an introduction", Oxford university press, |
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136 2010 |
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137 |
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138 Examples |
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139 -------- |
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140 >>> G = nx.karate_club_graph() |
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141 >>> H = nx.spectral_graph_forge(G, 0.3) |
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142 >>> |
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143 """ |
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144 |
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145 import numpy as np |
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146 import scipy.stats as stats |
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147 |
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148 available_transformations = ["identity", "modularity"] |
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149 alpha = np.clip(alpha, 0, 1) |
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150 A = nx.to_numpy_array(G) |
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151 n = A.shape[1] |
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152 level = int(round(n * alpha)) |
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153 |
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154 if transformation not in available_transformations: |
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155 msg = f"'{transformation}' is not a valid transformation. " |
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156 msg += f"Transformations: {available_transformations}" |
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157 raise nx.NetworkXError(msg) |
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158 |
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159 K = np.ones((1, n)) @ A |
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160 |
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161 B = A |
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162 if transformation == "modularity": |
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163 B -= K.T @ K / K.sum() |
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164 |
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165 B = _mat_spect_approx(B, level, sorteigs=True, absolute=True) |
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166 |
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167 if transformation == "modularity": |
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168 B += K.T @ K / K.sum() |
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169 |
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170 B = np.clip(B, 0, 1) |
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171 np.fill_diagonal(B, 0) |
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172 |
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173 for i in range(n - 1): |
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174 B[i, i + 1 :] = stats.bernoulli.rvs(B[i, i + 1 :], random_state=seed) |
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175 B[i + 1 :, i] = np.transpose(B[i, i + 1 :]) |
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176 |
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177 H = nx.from_numpy_array(B) |
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178 |
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179 return H |