Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/linalg/laplacianmatrix.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 """Laplacian 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__ = [ |
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7 "laplacian_matrix", |
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8 "normalized_laplacian_matrix", |
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9 "directed_laplacian_matrix", |
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10 "directed_combinatorial_laplacian_matrix", |
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11 ] |
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12 |
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13 |
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14 @not_implemented_for("directed") |
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15 def laplacian_matrix(G, nodelist=None, weight="weight"): |
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16 """Returns the Laplacian matrix of G. |
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17 |
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18 The graph Laplacian is the matrix L = D - A, where |
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19 A is the adjacency matrix and D is the diagonal matrix of node degrees. |
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20 |
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21 Parameters |
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22 ---------- |
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23 G : graph |
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24 A NetworkX graph |
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25 |
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26 nodelist : list, optional |
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27 The rows and columns are ordered according to the nodes in nodelist. |
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28 If nodelist is None, then the ordering is produced by G.nodes(). |
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29 |
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30 weight : string or None, optional (default='weight') |
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31 The edge data key used to compute each value in the matrix. |
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32 If None, then each edge has weight 1. |
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33 |
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34 Returns |
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35 ------- |
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36 L : SciPy sparse matrix |
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37 The Laplacian matrix of G. |
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38 |
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39 Notes |
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40 ----- |
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41 For MultiGraph/MultiDiGraph, the edges weights are summed. |
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42 |
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43 See Also |
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44 -------- |
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45 to_numpy_array |
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46 normalized_laplacian_matrix |
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47 laplacian_spectrum |
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48 """ |
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49 import scipy.sparse |
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50 |
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51 if nodelist is None: |
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52 nodelist = list(G) |
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53 A = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, weight=weight, format="csr") |
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54 n, m = A.shape |
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55 diags = A.sum(axis=1) |
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56 D = scipy.sparse.spdiags(diags.flatten(), [0], m, n, format="csr") |
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57 return D - A |
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58 |
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59 |
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60 @not_implemented_for("directed") |
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61 def normalized_laplacian_matrix(G, nodelist=None, weight="weight"): |
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62 r"""Returns the normalized Laplacian matrix of G. |
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63 |
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64 The normalized graph Laplacian is the matrix |
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65 |
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66 .. math:: |
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67 |
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68 N = D^{-1/2} L D^{-1/2} |
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69 |
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70 where `L` is the graph Laplacian and `D` is the diagonal matrix of |
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71 node degrees. |
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72 |
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73 Parameters |
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74 ---------- |
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75 G : graph |
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76 A NetworkX graph |
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77 |
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78 nodelist : list, optional |
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79 The rows and columns are ordered according to the nodes in nodelist. |
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80 If nodelist is None, then the ordering is produced by G.nodes(). |
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81 |
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82 weight : string or None, optional (default='weight') |
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83 The edge data key used to compute each value in the matrix. |
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84 If None, then each edge has weight 1. |
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85 |
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86 Returns |
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87 ------- |
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88 N : Scipy sparse matrix |
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89 The normalized Laplacian matrix of G. |
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90 |
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91 Notes |
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92 ----- |
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93 For MultiGraph/MultiDiGraph, the edges weights are summed. |
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94 See to_numpy_array for other options. |
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95 |
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96 If the Graph contains selfloops, D is defined as diag(sum(A,1)), where A is |
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97 the adjacency matrix [2]_. |
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98 |
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99 See Also |
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100 -------- |
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101 laplacian_matrix |
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102 normalized_laplacian_spectrum |
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103 |
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104 References |
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105 ---------- |
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106 .. [1] Fan Chung-Graham, Spectral Graph Theory, |
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107 CBMS Regional Conference Series in Mathematics, Number 92, 1997. |
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108 .. [2] Steve Butler, Interlacing For Weighted Graphs Using The Normalized |
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109 Laplacian, Electronic Journal of Linear Algebra, Volume 16, pp. 90-98, |
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110 March 2007. |
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111 """ |
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112 import numpy as np |
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113 import scipy |
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114 import scipy.sparse |
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115 |
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116 if nodelist is None: |
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117 nodelist = list(G) |
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118 A = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, weight=weight, format="csr") |
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119 n, m = A.shape |
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120 diags = A.sum(axis=1).flatten() |
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121 D = scipy.sparse.spdiags(diags, [0], m, n, format="csr") |
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122 L = D - A |
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123 with scipy.errstate(divide="ignore"): |
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124 diags_sqrt = 1.0 / np.sqrt(diags) |
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125 diags_sqrt[np.isinf(diags_sqrt)] = 0 |
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126 DH = scipy.sparse.spdiags(diags_sqrt, [0], m, n, format="csr") |
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127 return DH.dot(L.dot(DH)) |
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128 |
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129 |
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130 ############################################################################### |
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131 # Code based on |
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132 # https://bitbucket.org/bedwards/networkx-community/src/370bd69fc02f/networkx/algorithms/community/ |
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133 |
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134 |
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135 @not_implemented_for("undirected") |
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136 @not_implemented_for("multigraph") |
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137 def directed_laplacian_matrix( |
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138 G, nodelist=None, weight="weight", walk_type=None, alpha=0.95 |
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139 ): |
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140 r"""Returns the directed Laplacian matrix of G. |
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141 |
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142 The graph directed Laplacian is the matrix |
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143 |
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144 .. math:: |
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145 |
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146 L = I - (\Phi^{1/2} P \Phi^{-1/2} + \Phi^{-1/2} P^T \Phi^{1/2} ) / 2 |
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147 |
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148 where `I` is the identity matrix, `P` is the transition matrix of the |
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149 graph, and `\Phi` a matrix with the Perron vector of `P` in the diagonal and |
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150 zeros elsewhere. |
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151 |
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152 Depending on the value of walk_type, `P` can be the transition matrix |
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153 induced by a random walk, a lazy random walk, or a random walk with |
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154 teleportation (PageRank). |
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155 |
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156 Parameters |
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157 ---------- |
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158 G : DiGraph |
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159 A NetworkX graph |
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160 |
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161 nodelist : list, optional |
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162 The rows and columns are ordered according to the nodes in nodelist. |
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163 If nodelist is None, then the ordering is produced by G.nodes(). |
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164 |
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165 weight : string or None, optional (default='weight') |
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166 The edge data key used to compute each value in the matrix. |
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167 If None, then each edge has weight 1. |
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168 |
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169 walk_type : string or None, optional (default=None) |
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170 If None, `P` is selected depending on the properties of the |
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171 graph. Otherwise is one of 'random', 'lazy', or 'pagerank' |
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172 |
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173 alpha : real |
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174 (1 - alpha) is the teleportation probability used with pagerank |
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175 |
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176 Returns |
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177 ------- |
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178 L : NumPy matrix |
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179 Normalized Laplacian of G. |
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180 |
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181 Notes |
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182 ----- |
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183 Only implemented for DiGraphs |
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184 |
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185 See Also |
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186 -------- |
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187 laplacian_matrix |
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188 |
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189 References |
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190 ---------- |
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191 .. [1] Fan Chung (2005). |
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192 Laplacians and the Cheeger inequality for directed graphs. |
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193 Annals of Combinatorics, 9(1), 2005 |
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194 """ |
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195 import numpy as np |
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196 from scipy.sparse import spdiags, linalg |
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197 |
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198 P = _transition_matrix( |
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199 G, nodelist=nodelist, weight=weight, walk_type=walk_type, alpha=alpha |
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200 ) |
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201 |
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202 n, m = P.shape |
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203 |
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204 evals, evecs = linalg.eigs(P.T, k=1) |
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205 v = evecs.flatten().real |
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206 p = v / v.sum() |
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207 sqrtp = np.sqrt(p) |
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208 Q = spdiags(sqrtp, [0], n, n) * P * spdiags(1.0 / sqrtp, [0], n, n) |
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209 I = np.identity(len(G)) |
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210 |
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211 return I - (Q + Q.T) / 2.0 |
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212 |
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213 |
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214 @not_implemented_for("undirected") |
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215 @not_implemented_for("multigraph") |
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216 def directed_combinatorial_laplacian_matrix( |
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217 G, nodelist=None, weight="weight", walk_type=None, alpha=0.95 |
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218 ): |
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219 r"""Return the directed combinatorial Laplacian matrix of G. |
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220 |
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221 The graph directed combinatorial Laplacian is the matrix |
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222 |
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223 .. math:: |
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224 |
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225 L = \Phi - (\Phi P + P^T \Phi) / 2 |
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226 |
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227 where `P` is the transition matrix of the graph and and `\Phi` a matrix |
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228 with the Perron vector of `P` in the diagonal and zeros elsewhere. |
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229 |
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230 Depending on the value of walk_type, `P` can be the transition matrix |
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231 induced by a random walk, a lazy random walk, or a random walk with |
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232 teleportation (PageRank). |
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233 |
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234 Parameters |
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235 ---------- |
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236 G : DiGraph |
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237 A NetworkX graph |
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238 |
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239 nodelist : list, optional |
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240 The rows and columns are ordered according to the nodes in nodelist. |
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241 If nodelist is None, then the ordering is produced by G.nodes(). |
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242 |
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243 weight : string or None, optional (default='weight') |
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244 The edge data key used to compute each value in the matrix. |
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245 If None, then each edge has weight 1. |
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246 |
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247 walk_type : string or None, optional (default=None) |
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248 If None, `P` is selected depending on the properties of the |
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249 graph. Otherwise is one of 'random', 'lazy', or 'pagerank' |
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250 |
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251 alpha : real |
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252 (1 - alpha) is the teleportation probability used with pagerank |
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253 |
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254 Returns |
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255 ------- |
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256 L : NumPy matrix |
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257 Combinatorial Laplacian of G. |
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258 |
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259 Notes |
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260 ----- |
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261 Only implemented for DiGraphs |
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262 |
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263 See Also |
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264 -------- |
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265 laplacian_matrix |
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266 |
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267 References |
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268 ---------- |
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269 .. [1] Fan Chung (2005). |
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270 Laplacians and the Cheeger inequality for directed graphs. |
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271 Annals of Combinatorics, 9(1), 2005 |
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272 """ |
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273 from scipy.sparse import spdiags, linalg |
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274 |
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275 P = _transition_matrix( |
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276 G, nodelist=nodelist, weight=weight, walk_type=walk_type, alpha=alpha |
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277 ) |
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278 |
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279 n, m = P.shape |
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280 |
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281 evals, evecs = linalg.eigs(P.T, k=1) |
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282 v = evecs.flatten().real |
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283 p = v / v.sum() |
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284 Phi = spdiags(p, [0], n, n) |
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285 |
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286 Phi = Phi.todense() |
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287 |
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288 return Phi - (Phi * P + P.T * Phi) / 2.0 |
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289 |
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290 |
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291 def _transition_matrix(G, nodelist=None, weight="weight", walk_type=None, alpha=0.95): |
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292 """Returns the transition matrix of G. |
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293 |
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294 This is a row stochastic giving the transition probabilities while |
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295 performing a random walk on the graph. Depending on the value of walk_type, |
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296 P can be the transition matrix induced by a random walk, a lazy random walk, |
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297 or a random walk with teleportation (PageRank). |
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298 |
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299 Parameters |
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300 ---------- |
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301 G : DiGraph |
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302 A NetworkX graph |
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303 |
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304 nodelist : list, optional |
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305 The rows and columns are ordered according to the nodes in nodelist. |
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306 If nodelist is None, then the ordering is produced by G.nodes(). |
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307 |
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308 weight : string or None, optional (default='weight') |
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309 The edge data key used to compute each value in the matrix. |
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310 If None, then each edge has weight 1. |
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311 |
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312 walk_type : string or None, optional (default=None) |
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313 If None, `P` is selected depending on the properties of the |
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314 graph. Otherwise is one of 'random', 'lazy', or 'pagerank' |
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315 |
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316 alpha : real |
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317 (1 - alpha) is the teleportation probability used with pagerank |
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318 |
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319 Returns |
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320 ------- |
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321 P : NumPy matrix |
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322 transition matrix of G. |
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323 |
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324 Raises |
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325 ------ |
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326 NetworkXError |
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327 If walk_type not specified or alpha not in valid range |
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328 """ |
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329 import numpy as np |
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330 from scipy.sparse import identity, spdiags |
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331 |
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332 if walk_type is None: |
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333 if nx.is_strongly_connected(G): |
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334 if nx.is_aperiodic(G): |
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335 walk_type = "random" |
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336 else: |
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337 walk_type = "lazy" |
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338 else: |
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339 walk_type = "pagerank" |
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340 |
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341 M = nx.to_scipy_sparse_matrix(G, nodelist=nodelist, weight=weight, dtype=float) |
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342 n, m = M.shape |
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343 if walk_type in ["random", "lazy"]: |
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344 DI = spdiags(1.0 / np.array(M.sum(axis=1).flat), [0], n, n) |
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345 if walk_type == "random": |
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346 P = DI * M |
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347 else: |
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348 I = identity(n) |
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349 P = (I + DI * M) / 2.0 |
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350 |
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351 elif walk_type == "pagerank": |
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352 if not (0 < alpha < 1): |
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353 raise nx.NetworkXError("alpha must be between 0 and 1") |
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354 # this is using a dense representation |
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355 M = M.todense() |
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356 # add constant to dangling nodes' row |
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357 dangling = np.where(M.sum(axis=1) == 0) |
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358 for d in dangling[0]: |
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359 M[d] = 1.0 / n |
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360 # normalize |
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361 M = M / M.sum(axis=1) |
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362 P = alpha * M + (1 - alpha) / n |
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363 else: |
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364 raise nx.NetworkXError("walk_type must be random, lazy, or pagerank") |
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365 |
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366 return P |