annotate env/lib/python3.9/site-packages/networkx/linalg/laplacianmatrix.py @ 0:4f3585e2f14b draft default tip

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author shellac
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