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annotate planemo/lib/python3.7/site-packages/networkx/algorithms/cuts.py @ 1:56ad4e20f292 draft

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"planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
author guerler
date Fri, 31 Jul 2020 00:32:28 -0400
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1 # -*- coding: utf-8 -*-
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2 # cuts.py - functions for computing and evaluating cuts
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3 #
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4 # Copyright 2011 Ben Edwards <bedwards@cs.unm.edu>.
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5 # Copyright 2011 Aric Hagberg <hagberg@lanl.gov>.
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6 # Copyright 2015 NetworkX developers.
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7 #
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8 # This file is part of NetworkX.
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9 #
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10 # NetworkX is distributed under a BSD license; see LICENSE.txt for more
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11 # information.
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12 """Functions for finding and evaluating cuts in a graph.
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13
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14 """
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15
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16 from itertools import chain
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17
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18 import networkx as nx
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19
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20 __all__ = ['boundary_expansion', 'conductance', 'cut_size', 'edge_expansion',
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21 'mixing_expansion', 'node_expansion', 'normalized_cut_size',
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22 'volume']
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23
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24
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25 # TODO STILL NEED TO UPDATE ALL THE DOCUMENTATION!
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26
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27 def cut_size(G, S, T=None, weight=None):
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28 """Returns the size of the cut between two sets of nodes.
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29
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30 A *cut* is a partition of the nodes of a graph into two sets. The
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31 *cut size* is the sum of the weights of the edges "between" the two
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32 sets of nodes.
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33
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34 Parameters
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35 ----------
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36 G : NetworkX graph
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37
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38 S : sequence
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39 A sequence of nodes in `G`.
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40
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41 T : sequence
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42 A sequence of nodes in `G`. If not specified, this is taken to
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43 be the set complement of `S`.
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44
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45 weight : object
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46 Edge attribute key to use as weight. If not specified, edges
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47 have weight one.
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48
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49 Returns
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50 -------
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51 number
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52 Total weight of all edges from nodes in set `S` to nodes in
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53 set `T` (and, in the case of directed graphs, all edges from
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54 nodes in `T` to nodes in `S`).
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55
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56 Examples
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57 --------
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58 In the graph with two cliques joined by a single edges, the natural
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59 bipartition of the graph into two blocks, one for each clique,
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60 yields a cut of weight one::
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61
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62 >>> G = nx.barbell_graph(3, 0)
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63 >>> S = {0, 1, 2}
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64 >>> T = {3, 4, 5}
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65 >>> nx.cut_size(G, S, T)
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66 1
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67
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68 Each parallel edge in a multigraph is counted when determining the
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69 cut size::
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70
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71 >>> G = nx.MultiGraph(['ab', 'ab'])
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72 >>> S = {'a'}
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73 >>> T = {'b'}
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74 >>> nx.cut_size(G, S, T)
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75 2
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76
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77 Notes
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78 -----
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79 In a multigraph, the cut size is the total weight of edges including
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80 multiplicity.
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81
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82 """
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83 edges = nx.edge_boundary(G, S, T, data=weight, default=1)
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84 if G.is_directed():
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85 edges = chain(edges, nx.edge_boundary(G, T, S, data=weight, default=1))
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86 return sum(weight for u, v, weight in edges)
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87
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88
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89 def volume(G, S, weight=None):
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90 """Returns the volume of a set of nodes.
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91
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92 The *volume* of a set *S* is the sum of the (out-)degrees of nodes
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93 in *S* (taking into account parallel edges in multigraphs). [1]
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94
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95 Parameters
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96 ----------
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97 G : NetworkX graph
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98
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99 S : sequence
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100 A sequence of nodes in `G`.
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101
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102 weight : object
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103 Edge attribute key to use as weight. If not specified, edges
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104 have weight one.
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105
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106 Returns
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107 -------
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108 number
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109 The volume of the set of nodes represented by `S` in the graph
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110 `G`.
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111
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112 See also
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113 --------
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114 conductance
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115 cut_size
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116 edge_expansion
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117 edge_boundary
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118 normalized_cut_size
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119
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120 References
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121 ----------
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122 .. [1] David Gleich.
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123 *Hierarchical Directed Spectral Graph Partitioning*.
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124 <https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20hierarchical%20directed%20spectral.pdf>
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125
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126 """
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127 degree = G.out_degree if G.is_directed() else G.degree
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128 return sum(d for v, d in degree(S, weight=weight))
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129
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130
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131 def normalized_cut_size(G, S, T=None, weight=None):
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132 """Returns the normalized size of the cut between two sets of nodes.
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133
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134 The *normalized cut size* is the cut size times the sum of the
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135 reciprocal sizes of the volumes of the two sets. [1]
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136
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137 Parameters
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138 ----------
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139 G : NetworkX graph
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140
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141 S : sequence
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142 A sequence of nodes in `G`.
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143
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144 T : sequence
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145 A sequence of nodes in `G`.
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146
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147 weight : object
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148 Edge attribute key to use as weight. If not specified, edges
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149 have weight one.
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150
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151 Returns
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152 -------
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153 number
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154 The normalized cut size between the two sets `S` and `T`.
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155
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156 Notes
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157 -----
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158 In a multigraph, the cut size is the total weight of edges including
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159 multiplicity.
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160
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161 See also
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162 --------
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163 conductance
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164 cut_size
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165 edge_expansion
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166 volume
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167
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168 References
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169 ----------
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170 .. [1] David Gleich.
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171 *Hierarchical Directed Spectral Graph Partitioning*.
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172 <https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20hierarchical%20directed%20spectral.pdf>
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173
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174 """
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175 if T is None:
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176 T = set(G) - set(S)
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177 num_cut_edges = cut_size(G, S, T=T, weight=weight)
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178 volume_S = volume(G, S, weight=weight)
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179 volume_T = volume(G, T, weight=weight)
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180 return num_cut_edges * ((1 / volume_S) + (1 / volume_T))
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181
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182
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183 def conductance(G, S, T=None, weight=None):
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184 """Returns the conductance of two sets of nodes.
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185
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186 The *conductance* is the quotient of the cut size and the smaller of
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187 the volumes of the two sets. [1]
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188
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189 Parameters
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190 ----------
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191 G : NetworkX graph
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192
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193 S : sequence
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194 A sequence of nodes in `G`.
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195
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196 T : sequence
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197 A sequence of nodes in `G`.
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198
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199 weight : object
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200 Edge attribute key to use as weight. If not specified, edges
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201 have weight one.
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202
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203 Returns
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204 -------
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205 number
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206 The conductance between the two sets `S` and `T`.
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207
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208 See also
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209 --------
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210 cut_size
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211 edge_expansion
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212 normalized_cut_size
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213 volume
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214
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215 References
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216 ----------
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217 .. [1] David Gleich.
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218 *Hierarchical Directed Spectral Graph Partitioning*.
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219 <https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20hierarchical%20directed%20spectral.pdf>
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220
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221 """
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222 if T is None:
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223 T = set(G) - set(S)
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224 num_cut_edges = cut_size(G, S, T, weight=weight)
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225 volume_S = volume(G, S, weight=weight)
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226 volume_T = volume(G, T, weight=weight)
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227 return num_cut_edges / min(volume_S, volume_T)
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228
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229
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230 def edge_expansion(G, S, T=None, weight=None):
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231 """Returns the edge expansion between two node sets.
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232
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233 The *edge expansion* is the quotient of the cut size and the smaller
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234 of the cardinalities of the two sets. [1]
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235
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236 Parameters
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237 ----------
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238 G : NetworkX graph
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239
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240 S : sequence
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241 A sequence of nodes in `G`.
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242
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243 T : sequence
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244 A sequence of nodes in `G`.
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245
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246 weight : object
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247 Edge attribute key to use as weight. If not specified, edges
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248 have weight one.
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249
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250 Returns
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251 -------
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252 number
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253 The edge expansion between the two sets `S` and `T`.
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254
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255 See also
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256 --------
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257 boundary_expansion
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258 mixing_expansion
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259 node_expansion
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260
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261 References
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262 ----------
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263 .. [1] Fan Chung.
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264 *Spectral Graph Theory*.
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265 (CBMS Regional Conference Series in Mathematics, No. 92),
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266 American Mathematical Society, 1997, ISBN 0-8218-0315-8
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267 <http://www.math.ucsd.edu/~fan/research/revised.html>
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268
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269 """
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270 if T is None:
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271 T = set(G) - set(S)
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272 num_cut_edges = cut_size(G, S, T=T, weight=weight)
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273 return num_cut_edges / min(len(S), len(T))
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274
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275
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276 def mixing_expansion(G, S, T=None, weight=None):
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277 """Returns the mixing expansion between two node sets.
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278
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279 The *mixing expansion* is the quotient of the cut size and twice the
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280 number of edges in the graph. [1]
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281
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282 Parameters
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283 ----------
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284 G : NetworkX graph
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285
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286 S : sequence
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287 A sequence of nodes in `G`.
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288
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289 T : sequence
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290 A sequence of nodes in `G`.
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291
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292 weight : object
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293 Edge attribute key to use as weight. If not specified, edges
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294 have weight one.
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295
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296 Returns
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297 -------
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298 number
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299 The mixing expansion between the two sets `S` and `T`.
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300
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301 See also
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302 --------
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303 boundary_expansion
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304 edge_expansion
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305 node_expansion
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306
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307 References
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308 ----------
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309 .. [1] Vadhan, Salil P.
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310 "Pseudorandomness."
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311 *Foundations and Trends
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312 in Theoretical Computer Science* 7.1–3 (2011): 1–336.
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313 <https://doi.org/10.1561/0400000010>
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314
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315 """
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316 num_cut_edges = cut_size(G, S, T=T, weight=weight)
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317 num_total_edges = G.number_of_edges()
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318 return num_cut_edges / (2 * num_total_edges)
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319
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320
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321 # TODO What is the generalization to two arguments, S and T? Does the
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322 # denominator become `min(len(S), len(T))`?
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323 def node_expansion(G, S):
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324 """Returns the node expansion of the set `S`.
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325
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326 The *node expansion* is the quotient of the size of the node
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327 boundary of *S* and the cardinality of *S*. [1]
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328
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329 Parameters
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330 ----------
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331 G : NetworkX graph
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332
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333 S : sequence
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334 A sequence of nodes in `G`.
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335
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336 Returns
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337 -------
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338 number
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339 The node expansion of the set `S`.
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340
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341 See also
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342 --------
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343 boundary_expansion
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344 edge_expansion
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345 mixing_expansion
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346
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347 References
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348 ----------
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349 .. [1] Vadhan, Salil P.
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350 "Pseudorandomness."
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351 *Foundations and Trends
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352 in Theoretical Computer Science* 7.1–3 (2011): 1–336.
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353 <https://doi.org/10.1561/0400000010>
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354
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355 """
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356 neighborhood = set(chain.from_iterable(G.neighbors(v) for v in S))
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357 return len(neighborhood) / len(S)
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358
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359
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360 # TODO What is the generalization to two arguments, S and T? Does the
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361 # denominator become `min(len(S), len(T))`?
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362 def boundary_expansion(G, S):
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363 """Returns the boundary expansion of the set `S`.
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364
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365 The *boundary expansion* is the quotient of the size of the edge
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366 boundary and the cardinality of *S*. [1]
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367
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368 Parameters
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369 ----------
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370 G : NetworkX graph
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371
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372 S : sequence
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373 A sequence of nodes in `G`.
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374
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375 Returns
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376 -------
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377 number
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378 The boundary expansion of the set `S`.
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379
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380 See also
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381 --------
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382 edge_expansion
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383 mixing_expansion
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384 node_expansion
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385
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386 References
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387 ----------
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388 .. [1] Vadhan, Salil P.
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389 "Pseudorandomness."
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390 *Foundations and Trends in Theoretical Computer Science*
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391 7.1–3 (2011): 1–336.
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392 <https://doi.org/10.1561/0400000010>
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393
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394 """
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395 return len(nx.node_boundary(G, S)) / len(S)