Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/cuts.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 """Functions for finding and evaluating cuts in a graph. |
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2 |
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3 """ |
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4 |
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5 from itertools import chain |
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6 |
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7 import networkx as nx |
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8 |
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9 __all__ = [ |
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10 "boundary_expansion", |
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11 "conductance", |
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12 "cut_size", |
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13 "edge_expansion", |
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14 "mixing_expansion", |
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15 "node_expansion", |
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16 "normalized_cut_size", |
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17 "volume", |
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18 ] |
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19 |
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20 |
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21 # TODO STILL NEED TO UPDATE ALL THE DOCUMENTATION! |
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22 |
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23 |
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24 def cut_size(G, S, T=None, weight=None): |
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25 """Returns the size of the cut between two sets of nodes. |
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26 |
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27 A *cut* is a partition of the nodes of a graph into two sets. The |
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28 *cut size* is the sum of the weights of the edges "between" the two |
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29 sets of nodes. |
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30 |
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31 Parameters |
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32 ---------- |
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33 G : NetworkX graph |
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34 |
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35 S : sequence |
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36 A sequence of nodes in `G`. |
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37 |
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38 T : sequence |
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39 A sequence of nodes in `G`. If not specified, this is taken to |
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40 be the set complement of `S`. |
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41 |
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42 weight : object |
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43 Edge attribute key to use as weight. If not specified, edges |
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44 have weight one. |
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45 |
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46 Returns |
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47 ------- |
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48 number |
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49 Total weight of all edges from nodes in set `S` to nodes in |
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50 set `T` (and, in the case of directed graphs, all edges from |
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51 nodes in `T` to nodes in `S`). |
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52 |
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53 Examples |
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54 -------- |
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55 In the graph with two cliques joined by a single edges, the natural |
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56 bipartition of the graph into two blocks, one for each clique, |
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57 yields a cut of weight one:: |
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58 |
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59 >>> G = nx.barbell_graph(3, 0) |
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60 >>> S = {0, 1, 2} |
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61 >>> T = {3, 4, 5} |
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62 >>> nx.cut_size(G, S, T) |
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63 1 |
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64 |
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65 Each parallel edge in a multigraph is counted when determining the |
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66 cut size:: |
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67 |
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68 >>> G = nx.MultiGraph(["ab", "ab"]) |
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69 >>> S = {"a"} |
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70 >>> T = {"b"} |
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71 >>> nx.cut_size(G, S, T) |
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72 2 |
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73 |
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74 Notes |
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75 ----- |
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76 In a multigraph, the cut size is the total weight of edges including |
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77 multiplicity. |
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78 |
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79 """ |
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80 edges = nx.edge_boundary(G, S, T, data=weight, default=1) |
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81 if G.is_directed(): |
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82 edges = chain(edges, nx.edge_boundary(G, T, S, data=weight, default=1)) |
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83 return sum(weight for u, v, weight in edges) |
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84 |
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85 |
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86 def volume(G, S, weight=None): |
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87 """Returns the volume of a set of nodes. |
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88 |
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89 The *volume* of a set *S* is the sum of the (out-)degrees of nodes |
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90 in *S* (taking into account parallel edges in multigraphs). [1] |
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91 |
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92 Parameters |
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93 ---------- |
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94 G : NetworkX graph |
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95 |
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96 S : sequence |
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97 A sequence of nodes in `G`. |
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98 |
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99 weight : object |
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100 Edge attribute key to use as weight. If not specified, edges |
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101 have weight one. |
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102 |
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103 Returns |
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104 ------- |
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105 number |
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106 The volume of the set of nodes represented by `S` in the graph |
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107 `G`. |
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108 |
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109 See also |
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110 -------- |
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111 conductance |
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112 cut_size |
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113 edge_expansion |
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114 edge_boundary |
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115 normalized_cut_size |
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116 |
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117 References |
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118 ---------- |
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119 .. [1] David Gleich. |
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120 *Hierarchical Directed Spectral Graph Partitioning*. |
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121 <https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20hierarchical%20directed%20spectral.pdf> |
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122 |
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123 """ |
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124 degree = G.out_degree if G.is_directed() else G.degree |
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125 return sum(d for v, d in degree(S, weight=weight)) |
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126 |
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127 |
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128 def normalized_cut_size(G, S, T=None, weight=None): |
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129 """Returns the normalized size of the cut between two sets of nodes. |
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130 |
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131 The *normalized cut size* is the cut size times the sum of the |
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132 reciprocal sizes of the volumes of the two sets. [1] |
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133 |
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134 Parameters |
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135 ---------- |
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136 G : NetworkX graph |
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137 |
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138 S : sequence |
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139 A sequence of nodes in `G`. |
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140 |
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141 T : sequence |
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142 A sequence of nodes in `G`. |
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143 |
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144 weight : object |
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145 Edge attribute key to use as weight. If not specified, edges |
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146 have weight one. |
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147 |
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148 Returns |
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149 ------- |
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150 number |
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151 The normalized cut size between the two sets `S` and `T`. |
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152 |
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153 Notes |
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154 ----- |
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155 In a multigraph, the cut size is the total weight of edges including |
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156 multiplicity. |
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157 |
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158 See also |
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159 -------- |
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160 conductance |
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161 cut_size |
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162 edge_expansion |
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163 volume |
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164 |
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165 References |
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166 ---------- |
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167 .. [1] David Gleich. |
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168 *Hierarchical Directed Spectral Graph Partitioning*. |
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169 <https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20hierarchical%20directed%20spectral.pdf> |
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170 |
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171 """ |
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172 if T is None: |
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173 T = set(G) - set(S) |
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174 num_cut_edges = cut_size(G, S, T=T, weight=weight) |
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175 volume_S = volume(G, S, weight=weight) |
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176 volume_T = volume(G, T, weight=weight) |
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177 return num_cut_edges * ((1 / volume_S) + (1 / volume_T)) |
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178 |
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179 |
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180 def conductance(G, S, T=None, weight=None): |
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181 """Returns the conductance of two sets of nodes. |
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182 |
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183 The *conductance* is the quotient of the cut size and the smaller of |
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184 the volumes of the two sets. [1] |
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185 |
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186 Parameters |
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187 ---------- |
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188 G : NetworkX graph |
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189 |
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190 S : sequence |
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191 A sequence of nodes in `G`. |
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192 |
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193 T : sequence |
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194 A sequence of nodes in `G`. |
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195 |
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196 weight : object |
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197 Edge attribute key to use as weight. If not specified, edges |
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198 have weight one. |
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199 |
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200 Returns |
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201 ------- |
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202 number |
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203 The conductance between the two sets `S` and `T`. |
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204 |
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205 See also |
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206 -------- |
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207 cut_size |
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208 edge_expansion |
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209 normalized_cut_size |
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210 volume |
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211 |
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212 References |
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213 ---------- |
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214 .. [1] David Gleich. |
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215 *Hierarchical Directed Spectral Graph Partitioning*. |
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216 <https://www.cs.purdue.edu/homes/dgleich/publications/Gleich%202005%20-%20hierarchical%20directed%20spectral.pdf> |
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217 |
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218 """ |
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219 if T is None: |
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220 T = set(G) - set(S) |
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221 num_cut_edges = cut_size(G, S, T, weight=weight) |
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222 volume_S = volume(G, S, weight=weight) |
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223 volume_T = volume(G, T, weight=weight) |
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224 return num_cut_edges / min(volume_S, volume_T) |
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225 |
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226 |
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227 def edge_expansion(G, S, T=None, weight=None): |
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228 """Returns the edge expansion between two node sets. |
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229 |
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230 The *edge expansion* is the quotient of the cut size and the smaller |
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231 of the cardinalities of the two sets. [1] |
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232 |
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233 Parameters |
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234 ---------- |
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235 G : NetworkX graph |
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236 |
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237 S : sequence |
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238 A sequence of nodes in `G`. |
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239 |
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240 T : sequence |
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241 A sequence of nodes in `G`. |
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242 |
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243 weight : object |
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244 Edge attribute key to use as weight. If not specified, edges |
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245 have weight one. |
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246 |
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247 Returns |
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248 ------- |
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249 number |
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250 The edge expansion between the two sets `S` and `T`. |
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251 |
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252 See also |
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253 -------- |
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254 boundary_expansion |
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255 mixing_expansion |
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256 node_expansion |
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257 |
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258 References |
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259 ---------- |
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260 .. [1] Fan Chung. |
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261 *Spectral Graph Theory*. |
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262 (CBMS Regional Conference Series in Mathematics, No. 92), |
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263 American Mathematical Society, 1997, ISBN 0-8218-0315-8 |
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264 <http://www.math.ucsd.edu/~fan/research/revised.html> |
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265 |
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266 """ |
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267 if T is None: |
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268 T = set(G) - set(S) |
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269 num_cut_edges = cut_size(G, S, T=T, weight=weight) |
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270 return num_cut_edges / min(len(S), len(T)) |
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271 |
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272 |
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273 def mixing_expansion(G, S, T=None, weight=None): |
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274 """Returns the mixing expansion between two node sets. |
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275 |
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276 The *mixing expansion* is the quotient of the cut size and twice the |
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277 number of edges in the graph. [1] |
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278 |
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279 Parameters |
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280 ---------- |
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281 G : NetworkX graph |
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282 |
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283 S : sequence |
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284 A sequence of nodes in `G`. |
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285 |
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286 T : sequence |
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287 A sequence of nodes in `G`. |
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288 |
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289 weight : object |
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290 Edge attribute key to use as weight. If not specified, edges |
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291 have weight one. |
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292 |
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293 Returns |
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294 ------- |
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295 number |
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296 The mixing expansion between the two sets `S` and `T`. |
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297 |
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298 See also |
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299 -------- |
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300 boundary_expansion |
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301 edge_expansion |
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302 node_expansion |
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303 |
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304 References |
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305 ---------- |
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306 .. [1] Vadhan, Salil P. |
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307 "Pseudorandomness." |
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308 *Foundations and Trends |
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309 in Theoretical Computer Science* 7.1–3 (2011): 1–336. |
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310 <https://doi.org/10.1561/0400000010> |
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311 |
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312 """ |
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313 num_cut_edges = cut_size(G, S, T=T, weight=weight) |
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314 num_total_edges = G.number_of_edges() |
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315 return num_cut_edges / (2 * num_total_edges) |
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316 |
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317 |
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318 # TODO What is the generalization to two arguments, S and T? Does the |
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319 # denominator become `min(len(S), len(T))`? |
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320 def node_expansion(G, S): |
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321 """Returns the node expansion of the set `S`. |
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322 |
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323 The *node expansion* is the quotient of the size of the node |
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324 boundary of *S* and the cardinality of *S*. [1] |
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325 |
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326 Parameters |
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327 ---------- |
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328 G : NetworkX graph |
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329 |
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330 S : sequence |
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331 A sequence of nodes in `G`. |
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332 |
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333 Returns |
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334 ------- |
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335 number |
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336 The node expansion of the set `S`. |
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337 |
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338 See also |
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339 -------- |
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340 boundary_expansion |
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341 edge_expansion |
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342 mixing_expansion |
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343 |
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344 References |
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345 ---------- |
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346 .. [1] Vadhan, Salil P. |
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347 "Pseudorandomness." |
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348 *Foundations and Trends |
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349 in Theoretical Computer Science* 7.1–3 (2011): 1–336. |
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350 <https://doi.org/10.1561/0400000010> |
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351 |
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352 """ |
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353 neighborhood = set(chain.from_iterable(G.neighbors(v) for v in S)) |
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354 return len(neighborhood) / len(S) |
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355 |
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356 |
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357 # TODO What is the generalization to two arguments, S and T? Does the |
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358 # denominator become `min(len(S), len(T))`? |
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359 def boundary_expansion(G, S): |
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360 """Returns the boundary expansion of the set `S`. |
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361 |
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362 The *boundary expansion* is the quotient of the size |
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363 of the node boundary and the cardinality of *S*. [1] |
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364 |
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365 Parameters |
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366 ---------- |
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367 G : NetworkX graph |
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368 |
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369 S : sequence |
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370 A sequence of nodes in `G`. |
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371 |
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372 Returns |
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373 ------- |
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374 number |
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375 The boundary expansion of the set `S`. |
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376 |
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377 See also |
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378 -------- |
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379 edge_expansion |
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380 mixing_expansion |
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381 node_expansion |
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382 |
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383 References |
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384 ---------- |
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385 .. [1] Vadhan, Salil P. |
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386 "Pseudorandomness." |
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387 *Foundations and Trends in Theoretical Computer Science* |
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388 7.1–3 (2011): 1–336. |
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389 <https://doi.org/10.1561/0400000010> |
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390 |
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391 """ |
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392 return len(nx.node_boundary(G, S)) / len(S) |