Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/minors.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 """Provides functions for computing minors of a graph.""" |
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2 from itertools import chain |
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3 from itertools import combinations |
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4 from itertools import permutations |
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5 from itertools import product |
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6 |
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7 import networkx as nx |
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8 from networkx import density |
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9 from networkx.exception import NetworkXException |
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10 from networkx.utils import arbitrary_element |
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11 |
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12 __all__ = ["contracted_edge", "contracted_nodes", "identified_nodes", "quotient_graph"] |
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13 |
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14 chaini = chain.from_iterable |
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15 |
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16 |
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17 def equivalence_classes(iterable, relation): |
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18 """Returns the set of equivalence classes of the given `iterable` under |
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19 the specified equivalence relation. |
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20 |
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21 `relation` must be a Boolean-valued function that takes two argument. It |
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22 must represent an equivalence relation (that is, the relation induced by |
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23 the function must be reflexive, symmetric, and transitive). |
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24 |
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25 The return value is a set of sets. It is a partition of the elements of |
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26 `iterable`; duplicate elements will be ignored so it makes the most sense |
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27 for `iterable` to be a :class:`set`. |
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28 |
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29 """ |
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30 # For simplicity of implementation, we initialize the return value as a |
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31 # list of lists, then convert it to a set of sets at the end of the |
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32 # function. |
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33 blocks = [] |
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34 # Determine the equivalence class for each element of the iterable. |
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35 for y in iterable: |
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36 # Each element y must be in *exactly one* equivalence class. |
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37 # |
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38 # Each block is guaranteed to be non-empty |
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39 for block in blocks: |
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40 x = arbitrary_element(block) |
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41 if relation(x, y): |
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42 block.append(y) |
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43 break |
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44 else: |
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45 # If the element y is not part of any known equivalence class, it |
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46 # must be in its own, so we create a new singleton equivalence |
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47 # class for it. |
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48 blocks.append([y]) |
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49 return {frozenset(block) for block in blocks} |
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50 |
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51 |
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52 def quotient_graph( |
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53 G, |
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54 partition, |
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55 edge_relation=None, |
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56 node_data=None, |
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57 edge_data=None, |
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58 relabel=False, |
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59 create_using=None, |
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60 ): |
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61 """Returns the quotient graph of `G` under the specified equivalence |
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62 relation on nodes. |
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63 |
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64 Parameters |
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65 ---------- |
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66 G : NetworkX graph |
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67 The graph for which to return the quotient graph with the |
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68 specified node relation. |
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69 |
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70 partition : function or list of sets |
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71 If a function, this function must represent an equivalence |
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72 relation on the nodes of `G`. It must take two arguments *u* |
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73 and *v* and return True exactly when *u* and *v* are in the |
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74 same equivalence class. The equivalence classes form the nodes |
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75 in the returned graph. |
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76 |
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77 If a list of sets, the list must form a valid partition of |
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78 the nodes of the graph. That is, each node must be in exactly |
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79 one block of the partition. |
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80 |
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81 edge_relation : Boolean function with two arguments |
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82 This function must represent an edge relation on the *blocks* of |
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83 `G` in the partition induced by `node_relation`. It must |
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84 take two arguments, *B* and *C*, each one a set of nodes, and |
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85 return True exactly when there should be an edge joining |
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86 block *B* to block *C* in the returned graph. |
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87 |
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88 If `edge_relation` is not specified, it is assumed to be the |
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89 following relation. Block *B* is related to block *C* if and |
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90 only if some node in *B* is adjacent to some node in *C*, |
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91 according to the edge set of `G`. |
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92 |
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93 edge_data : function |
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94 This function takes two arguments, *B* and *C*, each one a set |
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95 of nodes, and must return a dictionary representing the edge |
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96 data attributes to set on the edge joining *B* and *C*, should |
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97 there be an edge joining *B* and *C* in the quotient graph (if |
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98 no such edge occurs in the quotient graph as determined by |
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99 `edge_relation`, then the output of this function is ignored). |
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100 |
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101 If the quotient graph would be a multigraph, this function is |
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102 not applied, since the edge data from each edge in the graph |
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103 `G` appears in the edges of the quotient graph. |
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104 |
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105 node_data : function |
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106 This function takes one argument, *B*, a set of nodes in `G`, |
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107 and must return a dictionary representing the node data |
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108 attributes to set on the node representing *B* in the quotient graph. |
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109 If None, the following node attributes will be set: |
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110 |
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111 * 'graph', the subgraph of the graph `G` that this block |
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112 represents, |
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113 * 'nnodes', the number of nodes in this block, |
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114 * 'nedges', the number of edges within this block, |
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115 * 'density', the density of the subgraph of `G` that this |
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116 block represents. |
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117 |
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118 relabel : bool |
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119 If True, relabel the nodes of the quotient graph to be |
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120 nonnegative integers. Otherwise, the nodes are identified with |
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121 :class:`frozenset` instances representing the blocks given in |
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122 `partition`. |
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123 |
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124 create_using : NetworkX graph constructor, optional (default=nx.Graph) |
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125 Graph type to create. If graph instance, then cleared before populated. |
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126 |
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127 Returns |
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128 ------- |
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129 NetworkX graph |
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130 The quotient graph of `G` under the equivalence relation |
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131 specified by `partition`. If the partition were given as a |
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132 list of :class:`set` instances and `relabel` is False, |
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133 each node will be a :class:`frozenset` corresponding to the same |
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134 :class:`set`. |
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135 |
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136 Raises |
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137 ------ |
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138 NetworkXException |
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139 If the given partition is not a valid partition of the nodes of |
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140 `G`. |
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141 |
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142 Examples |
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143 -------- |
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144 The quotient graph of the complete bipartite graph under the "same |
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145 neighbors" equivalence relation is `K_2`. Under this relation, two nodes |
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146 are equivalent if they are not adjacent but have the same neighbor set:: |
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147 |
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148 >>> G = nx.complete_bipartite_graph(2, 3) |
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149 >>> same_neighbors = lambda u, v: ( |
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150 ... u not in G[v] and v not in G[u] and G[u] == G[v] |
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151 ... ) |
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152 >>> Q = nx.quotient_graph(G, same_neighbors) |
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153 >>> K2 = nx.complete_graph(2) |
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154 >>> nx.is_isomorphic(Q, K2) |
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155 True |
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156 |
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157 The quotient graph of a directed graph under the "same strongly connected |
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158 component" equivalence relation is the condensation of the graph (see |
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159 :func:`condensation`). This example comes from the Wikipedia article |
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160 *`Strongly connected component`_*:: |
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161 |
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162 >>> G = nx.DiGraph() |
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163 >>> edges = [ |
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164 ... "ab", |
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165 ... "be", |
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166 ... "bf", |
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167 ... "bc", |
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168 ... "cg", |
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169 ... "cd", |
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170 ... "dc", |
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171 ... "dh", |
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172 ... "ea", |
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173 ... "ef", |
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174 ... "fg", |
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175 ... "gf", |
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176 ... "hd", |
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177 ... "hf", |
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178 ... ] |
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179 >>> G.add_edges_from(tuple(x) for x in edges) |
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180 >>> components = list(nx.strongly_connected_components(G)) |
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181 >>> sorted(sorted(component) for component in components) |
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182 [['a', 'b', 'e'], ['c', 'd', 'h'], ['f', 'g']] |
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183 >>> |
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184 >>> C = nx.condensation(G, components) |
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185 >>> component_of = C.graph["mapping"] |
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186 >>> same_component = lambda u, v: component_of[u] == component_of[v] |
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187 >>> Q = nx.quotient_graph(G, same_component) |
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188 >>> nx.is_isomorphic(C, Q) |
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189 True |
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190 |
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191 Node identification can be represented as the quotient of a graph under the |
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192 equivalence relation that places the two nodes in one block and each other |
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193 node in its own singleton block:: |
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194 |
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195 >>> K24 = nx.complete_bipartite_graph(2, 4) |
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196 >>> K34 = nx.complete_bipartite_graph(3, 4) |
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197 >>> C = nx.contracted_nodes(K34, 1, 2) |
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198 >>> nodes = {1, 2} |
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199 >>> is_contracted = lambda u, v: u in nodes and v in nodes |
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200 >>> Q = nx.quotient_graph(K34, is_contracted) |
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201 >>> nx.is_isomorphic(Q, C) |
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202 True |
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203 >>> nx.is_isomorphic(Q, K24) |
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204 True |
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205 |
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206 The blockmodeling technique described in [1]_ can be implemented as a |
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207 quotient graph:: |
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208 |
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209 >>> G = nx.path_graph(6) |
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210 >>> partition = [{0, 1}, {2, 3}, {4, 5}] |
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211 >>> M = nx.quotient_graph(G, partition, relabel=True) |
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212 >>> list(M.edges()) |
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213 [(0, 1), (1, 2)] |
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214 |
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215 .. _Strongly connected component: https://en.wikipedia.org/wiki/Strongly_connected_component |
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216 |
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217 References |
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218 ---------- |
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219 .. [1] Patrick Doreian, Vladimir Batagelj, and Anuska Ferligoj. |
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220 *Generalized Blockmodeling*. |
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221 Cambridge University Press, 2004. |
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222 |
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223 """ |
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224 # If the user provided an equivalence relation as a function compute |
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225 # the blocks of the partition on the nodes of G induced by the |
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226 # equivalence relation. |
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227 if callable(partition): |
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228 # equivalence_classes always return partition of whole G. |
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229 partition = equivalence_classes(G, partition) |
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230 return _quotient_graph( |
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231 G, partition, edge_relation, node_data, edge_data, relabel, create_using |
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232 ) |
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233 |
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234 # If the user provided partition as a collection of sets. Then we |
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235 # need to check if partition covers all of G nodes. If the answer |
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236 # is 'No' then we need to prepare suitable subgraph view. |
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237 partition_nodes = set().union(*partition) |
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238 if len(partition_nodes) != len(G): |
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239 G = G.subgraph(partition_nodes) |
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240 return _quotient_graph( |
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241 G, partition, edge_relation, node_data, edge_data, relabel, create_using |
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242 ) |
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243 |
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244 |
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245 def _quotient_graph( |
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246 G, |
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247 partition, |
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248 edge_relation=None, |
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249 node_data=None, |
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250 edge_data=None, |
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251 relabel=False, |
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252 create_using=None, |
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253 ): |
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254 # Each node in the graph must be in exactly one block. |
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255 if any(sum(1 for b in partition if v in b) != 1 for v in G): |
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256 raise NetworkXException("each node must be in exactly one block") |
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257 if create_using is None: |
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258 H = G.__class__() |
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259 else: |
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260 H = nx.empty_graph(0, create_using) |
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261 # By default set some basic information about the subgraph that each block |
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262 # represents on the nodes in the quotient graph. |
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263 if node_data is None: |
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264 |
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265 def node_data(b): |
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266 S = G.subgraph(b) |
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267 return dict( |
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268 graph=S, nnodes=len(S), nedges=S.number_of_edges(), density=density(S) |
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269 ) |
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270 |
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271 # Each block of the partition becomes a node in the quotient graph. |
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272 partition = [frozenset(b) for b in partition] |
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273 H.add_nodes_from((b, node_data(b)) for b in partition) |
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274 # By default, the edge relation is the relation defined as follows. B is |
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275 # adjacent to C if a node in B is adjacent to a node in C, according to the |
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276 # edge set of G. |
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277 # |
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278 # This is not a particularly efficient implementation of this relation: |
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279 # there are O(n^2) pairs to check and each check may require O(log n) time |
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280 # (to check set membership). This can certainly be parallelized. |
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281 if edge_relation is None: |
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282 |
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283 def edge_relation(b, c): |
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284 return any(v in G[u] for u, v in product(b, c)) |
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285 |
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286 # By default, sum the weights of the edges joining pairs of nodes across |
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287 # blocks to get the weight of the edge joining those two blocks. |
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288 if edge_data is None: |
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289 |
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290 def edge_data(b, c): |
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291 edgedata = ( |
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292 d |
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293 for u, v, d in G.edges(b | c, data=True) |
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294 if (u in b and v in c) or (u in c and v in b) |
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295 ) |
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296 return {"weight": sum(d.get("weight", 1) for d in edgedata)} |
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297 |
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298 block_pairs = permutations(H, 2) if H.is_directed() else combinations(H, 2) |
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299 # In a multigraph, add one edge in the quotient graph for each edge |
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300 # in the original graph. |
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301 if H.is_multigraph(): |
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302 edges = chaini( |
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303 ( |
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304 (b, c, G.get_edge_data(u, v, default={})) |
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305 for u, v in product(b, c) |
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306 if v in G[u] |
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307 ) |
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308 for b, c in block_pairs |
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309 if edge_relation(b, c) |
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310 ) |
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311 # In a simple graph, apply the edge data function to each pair of |
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312 # blocks to determine the edge data attributes to apply to each edge |
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313 # in the quotient graph. |
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314 else: |
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315 edges = ( |
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316 (b, c, edge_data(b, c)) for (b, c) in block_pairs if edge_relation(b, c) |
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317 ) |
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318 H.add_edges_from(edges) |
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319 # If requested by the user, relabel the nodes to be integers, |
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320 # numbered in increasing order from zero in the same order as the |
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321 # iteration order of `partition`. |
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322 if relabel: |
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323 # Can't use nx.convert_node_labels_to_integers() here since we |
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324 # want the order of iteration to be the same for backward |
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325 # compatibility with the nx.blockmodel() function. |
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326 labels = {b: i for i, b in enumerate(partition)} |
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327 H = nx.relabel_nodes(H, labels) |
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328 return H |
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329 |
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330 |
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331 def contracted_nodes(G, u, v, self_loops=True, copy=True): |
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332 """Returns the graph that results from contracting `u` and `v`. |
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333 |
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334 Node contraction identifies the two nodes as a single node incident to any |
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335 edge that was incident to the original two nodes. |
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336 |
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337 Parameters |
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338 ---------- |
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339 G : NetworkX graph |
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340 The graph whose nodes will be contracted. |
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341 |
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342 u, v : nodes |
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343 Must be nodes in `G`. |
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344 |
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345 self_loops : Boolean |
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346 If this is True, any edges joining `u` and `v` in `G` become |
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347 self-loops on the new node in the returned graph. |
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348 |
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349 copy : Boolean |
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350 If this is True (default True), make a copy of |
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351 `G` and return that instead of directly changing `G`. |
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352 |
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353 Returns |
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354 ------- |
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355 Networkx graph |
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356 If Copy is True: |
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357 A new graph object of the same type as `G` (leaving `G` unmodified) |
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358 with `u` and `v` identified in a single node. The right node `v` |
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359 will be merged into the node `u`, so only `u` will appear in the |
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360 returned graph. |
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361 if Copy is False: |
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362 Modifies `G` with `u` and `v` identified in a single node. |
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363 The right node `v` will be merged into the node `u`, so |
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364 only `u` will appear in the returned graph. |
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365 |
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366 Notes |
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367 ----- |
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368 For multigraphs, the edge keys for the realigned edges may |
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369 not be the same as the edge keys for the old edges. This is |
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370 natural because edge keys are unique only within each pair of nodes. |
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371 |
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372 Examples |
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373 -------- |
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374 Contracting two nonadjacent nodes of the cycle graph on four nodes `C_4` |
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375 yields the path graph (ignoring parallel edges):: |
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376 |
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377 >>> G = nx.cycle_graph(4) |
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378 >>> M = nx.contracted_nodes(G, 1, 3) |
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379 >>> P3 = nx.path_graph(3) |
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380 >>> nx.is_isomorphic(M, P3) |
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381 True |
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382 |
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383 >>> G = nx.MultiGraph(P3) |
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384 >>> M = nx.contracted_nodes(G, 0, 2) |
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385 >>> M.edges |
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386 MultiEdgeView([(0, 1, 0), (0, 1, 1)]) |
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387 |
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388 >>> G = nx.Graph([(1, 2), (2, 2)]) |
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389 >>> H = nx.contracted_nodes(G, 1, 2, self_loops=False) |
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390 >>> list(H.nodes()) |
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391 [1] |
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392 >>> list(H.edges()) |
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393 [(1, 1)] |
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394 |
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395 See also |
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396 -------- |
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397 contracted_edge |
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398 quotient_graph |
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399 |
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400 Notes |
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401 ----- |
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402 This function is also available as `identified_nodes`. |
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403 """ |
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404 # Copying has significant overhead and can be disabled if needed |
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405 if copy: |
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406 H = G.copy() |
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407 else: |
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408 H = G |
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409 |
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410 # edge code uses G.edges(v) instead of G.adj[v] to handle multiedges |
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411 if H.is_directed(): |
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412 in_edges = ( |
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413 (w if w != v else u, u, d) |
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414 for w, x, d in G.in_edges(v, data=True) |
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415 if self_loops or w != u |
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416 ) |
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417 out_edges = ( |
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418 (u, w if w != v else u, d) |
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419 for x, w, d in G.out_edges(v, data=True) |
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420 if self_loops or w != u |
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421 ) |
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422 new_edges = chain(in_edges, out_edges) |
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423 else: |
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424 new_edges = ( |
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425 (u, w if w != v else u, d) |
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426 for x, w, d in G.edges(v, data=True) |
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427 if self_loops or w != u |
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428 ) |
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429 |
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430 # If the H=G, the generators change as H changes |
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431 # This makes the new_edges independent of H |
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432 if not copy: |
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433 new_edges = list(new_edges) |
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434 |
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435 v_data = H.nodes[v] |
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436 H.remove_node(v) |
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437 H.add_edges_from(new_edges) |
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438 |
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439 if "contraction" in H.nodes[u]: |
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440 H.nodes[u]["contraction"][v] = v_data |
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441 else: |
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442 H.nodes[u]["contraction"] = {v: v_data} |
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443 return H |
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444 |
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445 |
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446 identified_nodes = contracted_nodes |
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447 |
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448 |
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449 def contracted_edge(G, edge, self_loops=True): |
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450 """Returns the graph that results from contracting the specified edge. |
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451 |
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452 Edge contraction identifies the two endpoints of the edge as a single node |
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453 incident to any edge that was incident to the original two nodes. A graph |
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454 that results from edge contraction is called a *minor* of the original |
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455 graph. |
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456 |
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457 Parameters |
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458 ---------- |
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459 G : NetworkX graph |
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460 The graph whose edge will be contracted. |
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461 |
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462 edge : tuple |
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463 Must be a pair of nodes in `G`. |
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464 |
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465 self_loops : Boolean |
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466 If this is True, any edges (including `edge`) joining the |
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467 endpoints of `edge` in `G` become self-loops on the new node in the |
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468 returned graph. |
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469 |
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470 Returns |
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471 ------- |
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472 Networkx graph |
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473 A new graph object of the same type as `G` (leaving `G` unmodified) |
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474 with endpoints of `edge` identified in a single node. The right node |
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475 of `edge` will be merged into the left one, so only the left one will |
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476 appear in the returned graph. |
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477 |
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478 Raises |
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479 ------ |
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480 ValueError |
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481 If `edge` is not an edge in `G`. |
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482 |
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483 Examples |
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484 -------- |
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485 Attempting to contract two nonadjacent nodes yields an error:: |
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486 |
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487 >>> G = nx.cycle_graph(4) |
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488 >>> nx.contracted_edge(G, (1, 3)) |
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489 Traceback (most recent call last): |
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490 ... |
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491 ValueError: Edge (1, 3) does not exist in graph G; cannot contract it |
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492 |
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493 Contracting two adjacent nodes in the cycle graph on *n* nodes yields the |
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494 cycle graph on *n - 1* nodes:: |
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495 |
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496 >>> C5 = nx.cycle_graph(5) |
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497 >>> C4 = nx.cycle_graph(4) |
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498 >>> M = nx.contracted_edge(C5, (0, 1), self_loops=False) |
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499 >>> nx.is_isomorphic(M, C4) |
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500 True |
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501 |
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502 See also |
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503 -------- |
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504 contracted_nodes |
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505 quotient_graph |
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506 |
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507 """ |
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508 if not G.has_edge(*edge): |
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509 raise ValueError(f"Edge {edge} does not exist in graph G; cannot contract it") |
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510 return contracted_nodes(G, *edge, self_loops=self_loops) |