Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/sitepackages/networkx/generators/line.py @ 0:4f3585e2f14b draft default tip
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author  shellac 

date  Mon, 22 Mar 2021 18:12:50 +0000 
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rev  line source 

0
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1 """Functions for generating line graphs.""" 
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2 from itertools import combinations 
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3 from collections import defaultdict 
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4 
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5 import networkx as nx 
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6 from networkx.utils import arbitrary_element, generate_unique_node 
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7 from networkx.utils.decorators import not_implemented_for 
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8 
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9 __all__ = ["line_graph", "inverse_line_graph"] 
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10 
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11 
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12 def line_graph(G, create_using=None): 
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13 r"""Returns the line graph of the graph or digraph `G`. 
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14 
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15 The line graph of a graph `G` has a node for each edge in `G` and an 
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16 edge joining those nodes if the two edges in `G` share a common node. For 
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17 directed graphs, nodes are adjacent exactly when the edges they represent 
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18 form a directed path of length two. 
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19 
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20 The nodes of the line graph are 2tuples of nodes in the original graph (or 
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21 3tuples for multigraphs, with the key of the edge as the third element). 
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22 
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23 For information about selfloops and more discussion, see the **Notes** 
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24 section below. 
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25 
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26 Parameters 
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27  
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28 G : graph 
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29 A NetworkX Graph, DiGraph, MultiGraph, or MultiDigraph. 
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30 create_using : NetworkX graph constructor, optional (default=nx.Graph) 
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31 Graph type to create. If graph instance, then cleared before populated. 
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32 
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33 Returns 
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34  
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35 L : graph 
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36 The line graph of G. 
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37 
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38 Examples 
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39  
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40 >>> G = nx.star_graph(3) 
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41 >>> L = nx.line_graph(G) 
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42 >>> print(sorted(map(sorted, L.edges()))) # makes a 3clique, K3 
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43 [[(0, 1), (0, 2)], [(0, 1), (0, 3)], [(0, 2), (0, 3)]] 
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44 
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45 Notes 
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46  
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47 Graph, node, and edge data are not propagated to the new graph. For 
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48 undirected graphs, the nodes in G must be sortable, otherwise the 
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49 constructed line graph may not be correct. 
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50 
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51 *Selfloops in undirected graphs* 
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52 
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53 For an undirected graph `G` without multiple edges, each edge can be 
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54 written as a set `\{u, v\}`. Its line graph `L` has the edges of `G` as 
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55 its nodes. If `x` and `y` are two nodes in `L`, then `\{x, y\}` is an edge 
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56 in `L` if and only if the intersection of `x` and `y` is nonempty. Thus, 
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57 the set of all edges is determined by the set of all pairwise intersections 
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58 of edges in `G`. 
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59 
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60 Trivially, every edge in G would have a nonzero intersection with itself, 
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61 and so every node in `L` should have a selfloop. This is not so 
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62 interesting, and the original context of line graphs was with simple 
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63 graphs, which had no selfloops or multiple edges. The line graph was also 
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64 meant to be a simple graph and thus, selfloops in `L` are not part of the 
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65 standard definition of a line graph. In a pairwise intersection matrix, 
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66 this is analogous to excluding the diagonal entries from the line graph 
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67 definition. 
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68 
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69 Selfloops and multiple edges in `G` add nodes to `L` in a natural way, and 
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70 do not require any fundamental changes to the definition. It might be 
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71 argued that the selfloops we excluded before should now be included. 
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72 However, the selfloops are still "trivial" in some sense and thus, are 
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73 usually excluded. 
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74 
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75 *Selfloops in directed graphs* 
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76 
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77 For a directed graph `G` without multiple edges, each edge can be written 
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78 as a tuple `(u, v)`. Its line graph `L` has the edges of `G` as its 
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79 nodes. If `x` and `y` are two nodes in `L`, then `(x, y)` is an edge in `L` 
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80 if and only if the tail of `x` matches the head of `y`, for example, if `x 
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81 = (a, b)` and `y = (b, c)` for some vertices `a`, `b`, and `c` in `G`. 
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82 
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83 Due to the directed nature of the edges, it is no longer the case that 
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84 every edge in `G` should have a selfloop in `L`. Now, the only time 
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85 selfloops arise is if a node in `G` itself has a selfloop. So such 
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86 selfloops are no longer "trivial" but instead, represent essential 
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87 features of the topology of `G`. For this reason, the historical 
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88 development of line digraphs is such that selfloops are included. When the 
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89 graph `G` has multiple edges, once again only superficial changes are 
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90 required to the definition. 
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91 
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92 References 
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93  
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94 * Harary, Frank, and Norman, Robert Z., "Some properties of line digraphs", 
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95 Rend. Circ. Mat. Palermo, II. Ser. 9 (1960), 161168. 
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96 * Hemminger, R. L.; Beineke, L. W. (1978), "Line graphs and line digraphs", 
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97 in Beineke, L. W.; Wilson, R. J., Selected Topics in Graph Theory, 
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98 Academic Press Inc., pp. 271305. 
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99 
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100 """ 
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101 if G.is_directed(): 
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102 L = _lg_directed(G, create_using=create_using) 
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103 else: 
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104 L = _lg_undirected(G, selfloops=False, create_using=create_using) 
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105 return L 
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106 
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107 
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108 def _node_func(G): 
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109 """Returns a function which returns a sorted node for line graphs. 
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110 
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111 When constructing a line graph for undirected graphs, we must normalize 
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112 the ordering of nodes as they appear in the edge. 
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113 
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114 """ 
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115 if G.is_multigraph(): 
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116 
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117 def sorted_node(u, v, key): 
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118 return (u, v, key) if u <= v else (v, u, key) 
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119 
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120 else: 
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121 
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122 def sorted_node(u, v): 
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123 return (u, v) if u <= v else (v, u) 
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124 
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125 return sorted_node 
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126 
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127 
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128 def _edge_func(G): 
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129 """Returns the edges from G, handling keys for multigraphs as necessary. 
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130 
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131 """ 
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132 if G.is_multigraph(): 
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133 
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134 def get_edges(nbunch=None): 
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135 return G.edges(nbunch, keys=True) 
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136 
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137 else: 
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138 
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139 def get_edges(nbunch=None): 
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140 return G.edges(nbunch) 
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141 
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142 return get_edges 
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143 
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144 
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145 def _sorted_edge(u, v): 
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146 """Returns a sorted edge. 
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147 
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148 During the construction of a line graph for undirected graphs, the data 
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149 structure can be a multigraph even though the line graph will never have 
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150 multiple edges between its nodes. For this reason, we must make sure not 
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151 to add any edge more than once. This requires that we build up a list of 
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152 edges to add and then remove all duplicates. And so, we must normalize 
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153 the representation of the edges. 
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154 
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155 """ 
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156 return (u, v) if u <= v else (v, u) 
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157 
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158 
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159 def _lg_directed(G, create_using=None): 
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160 """Returns the line graph L of the (multi)digraph G. 
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161 
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162 Edges in G appear as nodes in L, represented as tuples of the form (u,v) 
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163 or (u,v,key) if G is a multidigraph. A node in L corresponding to the edge 
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164 (u,v) is connected to every node corresponding to an edge (v,w). 
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165 
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166 Parameters 
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167  
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168 G : digraph 
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169 A directed graph or directed multigraph. 
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170 create_using : NetworkX graph constructor, optional 
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171 Graph type to create. If graph instance, then cleared before populated. 
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172 Default is to use the same graph class as `G`. 
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173 
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174 """ 
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175 L = nx.empty_graph(0, create_using, default=G.__class__) 
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176 
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177 # Create a graph specific edge function. 
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178 get_edges = _edge_func(G) 
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179 
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180 for from_node in get_edges(): 
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181 # from_node is: (u,v) or (u,v,key) 
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182 L.add_node(from_node) 
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183 for to_node in get_edges(from_node[1]): 
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184 L.add_edge(from_node, to_node) 
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185 
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186 return L 
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187 
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188 
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189 def _lg_undirected(G, selfloops=False, create_using=None): 
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190 """Returns the line graph L of the (multi)graph G. 
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191 
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192 Edges in G appear as nodes in L, represented as sorted tuples of the form 
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193 (u,v), or (u,v,key) if G is a multigraph. A node in L corresponding to 
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194 the edge {u,v} is connected to every node corresponding to an edge that 
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195 involves u or v. 
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196 
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197 Parameters 
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198  
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199 G : graph 
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200 An undirected graph or multigraph. 
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201 selfloops : bool 
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202 If `True`, then selfloops are included in the line graph. If `False`, 
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203 they are excluded. 
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204 create_using : NetworkX graph constructor, optional (default=nx.Graph) 
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205 Graph type to create. If graph instance, then cleared before populated. 
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206 
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207 Notes 
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208  
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209 The standard algorithm for line graphs of undirected graphs does not 
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210 produce selfloops. 
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211 
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212 """ 
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213 L = nx.empty_graph(0, create_using, default=G.__class__) 
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214 
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215 # Graph specific functions for edges and sorted nodes. 
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216 get_edges = _edge_func(G) 
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217 sorted_node = _node_func(G) 
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218 
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219 # Determine if we include selfloops or not. 
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220 shift = 0 if selfloops else 1 
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221 
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222 edges = set() 
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223 for u in G: 
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224 # Label nodes as a sorted tuple of nodes in original graph. 
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225 nodes = [sorted_node(*x) for x in get_edges(u)] 
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226 
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227 if len(nodes) == 1: 
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228 # Then the edge will be an isolated node in L. 
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229 L.add_node(nodes[0]) 
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230 
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231 # Add a clique of `nodes` to graph. To prevent double adding edges, 
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232 # especially important for multigraphs, we store the edges in 
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233 # canonical form in a set. 
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234 for i, a in enumerate(nodes): 
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235 edges.update([_sorted_edge(a, b) for b in nodes[i + shift :]]) 
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236 
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237 L.add_edges_from(edges) 
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238 return L 
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239 
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240 
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241 @not_implemented_for("directed") 
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242 @not_implemented_for("multigraph") 
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243 def inverse_line_graph(G): 
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244 """ Returns the inverse line graph of graph G. 
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245 
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246 If H is a graph, and G is the line graph of H, such that H = L(G). 
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247 Then H is the inverse line graph of G. 
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248 
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249 Not all graphs are line graphs and these do not have an inverse line graph. 
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250 In these cases this generator returns a NetworkXError. 
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251 
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252 Parameters 
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253  
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254 G : graph 
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255 A NetworkX Graph 
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256 
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257 Returns 
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258  
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259 H : graph 
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260 The inverse line graph of G. 
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261 
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262 Raises 
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263  
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264 NetworkXNotImplemented 
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265 If G is directed or a multigraph 
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266 
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267 NetworkXError 
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268 If G is not a line graph 
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269 
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270 Notes 
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271  
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272 This is an implementation of the Roussopoulos algorithm. 
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273 
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274 If G consists of multiple components, then the algorithm doesn't work. 
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275 You should invert every component seperately: 
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276 
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277 >>> K5 = nx.complete_graph(5) 
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278 >>> P4 = nx.Graph([("a", "b"), ("b", "c"), ("c", "d")]) 
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279 >>> G = nx.union(K5, P4) 
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280 >>> root_graphs = [] 
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281 >>> for comp in nx.connected_components(G): 
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282 ... root_graphs.append(nx.inverse_line_graph(G.subgraph(comp))) 
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283 >>> len(root_graphs) 
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284 2 
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285 
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286 References 
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287  
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288 * Roussopolous, N, "A max {m, n} algorithm for determining the graph H from 
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289 its line graph G", Information Processing Letters 2, (1973), 108112. 
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290 
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291 """ 
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292 if G.number_of_nodes() == 0: 
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293 a = generate_unique_node() 
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294 H = nx.Graph() 
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295 H.add_node(a) 
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296 return H 
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297 elif G.number_of_nodes() == 1: 
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298 v = list(G)[0] 
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299 a = (v, 0) 
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300 b = (v, 1) 
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301 H = nx.Graph([(a, b)]) 
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302 return H 
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303 elif G.number_of_nodes() > 1 and G.number_of_edges() == 0: 
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304 msg = ( 
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305 "inverse_line_graph() doesn't work on an edgeless graph. " 
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306 "Please use this function on each component seperately." 
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307 ) 
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308 raise nx.NetworkXError(msg) 
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309 
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310 starting_cell = _select_starting_cell(G) 
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311 P = _find_partition(G, starting_cell) 
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312 # count how many times each vertex appears in the partition set 
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313 P_count = {u: 0 for u in G.nodes()} 
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314 for p in P: 
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315 for u in p: 
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316 P_count[u] += 1 
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317 
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318 if max(P_count.values()) > 2: 
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319 msg = "G is not a line graph (vertex found in more " "than two partition cells)" 
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320 raise nx.NetworkXError(msg) 
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321 W = tuple([(u,) for u in P_count if P_count[u] == 1]) 
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322 H = nx.Graph() 
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323 H.add_nodes_from(P) 
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324 H.add_nodes_from(W) 
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325 for a, b in combinations(H.nodes(), 2): 
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326 if len(set(a).intersection(set(b))) > 0: 
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327 H.add_edge(a, b) 
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328 return H 
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329 
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330 
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331 def _triangles(G, e): 
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332 """ Return list of all triangles containing edge e""" 
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333 u, v = e 
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334 if u not in G: 
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335 raise nx.NetworkXError(f"Vertex {u} not in graph") 
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336 if v not in G[u]: 
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337 raise nx.NetworkXError(f"Edge ({u}, {v}) not in graph") 
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338 triangle_list = [] 
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339 for x in G[u]: 
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340 if x in G[v]: 
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341 triangle_list.append((u, v, x)) 
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342 return triangle_list 
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343 
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344 
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345 def _odd_triangle(G, T): 
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346 """ Test whether T is an odd triangle in G 
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347 
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348 Parameters 
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349  
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350 G : NetworkX Graph 
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351 T : 3tuple of vertices forming triangle in G 
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352 
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353 Returns 
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354  
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355 True is T is an odd triangle 
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356 False otherwise 
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357 
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358 Raises 
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359  
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360 NetworkXError 
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361 T is not a triangle in G 
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362 
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363 Notes 
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364  
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365 An odd triangle is one in which there exists another vertex in G which is 
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366 adjacent to either exactly one or exactly all three of the vertices in the 
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367 triangle. 
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368 
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369 """ 
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370 for u in T: 
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371 if u not in G.nodes(): 
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372 raise nx.NetworkXError(f"Vertex {u} not in graph") 
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373 for e in list(combinations(T, 2)): 
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374 if e[0] not in G[e[1]]: 
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375 raise nx.NetworkXError(f"Edge ({e[0]}, {e[1]}) not in graph") 
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376 
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377 T_neighbors = defaultdict(int) 
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378 for t in T: 
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379 for v in G[t]: 
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380 if v not in T: 
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381 T_neighbors[v] += 1 
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382 for v in T_neighbors: 
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383 if T_neighbors[v] in [1, 3]: 
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384 return True 
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385 return False 
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386 
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387 
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388 def _find_partition(G, starting_cell): 
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389 """ Find a partition of the vertices of G into cells of complete graphs 
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390 
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391 Parameters 
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392  
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393 G : NetworkX Graph 
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394 starting_cell : tuple of vertices in G which form a cell 
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395 
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396 Returns 
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397  
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398 List of tuples of vertices of G 
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399 
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400 Raises 
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401  
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402 NetworkXError 
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403 If a cell is not a complete subgraph then G is not a line graph 
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404 """ 
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405 G_partition = G.copy() 
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406 P = [starting_cell] # partition set 
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407 G_partition.remove_edges_from(list(combinations(starting_cell, 2))) 
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408 # keep list of partitioned nodes which might have an edge in G_partition 
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409 partitioned_vertices = list(starting_cell) 
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410 while G_partition.number_of_edges() > 0: 
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411 # there are still edges left and so more cells to be made 
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412 u = partitioned_vertices[1] 
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413 deg_u = len(G_partition[u]) 
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414 if deg_u == 0: 
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415 # if u has no edges left in G_partition then we have found 
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416 # all of its cells so we do not need to keep looking 
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417 partitioned_vertices.pop() 
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418 else: 
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419 # if u still has edges then we need to find its other cell 
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420 # this other cell must be a complete subgraph or else G is 
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421 # not a line graph 
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422 new_cell = [u] + list(G_partition[u]) 
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423 for u in new_cell: 
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424 for v in new_cell: 
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425 if (u != v) and (v not in G_partition[u]): 
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426 msg = ( 
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427 "G is not a line graph" 
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428 "(partition cell not a complete subgraph)" 
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429 ) 
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430 raise nx.NetworkXError(msg) 
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431 P.append(tuple(new_cell)) 
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432 G_partition.remove_edges_from(list(combinations(new_cell, 2))) 
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433 partitioned_vertices += new_cell 
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434 return P 
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435 
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436 
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437 def _select_starting_cell(G, starting_edge=None): 
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438 """ Select a cell to initiate _find_partition 
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439 
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440 Parameters 
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441  
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442 G : NetworkX Graph 
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443 starting_edge: an edge to build the starting cell from 
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444 
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445 Returns 
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446  
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447 Tuple of vertices in G 
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448 
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449 Raises 
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450  
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451 NetworkXError 
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452 If it is determined that G is not a line graph 
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453 
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454 Notes 
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455  
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456 If starting edge not specified then pick an arbitrary edge  doesn't 
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457 matter which. However, this function may call itself requiring a 
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458 specific starting edge. Note that the r, s notation for counting 
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459 triangles is the same as in the Roussopoulos paper cited above. 
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460 """ 
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461 if starting_edge is None: 
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462 e = arbitrary_element(list(G.edges())) 
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463 else: 
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464 e = starting_edge 
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465 if e[0] not in G[e[1]]: 
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466 msg = f"starting_edge ({e[0]}, {e[1]}) is not in the Graph" 
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467 raise nx.NetworkXError(msg) 
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468 e_triangles = _triangles(G, e) 
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469 r = len(e_triangles) 
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470 if r == 0: 
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471 # there are no triangles containing e, so the starting cell is just e 
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472 starting_cell = e 
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473 elif r == 1: 
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474 # there is exactly one triangle, T, containing e. If other 2 edges 
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475 # of T belong only to this triangle then T is starting cell 
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476 T = e_triangles[0] 
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477 a, b, c = T 
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478 # ab was original edge so check the other 2 edges 
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479 ac_edges = [x for x in _triangles(G, (a, c))] 
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480 bc_edges = [x for x in _triangles(G, (b, c))] 
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481 if len(ac_edges) == 1: 
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482 if len(bc_edges) == 1: 
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483 starting_cell = T 
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484 else: 
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485 return _select_starting_cell(G, starting_edge=(b, c)) 
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486 else: 
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487 return _select_starting_cell(G, starting_edge=(a, c)) 
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488 else: 
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489 # r >= 2 so we need to count the number of odd triangles, s 
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490 s = 0 
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491 odd_triangles = [] 
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492 for T in e_triangles: 
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493 if _odd_triangle(G, T): 
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494 s += 1 
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495 odd_triangles.append(T) 
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496 if r == 2 and s == 0: 
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497 # in this case either triangle works, so just use T 
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498 starting_cell = T 
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499 elif r  1 <= s <= r: 
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500 # check if odd triangles containing e form complete subgraph 
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501 # there must be exactly s+2 of them 
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502 # and they must all be connected 
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503 triangle_nodes = set() 
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504 for T in odd_triangles: 
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505 for x in T: 
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506 triangle_nodes.add(x) 
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507 if len(triangle_nodes) == s + 2: 
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508 for u in triangle_nodes: 
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509 for v in triangle_nodes: 
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510 if u != v and (v not in G[u]): 
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511 msg = ( 
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512 "G is not a line graph (odd triangles " 
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513 "do not form complete subgraph)" 
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514 ) 
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515 raise nx.NetworkXError(msg) 
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516 # otherwise then we can use this as the starting cell 
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517 starting_cell = tuple(triangle_nodes) 
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518 else: 
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519 msg = ( 
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520 "G is not a line graph (odd triangles " 
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521 "do not form complete subgraph)" 
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522 ) 
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523 raise nx.NetworkXError(msg) 
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524 else: 
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525 msg = ( 
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526 "G is not a line graph (incorrect number of " 
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527 "odd triangles around starting edge)" 
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528 ) 
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529 raise nx.NetworkXError(msg) 
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530 return starting_cell 