Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/similarity.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 measuring similarity using graph edit distance. |
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2 |
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3 The graph edit distance is the number of edge/node changes needed |
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4 to make two graphs isomorphic. |
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5 |
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6 The default algorithm/implementation is sub-optimal for some graphs. |
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7 The problem of finding the exact Graph Edit Distance (GED) is NP-hard |
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8 so it is often slow. If the simple interface `graph_edit_distance` |
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9 takes too long for your graph, try `optimize_graph_edit_distance` |
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10 and/or `optimize_edit_paths`. |
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11 |
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12 At the same time, I encourage capable people to investigate |
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13 alternative GED algorithms, in order to improve the choices available. |
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14 """ |
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15 import time |
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16 from itertools import product |
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17 import networkx as nx |
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18 |
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19 __all__ = [ |
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20 "graph_edit_distance", |
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21 "optimal_edit_paths", |
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22 "optimize_graph_edit_distance", |
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23 "optimize_edit_paths", |
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24 "simrank_similarity", |
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25 "simrank_similarity_numpy", |
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26 ] |
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27 |
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28 |
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29 def debug_print(*args, **kwargs): |
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30 print(*args, **kwargs) |
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31 |
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32 |
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33 def graph_edit_distance( |
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34 G1, |
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35 G2, |
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36 node_match=None, |
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37 edge_match=None, |
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38 node_subst_cost=None, |
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39 node_del_cost=None, |
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40 node_ins_cost=None, |
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41 edge_subst_cost=None, |
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42 edge_del_cost=None, |
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43 edge_ins_cost=None, |
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44 roots=None, |
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45 upper_bound=None, |
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46 timeout=None, |
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47 ): |
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48 """Returns GED (graph edit distance) between graphs G1 and G2. |
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49 |
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50 Graph edit distance is a graph similarity measure analogous to |
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51 Levenshtein distance for strings. It is defined as minimum cost |
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52 of edit path (sequence of node and edge edit operations) |
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53 transforming graph G1 to graph isomorphic to G2. |
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54 |
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55 Parameters |
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56 ---------- |
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57 G1, G2: graphs |
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58 The two graphs G1 and G2 must be of the same type. |
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59 |
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60 node_match : callable |
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61 A function that returns True if node n1 in G1 and n2 in G2 |
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62 should be considered equal during matching. |
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63 |
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64 The function will be called like |
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65 |
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66 node_match(G1.nodes[n1], G2.nodes[n2]). |
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67 |
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68 That is, the function will receive the node attribute |
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69 dictionaries for n1 and n2 as inputs. |
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70 |
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71 Ignored if node_subst_cost is specified. If neither |
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72 node_match nor node_subst_cost are specified then node |
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73 attributes are not considered. |
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74 |
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75 edge_match : callable |
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76 A function that returns True if the edge attribute dictionaries |
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77 for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should |
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78 be considered equal during matching. |
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79 |
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80 The function will be called like |
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81 |
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82 edge_match(G1[u1][v1], G2[u2][v2]). |
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83 |
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84 That is, the function will receive the edge attribute |
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85 dictionaries of the edges under consideration. |
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86 |
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87 Ignored if edge_subst_cost is specified. If neither |
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88 edge_match nor edge_subst_cost are specified then edge |
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89 attributes are not considered. |
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90 |
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91 node_subst_cost, node_del_cost, node_ins_cost : callable |
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92 Functions that return the costs of node substitution, node |
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93 deletion, and node insertion, respectively. |
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94 |
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95 The functions will be called like |
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96 |
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97 node_subst_cost(G1.nodes[n1], G2.nodes[n2]), |
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98 node_del_cost(G1.nodes[n1]), |
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99 node_ins_cost(G2.nodes[n2]). |
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100 |
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101 That is, the functions will receive the node attribute |
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102 dictionaries as inputs. The functions are expected to return |
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103 positive numeric values. |
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104 |
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105 Function node_subst_cost overrides node_match if specified. |
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106 If neither node_match nor node_subst_cost are specified then |
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107 default node substitution cost of 0 is used (node attributes |
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108 are not considered during matching). |
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109 |
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110 If node_del_cost is not specified then default node deletion |
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111 cost of 1 is used. If node_ins_cost is not specified then |
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112 default node insertion cost of 1 is used. |
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113 |
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114 edge_subst_cost, edge_del_cost, edge_ins_cost : callable |
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115 Functions that return the costs of edge substitution, edge |
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116 deletion, and edge insertion, respectively. |
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117 |
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118 The functions will be called like |
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119 |
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120 edge_subst_cost(G1[u1][v1], G2[u2][v2]), |
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121 edge_del_cost(G1[u1][v1]), |
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122 edge_ins_cost(G2[u2][v2]). |
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123 |
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124 That is, the functions will receive the edge attribute |
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125 dictionaries as inputs. The functions are expected to return |
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126 positive numeric values. |
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127 |
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128 Function edge_subst_cost overrides edge_match if specified. |
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129 If neither edge_match nor edge_subst_cost are specified then |
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130 default edge substitution cost of 0 is used (edge attributes |
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131 are not considered during matching). |
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132 |
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133 If edge_del_cost is not specified then default edge deletion |
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134 cost of 1 is used. If edge_ins_cost is not specified then |
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135 default edge insertion cost of 1 is used. |
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136 |
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137 roots : 2-tuple |
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138 Tuple where first element is a node in G1 and the second |
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139 is a node in G2. |
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140 These nodes are forced to be matched in the comparison to |
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141 allow comparison between rooted graphs. |
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142 |
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143 upper_bound : numeric |
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144 Maximum edit distance to consider. Return None if no edit |
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145 distance under or equal to upper_bound exists. |
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146 |
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147 timeout : numeric |
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148 Maximum number of seconds to execute. |
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149 After timeout is met, the current best GED is returned. |
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150 |
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151 Examples |
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152 -------- |
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153 >>> G1 = nx.cycle_graph(6) |
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154 >>> G2 = nx.wheel_graph(7) |
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155 >>> nx.graph_edit_distance(G1, G2) |
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156 7.0 |
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157 |
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158 >>> G1 = nx.star_graph(5) |
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159 >>> G2 = nx.star_graph(5) |
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160 >>> nx.graph_edit_distance(G1, G2, roots=(0, 0)) |
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161 0.0 |
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162 >>> nx.graph_edit_distance(G1, G2, roots=(1, 0)) |
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163 8.0 |
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164 |
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165 See Also |
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166 -------- |
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167 optimal_edit_paths, optimize_graph_edit_distance, |
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168 |
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169 is_isomorphic (test for graph edit distance of 0) |
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170 |
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171 References |
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172 ---------- |
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173 .. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick |
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174 Martineau. An Exact Graph Edit Distance Algorithm for Solving |
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175 Pattern Recognition Problems. 4th International Conference on |
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176 Pattern Recognition Applications and Methods 2015, Jan 2015, |
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177 Lisbon, Portugal. 2015, |
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178 <10.5220/0005209202710278>. <hal-01168816> |
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179 https://hal.archives-ouvertes.fr/hal-01168816 |
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180 |
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181 """ |
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182 bestcost = None |
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183 for vertex_path, edge_path, cost in optimize_edit_paths( |
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184 G1, |
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185 G2, |
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186 node_match, |
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187 edge_match, |
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188 node_subst_cost, |
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189 node_del_cost, |
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190 node_ins_cost, |
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191 edge_subst_cost, |
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192 edge_del_cost, |
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193 edge_ins_cost, |
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194 upper_bound, |
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195 True, |
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196 roots, |
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197 timeout, |
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198 ): |
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199 # assert bestcost is None or cost < bestcost |
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200 bestcost = cost |
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201 return bestcost |
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202 |
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203 |
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204 def optimal_edit_paths( |
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205 G1, |
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206 G2, |
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207 node_match=None, |
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208 edge_match=None, |
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209 node_subst_cost=None, |
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210 node_del_cost=None, |
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211 node_ins_cost=None, |
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212 edge_subst_cost=None, |
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213 edge_del_cost=None, |
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214 edge_ins_cost=None, |
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215 upper_bound=None, |
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216 ): |
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217 """Returns all minimum-cost edit paths transforming G1 to G2. |
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218 |
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219 Graph edit path is a sequence of node and edge edit operations |
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220 transforming graph G1 to graph isomorphic to G2. Edit operations |
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221 include substitutions, deletions, and insertions. |
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222 |
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223 Parameters |
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224 ---------- |
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225 G1, G2: graphs |
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226 The two graphs G1 and G2 must be of the same type. |
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227 |
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228 node_match : callable |
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229 A function that returns True if node n1 in G1 and n2 in G2 |
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230 should be considered equal during matching. |
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231 |
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232 The function will be called like |
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233 |
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234 node_match(G1.nodes[n1], G2.nodes[n2]). |
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235 |
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236 That is, the function will receive the node attribute |
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237 dictionaries for n1 and n2 as inputs. |
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238 |
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239 Ignored if node_subst_cost is specified. If neither |
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240 node_match nor node_subst_cost are specified then node |
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241 attributes are not considered. |
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242 |
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243 edge_match : callable |
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244 A function that returns True if the edge attribute dictionaries |
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245 for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should |
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246 be considered equal during matching. |
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247 |
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248 The function will be called like |
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249 |
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250 edge_match(G1[u1][v1], G2[u2][v2]). |
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251 |
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252 That is, the function will receive the edge attribute |
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253 dictionaries of the edges under consideration. |
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254 |
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255 Ignored if edge_subst_cost is specified. If neither |
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256 edge_match nor edge_subst_cost are specified then edge |
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257 attributes are not considered. |
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258 |
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259 node_subst_cost, node_del_cost, node_ins_cost : callable |
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260 Functions that return the costs of node substitution, node |
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261 deletion, and node insertion, respectively. |
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262 |
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263 The functions will be called like |
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264 |
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265 node_subst_cost(G1.nodes[n1], G2.nodes[n2]), |
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266 node_del_cost(G1.nodes[n1]), |
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267 node_ins_cost(G2.nodes[n2]). |
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268 |
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269 That is, the functions will receive the node attribute |
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270 dictionaries as inputs. The functions are expected to return |
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271 positive numeric values. |
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272 |
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273 Function node_subst_cost overrides node_match if specified. |
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274 If neither node_match nor node_subst_cost are specified then |
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275 default node substitution cost of 0 is used (node attributes |
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276 are not considered during matching). |
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277 |
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278 If node_del_cost is not specified then default node deletion |
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279 cost of 1 is used. If node_ins_cost is not specified then |
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280 default node insertion cost of 1 is used. |
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281 |
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282 edge_subst_cost, edge_del_cost, edge_ins_cost : callable |
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283 Functions that return the costs of edge substitution, edge |
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284 deletion, and edge insertion, respectively. |
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285 |
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286 The functions will be called like |
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287 |
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288 edge_subst_cost(G1[u1][v1], G2[u2][v2]), |
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289 edge_del_cost(G1[u1][v1]), |
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290 edge_ins_cost(G2[u2][v2]). |
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291 |
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292 That is, the functions will receive the edge attribute |
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293 dictionaries as inputs. The functions are expected to return |
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294 positive numeric values. |
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295 |
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296 Function edge_subst_cost overrides edge_match if specified. |
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297 If neither edge_match nor edge_subst_cost are specified then |
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298 default edge substitution cost of 0 is used (edge attributes |
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299 are not considered during matching). |
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300 |
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301 If edge_del_cost is not specified then default edge deletion |
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302 cost of 1 is used. If edge_ins_cost is not specified then |
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303 default edge insertion cost of 1 is used. |
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304 |
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305 upper_bound : numeric |
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306 Maximum edit distance to consider. |
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307 |
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308 Returns |
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309 ------- |
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310 edit_paths : list of tuples (node_edit_path, edge_edit_path) |
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311 node_edit_path : list of tuples (u, v) |
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312 edge_edit_path : list of tuples ((u1, v1), (u2, v2)) |
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313 |
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314 cost : numeric |
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315 Optimal edit path cost (graph edit distance). |
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316 |
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317 Examples |
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318 -------- |
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319 >>> G1 = nx.cycle_graph(4) |
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320 >>> G2 = nx.wheel_graph(5) |
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321 >>> paths, cost = nx.optimal_edit_paths(G1, G2) |
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322 >>> len(paths) |
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323 40 |
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324 >>> cost |
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325 5.0 |
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326 |
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327 See Also |
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328 -------- |
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329 graph_edit_distance, optimize_edit_paths |
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330 |
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331 References |
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332 ---------- |
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333 .. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick |
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334 Martineau. An Exact Graph Edit Distance Algorithm for Solving |
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335 Pattern Recognition Problems. 4th International Conference on |
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336 Pattern Recognition Applications and Methods 2015, Jan 2015, |
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337 Lisbon, Portugal. 2015, |
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338 <10.5220/0005209202710278>. <hal-01168816> |
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339 https://hal.archives-ouvertes.fr/hal-01168816 |
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340 |
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341 """ |
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342 paths = list() |
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343 bestcost = None |
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344 for vertex_path, edge_path, cost in optimize_edit_paths( |
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345 G1, |
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346 G2, |
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347 node_match, |
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348 edge_match, |
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349 node_subst_cost, |
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350 node_del_cost, |
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351 node_ins_cost, |
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352 edge_subst_cost, |
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353 edge_del_cost, |
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354 edge_ins_cost, |
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355 upper_bound, |
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356 False, |
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357 ): |
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358 # assert bestcost is None or cost <= bestcost |
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359 if bestcost is not None and cost < bestcost: |
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360 paths = list() |
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361 paths.append((vertex_path, edge_path)) |
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362 bestcost = cost |
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363 return paths, bestcost |
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364 |
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365 |
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366 def optimize_graph_edit_distance( |
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367 G1, |
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368 G2, |
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369 node_match=None, |
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370 edge_match=None, |
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371 node_subst_cost=None, |
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372 node_del_cost=None, |
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373 node_ins_cost=None, |
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374 edge_subst_cost=None, |
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375 edge_del_cost=None, |
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376 edge_ins_cost=None, |
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377 upper_bound=None, |
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378 ): |
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379 """Returns consecutive approximations of GED (graph edit distance) |
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380 between graphs G1 and G2. |
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381 |
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382 Graph edit distance is a graph similarity measure analogous to |
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383 Levenshtein distance for strings. It is defined as minimum cost |
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384 of edit path (sequence of node and edge edit operations) |
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385 transforming graph G1 to graph isomorphic to G2. |
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386 |
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387 Parameters |
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388 ---------- |
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389 G1, G2: graphs |
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390 The two graphs G1 and G2 must be of the same type. |
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391 |
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392 node_match : callable |
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393 A function that returns True if node n1 in G1 and n2 in G2 |
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394 should be considered equal during matching. |
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395 |
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396 The function will be called like |
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397 |
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398 node_match(G1.nodes[n1], G2.nodes[n2]). |
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399 |
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400 That is, the function will receive the node attribute |
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401 dictionaries for n1 and n2 as inputs. |
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402 |
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403 Ignored if node_subst_cost is specified. If neither |
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404 node_match nor node_subst_cost are specified then node |
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405 attributes are not considered. |
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406 |
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407 edge_match : callable |
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408 A function that returns True if the edge attribute dictionaries |
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409 for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should |
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410 be considered equal during matching. |
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411 |
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412 The function will be called like |
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413 |
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414 edge_match(G1[u1][v1], G2[u2][v2]). |
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415 |
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416 That is, the function will receive the edge attribute |
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417 dictionaries of the edges under consideration. |
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418 |
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419 Ignored if edge_subst_cost is specified. If neither |
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420 edge_match nor edge_subst_cost are specified then edge |
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421 attributes are not considered. |
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422 |
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423 node_subst_cost, node_del_cost, node_ins_cost : callable |
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424 Functions that return the costs of node substitution, node |
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425 deletion, and node insertion, respectively. |
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426 |
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427 The functions will be called like |
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428 |
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429 node_subst_cost(G1.nodes[n1], G2.nodes[n2]), |
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430 node_del_cost(G1.nodes[n1]), |
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431 node_ins_cost(G2.nodes[n2]). |
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432 |
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433 That is, the functions will receive the node attribute |
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434 dictionaries as inputs. The functions are expected to return |
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435 positive numeric values. |
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436 |
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437 Function node_subst_cost overrides node_match if specified. |
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438 If neither node_match nor node_subst_cost are specified then |
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439 default node substitution cost of 0 is used (node attributes |
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440 are not considered during matching). |
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441 |
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442 If node_del_cost is not specified then default node deletion |
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443 cost of 1 is used. If node_ins_cost is not specified then |
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444 default node insertion cost of 1 is used. |
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445 |
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446 edge_subst_cost, edge_del_cost, edge_ins_cost : callable |
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447 Functions that return the costs of edge substitution, edge |
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448 deletion, and edge insertion, respectively. |
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449 |
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450 The functions will be called like |
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451 |
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452 edge_subst_cost(G1[u1][v1], G2[u2][v2]), |
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453 edge_del_cost(G1[u1][v1]), |
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454 edge_ins_cost(G2[u2][v2]). |
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455 |
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456 That is, the functions will receive the edge attribute |
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457 dictionaries as inputs. The functions are expected to return |
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458 positive numeric values. |
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459 |
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460 Function edge_subst_cost overrides edge_match if specified. |
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461 If neither edge_match nor edge_subst_cost are specified then |
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462 default edge substitution cost of 0 is used (edge attributes |
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463 are not considered during matching). |
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464 |
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465 If edge_del_cost is not specified then default edge deletion |
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466 cost of 1 is used. If edge_ins_cost is not specified then |
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467 default edge insertion cost of 1 is used. |
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468 |
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469 upper_bound : numeric |
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470 Maximum edit distance to consider. |
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471 |
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472 Returns |
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473 ------- |
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474 Generator of consecutive approximations of graph edit distance. |
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475 |
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476 Examples |
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477 -------- |
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478 >>> G1 = nx.cycle_graph(6) |
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479 >>> G2 = nx.wheel_graph(7) |
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480 >>> for v in nx.optimize_graph_edit_distance(G1, G2): |
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481 ... minv = v |
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482 >>> minv |
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483 7.0 |
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484 |
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485 See Also |
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486 -------- |
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487 graph_edit_distance, optimize_edit_paths |
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488 |
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489 References |
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490 ---------- |
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491 .. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick |
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492 Martineau. An Exact Graph Edit Distance Algorithm for Solving |
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493 Pattern Recognition Problems. 4th International Conference on |
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494 Pattern Recognition Applications and Methods 2015, Jan 2015, |
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495 Lisbon, Portugal. 2015, |
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496 <10.5220/0005209202710278>. <hal-01168816> |
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497 https://hal.archives-ouvertes.fr/hal-01168816 |
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498 """ |
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499 for vertex_path, edge_path, cost in optimize_edit_paths( |
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500 G1, |
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501 G2, |
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502 node_match, |
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503 edge_match, |
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504 node_subst_cost, |
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505 node_del_cost, |
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506 node_ins_cost, |
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507 edge_subst_cost, |
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508 edge_del_cost, |
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509 edge_ins_cost, |
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510 upper_bound, |
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511 True, |
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512 ): |
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513 yield cost |
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514 |
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515 |
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516 def optimize_edit_paths( |
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517 G1, |
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518 G2, |
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519 node_match=None, |
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520 edge_match=None, |
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521 node_subst_cost=None, |
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522 node_del_cost=None, |
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523 node_ins_cost=None, |
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524 edge_subst_cost=None, |
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525 edge_del_cost=None, |
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526 edge_ins_cost=None, |
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527 upper_bound=None, |
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528 strictly_decreasing=True, |
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529 roots=None, |
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530 timeout=None, |
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531 ): |
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532 """GED (graph edit distance) calculation: advanced interface. |
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533 |
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534 Graph edit path is a sequence of node and edge edit operations |
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535 transforming graph G1 to graph isomorphic to G2. Edit operations |
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536 include substitutions, deletions, and insertions. |
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537 |
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538 Graph edit distance is defined as minimum cost of edit path. |
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539 |
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540 Parameters |
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541 ---------- |
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542 G1, G2: graphs |
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543 The two graphs G1 and G2 must be of the same type. |
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544 |
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545 node_match : callable |
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546 A function that returns True if node n1 in G1 and n2 in G2 |
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547 should be considered equal during matching. |
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548 |
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549 The function will be called like |
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550 |
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551 node_match(G1.nodes[n1], G2.nodes[n2]). |
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552 |
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553 That is, the function will receive the node attribute |
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554 dictionaries for n1 and n2 as inputs. |
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555 |
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556 Ignored if node_subst_cost is specified. If neither |
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557 node_match nor node_subst_cost are specified then node |
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558 attributes are not considered. |
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559 |
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560 edge_match : callable |
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561 A function that returns True if the edge attribute dictionaries |
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562 for the pair of nodes (u1, v1) in G1 and (u2, v2) in G2 should |
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563 be considered equal during matching. |
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564 |
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565 The function will be called like |
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566 |
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567 edge_match(G1[u1][v1], G2[u2][v2]). |
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568 |
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569 That is, the function will receive the edge attribute |
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570 dictionaries of the edges under consideration. |
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571 |
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572 Ignored if edge_subst_cost is specified. If neither |
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573 edge_match nor edge_subst_cost are specified then edge |
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574 attributes are not considered. |
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575 |
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576 node_subst_cost, node_del_cost, node_ins_cost : callable |
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577 Functions that return the costs of node substitution, node |
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578 deletion, and node insertion, respectively. |
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579 |
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580 The functions will be called like |
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581 |
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582 node_subst_cost(G1.nodes[n1], G2.nodes[n2]), |
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583 node_del_cost(G1.nodes[n1]), |
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584 node_ins_cost(G2.nodes[n2]). |
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585 |
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586 That is, the functions will receive the node attribute |
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587 dictionaries as inputs. The functions are expected to return |
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588 positive numeric values. |
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589 |
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590 Function node_subst_cost overrides node_match if specified. |
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591 If neither node_match nor node_subst_cost are specified then |
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592 default node substitution cost of 0 is used (node attributes |
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593 are not considered during matching). |
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594 |
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595 If node_del_cost is not specified then default node deletion |
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596 cost of 1 is used. If node_ins_cost is not specified then |
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597 default node insertion cost of 1 is used. |
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598 |
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599 edge_subst_cost, edge_del_cost, edge_ins_cost : callable |
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600 Functions that return the costs of edge substitution, edge |
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601 deletion, and edge insertion, respectively. |
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602 |
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603 The functions will be called like |
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604 |
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605 edge_subst_cost(G1[u1][v1], G2[u2][v2]), |
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606 edge_del_cost(G1[u1][v1]), |
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607 edge_ins_cost(G2[u2][v2]). |
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608 |
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609 That is, the functions will receive the edge attribute |
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610 dictionaries as inputs. The functions are expected to return |
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611 positive numeric values. |
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612 |
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613 Function edge_subst_cost overrides edge_match if specified. |
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614 If neither edge_match nor edge_subst_cost are specified then |
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615 default edge substitution cost of 0 is used (edge attributes |
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616 are not considered during matching). |
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617 |
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618 If edge_del_cost is not specified then default edge deletion |
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619 cost of 1 is used. If edge_ins_cost is not specified then |
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620 default edge insertion cost of 1 is used. |
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621 |
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622 upper_bound : numeric |
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623 Maximum edit distance to consider. |
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624 |
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625 strictly_decreasing : bool |
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626 If True, return consecutive approximations of strictly |
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627 decreasing cost. Otherwise, return all edit paths of cost |
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628 less than or equal to the previous minimum cost. |
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629 |
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630 roots : 2-tuple |
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631 Tuple where first element is a node in G1 and the second |
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632 is a node in G2. |
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633 These nodes are forced to be matched in the comparison to |
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634 allow comparison between rooted graphs. |
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635 |
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636 timeout : numeric |
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637 Maximum number of seconds to execute. |
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638 After timeout is met, the current best GED is returned. |
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639 |
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640 Returns |
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641 ------- |
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642 Generator of tuples (node_edit_path, edge_edit_path, cost) |
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643 node_edit_path : list of tuples (u, v) |
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644 edge_edit_path : list of tuples ((u1, v1), (u2, v2)) |
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645 cost : numeric |
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646 |
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647 See Also |
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648 -------- |
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649 graph_edit_distance, optimize_graph_edit_distance, optimal_edit_paths |
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650 |
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651 References |
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652 ---------- |
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653 .. [1] Zeina Abu-Aisheh, Romain Raveaux, Jean-Yves Ramel, Patrick |
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654 Martineau. An Exact Graph Edit Distance Algorithm for Solving |
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655 Pattern Recognition Problems. 4th International Conference on |
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656 Pattern Recognition Applications and Methods 2015, Jan 2015, |
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657 Lisbon, Portugal. 2015, |
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658 <10.5220/0005209202710278>. <hal-01168816> |
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659 https://hal.archives-ouvertes.fr/hal-01168816 |
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660 |
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661 """ |
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662 # TODO: support DiGraph |
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663 |
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664 import numpy as np |
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665 from scipy.optimize import linear_sum_assignment |
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666 |
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667 class CostMatrix: |
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668 def __init__(self, C, lsa_row_ind, lsa_col_ind, ls): |
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669 # assert C.shape[0] == len(lsa_row_ind) |
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670 # assert C.shape[1] == len(lsa_col_ind) |
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671 # assert len(lsa_row_ind) == len(lsa_col_ind) |
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672 # assert set(lsa_row_ind) == set(range(len(lsa_row_ind))) |
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673 # assert set(lsa_col_ind) == set(range(len(lsa_col_ind))) |
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674 # assert ls == C[lsa_row_ind, lsa_col_ind].sum() |
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675 self.C = C |
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676 self.lsa_row_ind = lsa_row_ind |
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677 self.lsa_col_ind = lsa_col_ind |
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678 self.ls = ls |
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679 |
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680 def make_CostMatrix(C, m, n): |
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681 # assert(C.shape == (m + n, m + n)) |
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682 lsa_row_ind, lsa_col_ind = linear_sum_assignment(C) |
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683 |
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684 # Fixup dummy assignments: |
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685 # each substitution i<->j should have dummy assignment m+j<->n+i |
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686 # NOTE: fast reduce of Cv relies on it |
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687 # assert len(lsa_row_ind) == len(lsa_col_ind) |
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688 indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind) |
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689 subst_ind = list(k for k, i, j in indexes if i < m and j < n) |
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690 indexes = zip(range(len(lsa_row_ind)), lsa_row_ind, lsa_col_ind) |
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691 dummy_ind = list(k for k, i, j in indexes if i >= m and j >= n) |
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692 # assert len(subst_ind) == len(dummy_ind) |
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693 lsa_row_ind[dummy_ind] = lsa_col_ind[subst_ind] + m |
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694 lsa_col_ind[dummy_ind] = lsa_row_ind[subst_ind] + n |
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695 |
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696 return CostMatrix( |
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697 C, lsa_row_ind, lsa_col_ind, C[lsa_row_ind, lsa_col_ind].sum() |
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698 ) |
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699 |
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700 def extract_C(C, i, j, m, n): |
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701 # assert(C.shape == (m + n, m + n)) |
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702 row_ind = [k in i or k - m in j for k in range(m + n)] |
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703 col_ind = [k in j or k - n in i for k in range(m + n)] |
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704 return C[row_ind, :][:, col_ind] |
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705 |
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706 def reduce_C(C, i, j, m, n): |
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707 # assert(C.shape == (m + n, m + n)) |
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708 row_ind = [k not in i and k - m not in j for k in range(m + n)] |
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709 col_ind = [k not in j and k - n not in i for k in range(m + n)] |
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710 return C[row_ind, :][:, col_ind] |
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711 |
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712 def reduce_ind(ind, i): |
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713 # assert set(ind) == set(range(len(ind))) |
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714 rind = ind[[k not in i for k in ind]] |
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715 for k in set(i): |
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716 rind[rind >= k] -= 1 |
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717 return rind |
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718 |
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719 def match_edges(u, v, pending_g, pending_h, Ce, matched_uv=[]): |
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720 """ |
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721 Parameters: |
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722 u, v: matched vertices, u=None or v=None for |
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723 deletion/insertion |
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724 pending_g, pending_h: lists of edges not yet mapped |
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725 Ce: CostMatrix of pending edge mappings |
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726 matched_uv: partial vertex edit path |
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727 list of tuples (u, v) of previously matched vertex |
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728 mappings u<->v, u=None or v=None for |
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729 deletion/insertion |
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730 |
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731 Returns: |
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732 list of (i, j): indices of edge mappings g<->h |
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733 localCe: local CostMatrix of edge mappings |
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734 (basically submatrix of Ce at cross of rows i, cols j) |
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735 """ |
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736 M = len(pending_g) |
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737 N = len(pending_h) |
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738 # assert Ce.C.shape == (M + N, M + N) |
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739 |
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740 g_ind = [ |
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741 i |
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742 for i in range(M) |
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743 if pending_g[i][:2] == (u, u) |
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744 or any(pending_g[i][:2] in ((p, u), (u, p)) for p, q in matched_uv) |
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745 ] |
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746 h_ind = [ |
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747 j |
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748 for j in range(N) |
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749 if pending_h[j][:2] == (v, v) |
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750 or any(pending_h[j][:2] in ((q, v), (v, q)) for p, q in matched_uv) |
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751 ] |
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752 m = len(g_ind) |
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753 n = len(h_ind) |
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754 |
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755 if m or n: |
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756 C = extract_C(Ce.C, g_ind, h_ind, M, N) |
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757 # assert C.shape == (m + n, m + n) |
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758 |
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759 # Forbid structurally invalid matches |
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760 # NOTE: inf remembered from Ce construction |
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761 for k, i in zip(range(m), g_ind): |
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762 g = pending_g[i][:2] |
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763 for l, j in zip(range(n), h_ind): |
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764 h = pending_h[j][:2] |
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765 if nx.is_directed(G1) or nx.is_directed(G2): |
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766 if any( |
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767 g == (p, u) and h == (q, v) or g == (u, p) and h == (v, q) |
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768 for p, q in matched_uv |
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769 ): |
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770 continue |
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771 else: |
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772 if any( |
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773 g in ((p, u), (u, p)) and h in ((q, v), (v, q)) |
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774 for p, q in matched_uv |
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775 ): |
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776 continue |
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777 if g == (u, u): |
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778 continue |
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779 if h == (v, v): |
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780 continue |
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781 C[k, l] = inf |
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782 |
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783 localCe = make_CostMatrix(C, m, n) |
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784 ij = list( |
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785 ( |
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786 g_ind[k] if k < m else M + h_ind[l], |
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787 h_ind[l] if l < n else N + g_ind[k], |
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788 ) |
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789 for k, l in zip(localCe.lsa_row_ind, localCe.lsa_col_ind) |
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790 if k < m or l < n |
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|
791 ) |
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|
792 |
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|
793 else: |
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|
794 ij = [] |
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|
795 localCe = CostMatrix(np.empty((0, 0)), [], [], 0) |
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|
796 |
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797 return ij, localCe |
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|
798 |
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|
799 def reduce_Ce(Ce, ij, m, n): |
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800 if len(ij): |
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801 i, j = zip(*ij) |
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802 m_i = m - sum(1 for t in i if t < m) |
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803 n_j = n - sum(1 for t in j if t < n) |
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804 return make_CostMatrix(reduce_C(Ce.C, i, j, m, n), m_i, n_j) |
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|
805 else: |
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806 return Ce |
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807 |
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808 def get_edit_ops( |
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809 matched_uv, pending_u, pending_v, Cv, pending_g, pending_h, Ce, matched_cost |
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810 ): |
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811 """ |
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|
812 Parameters: |
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|
813 matched_uv: partial vertex edit path |
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814 list of tuples (u, v) of vertex mappings u<->v, |
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815 u=None or v=None for deletion/insertion |
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816 pending_u, pending_v: lists of vertices not yet mapped |
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817 Cv: CostMatrix of pending vertex mappings |
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818 pending_g, pending_h: lists of edges not yet mapped |
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819 Ce: CostMatrix of pending edge mappings |
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820 matched_cost: cost of partial edit path |
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821 |
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822 Returns: |
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823 sequence of |
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824 (i, j): indices of vertex mapping u<->v |
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825 Cv_ij: reduced CostMatrix of pending vertex mappings |
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826 (basically Cv with row i, col j removed) |
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827 list of (x, y): indices of edge mappings g<->h |
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828 Ce_xy: reduced CostMatrix of pending edge mappings |
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829 (basically Ce with rows x, cols y removed) |
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830 cost: total cost of edit operation |
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831 NOTE: most promising ops first |
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832 """ |
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833 m = len(pending_u) |
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834 n = len(pending_v) |
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835 # assert Cv.C.shape == (m + n, m + n) |
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836 |
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837 # 1) a vertex mapping from optimal linear sum assignment |
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838 i, j = min( |
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839 (k, l) for k, l in zip(Cv.lsa_row_ind, Cv.lsa_col_ind) if k < m or l < n |
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840 ) |
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841 xy, localCe = match_edges( |
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842 pending_u[i] if i < m else None, |
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843 pending_v[j] if j < n else None, |
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844 pending_g, |
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845 pending_h, |
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846 Ce, |
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847 matched_uv, |
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|
848 ) |
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849 Ce_xy = reduce_Ce(Ce, xy, len(pending_g), len(pending_h)) |
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850 # assert Ce.ls <= localCe.ls + Ce_xy.ls |
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851 if prune(matched_cost + Cv.ls + localCe.ls + Ce_xy.ls): |
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|
852 pass |
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853 else: |
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854 # get reduced Cv efficiently |
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855 Cv_ij = CostMatrix( |
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856 reduce_C(Cv.C, (i,), (j,), m, n), |
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parents:
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|
857 reduce_ind(Cv.lsa_row_ind, (i, m + j)), |
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858 reduce_ind(Cv.lsa_col_ind, (j, n + i)), |
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|
859 Cv.ls - Cv.C[i, j], |
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|
860 ) |
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861 yield (i, j), Cv_ij, xy, Ce_xy, Cv.C[i, j] + localCe.ls |
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|
862 |
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863 # 2) other candidates, sorted by lower-bound cost estimate |
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864 other = list() |
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865 fixed_i, fixed_j = i, j |
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866 if m <= n: |
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867 candidates = ( |
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868 (t, fixed_j) |
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869 for t in range(m + n) |
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870 if t != fixed_i and (t < m or t == m + fixed_j) |
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|
871 ) |
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872 else: |
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873 candidates = ( |
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874 (fixed_i, t) |
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875 for t in range(m + n) |
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876 if t != fixed_j and (t < n or t == n + fixed_i) |
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|
877 ) |
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878 for i, j in candidates: |
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879 if prune(matched_cost + Cv.C[i, j] + Ce.ls): |
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|
880 continue |
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881 Cv_ij = make_CostMatrix( |
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882 reduce_C(Cv.C, (i,), (j,), m, n), |
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883 m - 1 if i < m else m, |
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884 n - 1 if j < n else n, |
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|
885 ) |
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886 # assert Cv.ls <= Cv.C[i, j] + Cv_ij.ls |
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887 if prune(matched_cost + Cv.C[i, j] + Cv_ij.ls + Ce.ls): |
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888 continue |
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889 xy, localCe = match_edges( |
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890 pending_u[i] if i < m else None, |
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891 pending_v[j] if j < n else None, |
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892 pending_g, |
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893 pending_h, |
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894 Ce, |
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895 matched_uv, |
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896 ) |
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897 if prune(matched_cost + Cv.C[i, j] + Cv_ij.ls + localCe.ls): |
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898 continue |
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parents:
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899 Ce_xy = reduce_Ce(Ce, xy, len(pending_g), len(pending_h)) |
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900 # assert Ce.ls <= localCe.ls + Ce_xy.ls |
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901 if prune(matched_cost + Cv.C[i, j] + Cv_ij.ls + localCe.ls + Ce_xy.ls): |
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|
902 continue |
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|
903 other.append(((i, j), Cv_ij, xy, Ce_xy, Cv.C[i, j] + localCe.ls)) |
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|
904 |
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|
905 yield from sorted(other, key=lambda t: t[4] + t[1].ls + t[3].ls) |
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906 |
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|
907 def get_edit_paths( |
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908 matched_uv, |
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|
909 pending_u, |
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|
910 pending_v, |
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|
911 Cv, |
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|
912 matched_gh, |
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913 pending_g, |
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914 pending_h, |
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915 Ce, |
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|
916 matched_cost, |
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917 ): |
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918 """ |
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|
919 Parameters: |
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920 matched_uv: partial vertex edit path |
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921 list of tuples (u, v) of vertex mappings u<->v, |
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922 u=None or v=None for deletion/insertion |
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923 pending_u, pending_v: lists of vertices not yet mapped |
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924 Cv: CostMatrix of pending vertex mappings |
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925 matched_gh: partial edge edit path |
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926 list of tuples (g, h) of edge mappings g<->h, |
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927 g=None or h=None for deletion/insertion |
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928 pending_g, pending_h: lists of edges not yet mapped |
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929 Ce: CostMatrix of pending edge mappings |
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930 matched_cost: cost of partial edit path |
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931 |
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932 Returns: |
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|
933 sequence of (vertex_path, edge_path, cost) |
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934 vertex_path: complete vertex edit path |
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935 list of tuples (u, v) of vertex mappings u<->v, |
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936 u=None or v=None for deletion/insertion |
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|
937 edge_path: complete edge edit path |
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938 list of tuples (g, h) of edge mappings g<->h, |
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939 g=None or h=None for deletion/insertion |
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940 cost: total cost of edit path |
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941 NOTE: path costs are non-increasing |
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942 """ |
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943 # debug_print('matched-uv:', matched_uv) |
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944 # debug_print('matched-gh:', matched_gh) |
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945 # debug_print('matched-cost:', matched_cost) |
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946 # debug_print('pending-u:', pending_u) |
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947 # debug_print('pending-v:', pending_v) |
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948 # debug_print(Cv.C) |
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949 # assert list(sorted(G1.nodes)) == list(sorted(list(u for u, v in matched_uv if u is not None) + pending_u)) |
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950 # assert list(sorted(G2.nodes)) == list(sorted(list(v for u, v in matched_uv if v is not None) + pending_v)) |
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951 # debug_print('pending-g:', pending_g) |
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952 # debug_print('pending-h:', pending_h) |
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953 # debug_print(Ce.C) |
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954 # assert list(sorted(G1.edges)) == list(sorted(list(g for g, h in matched_gh if g is not None) + pending_g)) |
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955 # assert list(sorted(G2.edges)) == list(sorted(list(h for g, h in matched_gh if h is not None) + pending_h)) |
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956 # debug_print() |
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957 |
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958 if prune(matched_cost + Cv.ls + Ce.ls): |
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|
959 return |
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|
960 |
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961 if not max(len(pending_u), len(pending_v)): |
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962 # assert not len(pending_g) |
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963 # assert not len(pending_h) |
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964 # path completed! |
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965 # assert matched_cost <= maxcost.value |
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966 maxcost.value = min(maxcost.value, matched_cost) |
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967 yield matched_uv, matched_gh, matched_cost |
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968 |
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969 else: |
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970 edit_ops = get_edit_ops( |
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971 matched_uv, |
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972 pending_u, |
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973 pending_v, |
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974 Cv, |
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975 pending_g, |
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976 pending_h, |
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977 Ce, |
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978 matched_cost, |
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979 ) |
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980 for ij, Cv_ij, xy, Ce_xy, edit_cost in edit_ops: |
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981 i, j = ij |
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982 # assert Cv.C[i, j] + sum(Ce.C[t] for t in xy) == edit_cost |
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983 if prune(matched_cost + edit_cost + Cv_ij.ls + Ce_xy.ls): |
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984 continue |
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985 |
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986 # dive deeper |
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987 u = pending_u.pop(i) if i < len(pending_u) else None |
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988 v = pending_v.pop(j) if j < len(pending_v) else None |
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989 matched_uv.append((u, v)) |
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990 for x, y in xy: |
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991 len_g = len(pending_g) |
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992 len_h = len(pending_h) |
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993 matched_gh.append( |
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994 ( |
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995 pending_g[x] if x < len_g else None, |
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996 pending_h[y] if y < len_h else None, |
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997 ) |
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998 ) |
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999 sortedx = list(sorted(x for x, y in xy)) |
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1000 sortedy = list(sorted(y for x, y in xy)) |
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1001 G = list( |
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1002 (pending_g.pop(x) if x < len(pending_g) else None) |
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1003 for x in reversed(sortedx) |
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1004 ) |
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1005 H = list( |
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1006 (pending_h.pop(y) if y < len(pending_h) else None) |
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1007 for y in reversed(sortedy) |
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1008 ) |
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1009 |
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1010 yield from get_edit_paths( |
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1011 matched_uv, |
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1012 pending_u, |
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1013 pending_v, |
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1014 Cv_ij, |
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1015 matched_gh, |
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1016 pending_g, |
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1017 pending_h, |
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1018 Ce_xy, |
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1019 matched_cost + edit_cost, |
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1020 ) |
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1021 |
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1022 # backtrack |
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1023 if u is not None: |
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1024 pending_u.insert(i, u) |
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1025 if v is not None: |
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1026 pending_v.insert(j, v) |
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1027 matched_uv.pop() |
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1028 for x, g in zip(sortedx, reversed(G)): |
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1029 if g is not None: |
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1030 pending_g.insert(x, g) |
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1031 for y, h in zip(sortedy, reversed(H)): |
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1032 if h is not None: |
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1033 pending_h.insert(y, h) |
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1034 for t in xy: |
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1035 matched_gh.pop() |
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1036 |
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1037 # Initialization |
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1038 |
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1039 pending_u = list(G1.nodes) |
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1040 pending_v = list(G2.nodes) |
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1041 |
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1042 initial_cost = 0 |
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1043 if roots: |
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1044 root_u, root_v = roots |
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1045 if root_u not in pending_u or root_v not in pending_v: |
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1046 raise nx.NodeNotFound("Root node not in graph.") |
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1047 |
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1048 # remove roots from pending |
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1049 pending_u.remove(root_u) |
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1050 pending_v.remove(root_v) |
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1051 |
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1052 # cost matrix of vertex mappings |
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1053 m = len(pending_u) |
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1054 n = len(pending_v) |
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1055 C = np.zeros((m + n, m + n)) |
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1056 if node_subst_cost: |
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1057 C[0:m, 0:n] = np.array( |
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1058 [ |
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1059 node_subst_cost(G1.nodes[u], G2.nodes[v]) |
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1060 for u in pending_u |
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1061 for v in pending_v |
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1062 ] |
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1063 ).reshape(m, n) |
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1064 if roots: |
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1065 initial_cost = node_subst_cost(G1.nodes[root_u], G2.nodes[root_v]) |
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1066 elif node_match: |
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1067 C[0:m, 0:n] = np.array( |
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1068 [ |
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1069 1 - int(node_match(G1.nodes[u], G2.nodes[v])) |
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1070 for u in pending_u |
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1071 for v in pending_v |
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1072 ] |
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1073 ).reshape(m, n) |
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1074 if roots: |
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1075 initial_cost = 1 - node_match(G1.nodes[root_u], G2.nodes[root_v]) |
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1076 else: |
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1077 # all zeroes |
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1078 pass |
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1079 # assert not min(m, n) or C[0:m, 0:n].min() >= 0 |
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1080 if node_del_cost: |
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1081 del_costs = [node_del_cost(G1.nodes[u]) for u in pending_u] |
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1082 else: |
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1083 del_costs = [1] * len(pending_u) |
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1084 # assert not m or min(del_costs) >= 0 |
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1085 if node_ins_cost: |
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1086 ins_costs = [node_ins_cost(G2.nodes[v]) for v in pending_v] |
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1087 else: |
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1088 ins_costs = [1] * len(pending_v) |
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1089 # assert not n or min(ins_costs) >= 0 |
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1090 inf = C[0:m, 0:n].sum() + sum(del_costs) + sum(ins_costs) + 1 |
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1091 C[0:m, n : n + m] = np.array( |
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1092 [del_costs[i] if i == j else inf for i in range(m) for j in range(m)] |
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1093 ).reshape(m, m) |
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1094 C[m : m + n, 0:n] = np.array( |
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1095 [ins_costs[i] if i == j else inf for i in range(n) for j in range(n)] |
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1096 ).reshape(n, n) |
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1097 Cv = make_CostMatrix(C, m, n) |
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1098 # debug_print(f"Cv: {m} x {n}") |
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1099 # debug_print(Cv.C) |
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1100 |
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1101 pending_g = list(G1.edges) |
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1102 pending_h = list(G2.edges) |
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1103 |
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1104 # cost matrix of edge mappings |
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1105 m = len(pending_g) |
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1106 n = len(pending_h) |
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1107 C = np.zeros((m + n, m + n)) |
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1108 if edge_subst_cost: |
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1109 C[0:m, 0:n] = np.array( |
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1110 [ |
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1111 edge_subst_cost(G1.edges[g], G2.edges[h]) |
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1112 for g in pending_g |
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1113 for h in pending_h |
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1114 ] |
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1115 ).reshape(m, n) |
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1116 elif edge_match: |
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1117 C[0:m, 0:n] = np.array( |
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1118 [ |
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1119 1 - int(edge_match(G1.edges[g], G2.edges[h])) |
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1120 for g in pending_g |
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1121 for h in pending_h |
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1122 ] |
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1123 ).reshape(m, n) |
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1124 else: |
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1125 # all zeroes |
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1126 pass |
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1127 # assert not min(m, n) or C[0:m, 0:n].min() >= 0 |
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1128 if edge_del_cost: |
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1129 del_costs = [edge_del_cost(G1.edges[g]) for g in pending_g] |
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1130 else: |
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1131 del_costs = [1] * len(pending_g) |
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1132 # assert not m or min(del_costs) >= 0 |
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1133 if edge_ins_cost: |
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1134 ins_costs = [edge_ins_cost(G2.edges[h]) for h in pending_h] |
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1135 else: |
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1136 ins_costs = [1] * len(pending_h) |
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1137 # assert not n or min(ins_costs) >= 0 |
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1138 inf = C[0:m, 0:n].sum() + sum(del_costs) + sum(ins_costs) + 1 |
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1139 C[0:m, n : n + m] = np.array( |
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1140 [del_costs[i] if i == j else inf for i in range(m) for j in range(m)] |
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1141 ).reshape(m, m) |
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1142 C[m : m + n, 0:n] = np.array( |
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1143 [ins_costs[i] if i == j else inf for i in range(n) for j in range(n)] |
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1144 ).reshape(n, n) |
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1145 Ce = make_CostMatrix(C, m, n) |
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1146 # debug_print(f'Ce: {m} x {n}') |
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1147 # debug_print(Ce.C) |
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1148 # debug_print() |
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1149 |
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1150 class MaxCost: |
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1151 def __init__(self): |
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1152 # initial upper-bound estimate |
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1153 # NOTE: should work for empty graph |
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1154 self.value = Cv.C.sum() + Ce.C.sum() + 1 |
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1155 |
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1156 maxcost = MaxCost() |
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1157 |
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1158 if timeout is not None: |
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1159 if timeout <= 0: |
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1160 raise nx.NetworkXError("Timeout value must be greater than 0") |
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1161 start = time.perf_counter() |
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1162 |
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1163 def prune(cost): |
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1164 if timeout is not None: |
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1165 if time.perf_counter() - start > timeout: |
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1166 return True |
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1167 if upper_bound is not None: |
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1168 if cost > upper_bound: |
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1169 return True |
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1170 if cost > maxcost.value: |
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1171 return True |
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1172 elif strictly_decreasing and cost >= maxcost.value: |
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1173 return True |
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1174 |
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1175 # Now go! |
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1176 |
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1177 done_uv = [] if roots is None else [roots] |
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1178 |
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1179 for vertex_path, edge_path, cost in get_edit_paths( |
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1180 done_uv, pending_u, pending_v, Cv, [], pending_g, pending_h, Ce, initial_cost |
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1181 ): |
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1182 # assert sorted(G1.nodes) == sorted(u for u, v in vertex_path if u is not None) |
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1183 # assert sorted(G2.nodes) == sorted(v for u, v in vertex_path if v is not None) |
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1184 # assert sorted(G1.edges) == sorted(g for g, h in edge_path if g is not None) |
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1185 # assert sorted(G2.edges) == sorted(h for g, h in edge_path if h is not None) |
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1186 # print(vertex_path, edge_path, cost, file = sys.stderr) |
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1187 # assert cost == maxcost.value |
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1188 yield list(vertex_path), list(edge_path), cost |
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1189 |
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1190 |
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1191 def _is_close(d1, d2, atolerance=0, rtolerance=0): |
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1192 """Determines whether two adjacency matrices are within |
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1193 a provided tolerance. |
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1194 |
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1195 Parameters |
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1196 ---------- |
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1197 d1 : dict |
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1198 Adjacency dictionary |
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1199 |
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1200 d2 : dict |
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1201 Adjacency dictionary |
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1202 |
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1203 atolerance : float |
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1204 Some scalar tolerance value to determine closeness |
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1205 |
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1206 rtolerance : float |
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1207 A scalar tolerance value that will be some proportion |
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1208 of ``d2``'s value |
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1209 |
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1210 Returns |
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1211 ------- |
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1212 closeness : bool |
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1213 If all of the nodes within ``d1`` and ``d2`` are within |
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1214 a predefined tolerance, they are considered "close" and |
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1215 this method will return True. Otherwise, this method will |
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1216 return False. |
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1217 |
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1218 """ |
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1219 # Pre-condition: d1 and d2 have the same keys at each level if they |
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1220 # are dictionaries. |
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1221 if not isinstance(d1, dict) and not isinstance(d2, dict): |
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1222 return abs(d1 - d2) <= atolerance + rtolerance * abs(d2) |
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1223 return all(all(_is_close(d1[u][v], d2[u][v]) for v in d1[u]) for u in d1) |
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1224 |
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1225 |
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1226 def simrank_similarity( |
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1227 G, |
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1228 source=None, |
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1229 target=None, |
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1230 importance_factor=0.9, |
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1231 max_iterations=100, |
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1232 tolerance=1e-4, |
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1233 ): |
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1234 """Returns the SimRank similarity of nodes in the graph ``G``. |
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1235 |
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1236 SimRank is a similarity metric that says "two objects are considered |
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1237 to be similar if they are referenced by similar objects." [1]_. |
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1238 |
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1239 The pseudo-code definition from the paper is:: |
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1240 |
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1241 def simrank(G, u, v): |
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1242 in_neighbors_u = G.predecessors(u) |
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1243 in_neighbors_v = G.predecessors(v) |
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1244 scale = C / (len(in_neighbors_u) * len(in_neighbors_v)) |
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1245 return scale * sum(simrank(G, w, x) |
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1246 for w, x in product(in_neighbors_u, |
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1247 in_neighbors_v)) |
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1248 |
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1249 where ``G`` is the graph, ``u`` is the source, ``v`` is the target, |
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1250 and ``C`` is a float decay or importance factor between 0 and 1. |
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1251 |
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1252 The SimRank algorithm for determining node similarity is defined in |
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1253 [2]_. |
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1254 |
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1255 Parameters |
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1256 ---------- |
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1257 G : NetworkX graph |
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1258 A NetworkX graph |
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1259 |
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1260 source : node |
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1261 If this is specified, the returned dictionary maps each node |
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1262 ``v`` in the graph to the similarity between ``source`` and |
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1263 ``v``. |
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1264 |
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1265 target : node |
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1266 If both ``source`` and ``target`` are specified, the similarity |
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1267 value between ``source`` and ``target`` is returned. If |
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1268 ``target`` is specified but ``source`` is not, this argument is |
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1269 ignored. |
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1270 |
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1271 importance_factor : float |
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1272 The relative importance of indirect neighbors with respect to |
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1273 direct neighbors. |
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1274 |
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1275 max_iterations : integer |
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1276 Maximum number of iterations. |
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1277 |
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1278 tolerance : float |
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1279 Error tolerance used to check convergence. When an iteration of |
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1280 the algorithm finds that no similarity value changes more than |
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1281 this amount, the algorithm halts. |
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1282 |
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1283 Returns |
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1284 ------- |
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1285 similarity : dictionary or float |
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1286 If ``source`` and ``target`` are both ``None``, this returns a |
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1287 dictionary of dictionaries, where keys are node pairs and value |
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1288 are similarity of the pair of nodes. |
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1289 |
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1290 If ``source`` is not ``None`` but ``target`` is, this returns a |
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1291 dictionary mapping node to the similarity of ``source`` and that |
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1292 node. |
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1293 |
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1294 If neither ``source`` nor ``target`` is ``None``, this returns |
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1295 the similarity value for the given pair of nodes. |
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1296 |
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1297 Examples |
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1298 -------- |
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1299 If the nodes of the graph are numbered from zero to *n - 1*, where *n* |
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1300 is the number of nodes in the graph, you can create a SimRank matrix |
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1301 from the return value of this function where the node numbers are |
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1302 the row and column indices of the matrix:: |
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1303 |
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1304 >>> from numpy import array |
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1305 >>> G = nx.cycle_graph(4) |
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1306 >>> sim = nx.simrank_similarity(G) |
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1307 >>> lol = [[sim[u][v] for v in sorted(sim[u])] for u in sorted(sim)] |
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1308 >>> sim_array = array(lol) |
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1309 |
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1310 References |
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1311 ---------- |
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1312 .. [1] https://en.wikipedia.org/wiki/SimRank |
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1313 .. [2] G. Jeh and J. Widom. |
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1314 "SimRank: a measure of structural-context similarity", |
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1315 In KDD'02: Proceedings of the Eighth ACM SIGKDD |
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1316 International Conference on Knowledge Discovery and Data Mining, |
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1317 pp. 538--543. ACM Press, 2002. |
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1318 """ |
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1319 prevsim = None |
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1320 |
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1321 # build up our similarity adjacency dictionary output |
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1322 newsim = {u: {v: 1 if u == v else 0 for v in G} for u in G} |
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1323 |
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1324 # These functions compute the update to the similarity value of the nodes |
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1325 # `u` and `v` with respect to the previous similarity values. |
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1326 def avg_sim(s): |
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1327 return sum(newsim[w][x] for (w, x) in s) / len(s) if s else 0.0 |
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1328 |
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1329 def sim(u, v): |
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1330 Gadj = G.pred if G.is_directed() else G.adj |
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1331 return importance_factor * avg_sim(list(product(Gadj[u], Gadj[v]))) |
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1332 |
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1333 for _ in range(max_iterations): |
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1334 if prevsim and _is_close(prevsim, newsim, tolerance): |
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1335 break |
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1336 prevsim = newsim |
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1337 newsim = { |
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1338 u: {v: sim(u, v) if u is not v else 1 for v in newsim[u]} for u in newsim |
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1339 } |
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1340 |
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1341 if source is not None and target is not None: |
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1342 return newsim[source][target] |
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1343 if source is not None: |
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1344 return newsim[source] |
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1345 return newsim |
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1346 |
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1347 |
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1348 def simrank_similarity_numpy( |
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1349 G, |
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1350 source=None, |
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1351 target=None, |
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1352 importance_factor=0.9, |
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1353 max_iterations=100, |
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1354 tolerance=1e-4, |
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1355 ): |
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1356 """Calculate SimRank of nodes in ``G`` using matrices with ``numpy``. |
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1357 |
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1358 The SimRank algorithm for determining node similarity is defined in |
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1359 [1]_. |
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1360 |
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1361 Parameters |
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1362 ---------- |
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1363 G : NetworkX graph |
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1364 A NetworkX graph |
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1365 |
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1366 source : node |
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1367 If this is specified, the returned dictionary maps each node |
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1368 ``v`` in the graph to the similarity between ``source`` and |
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1369 ``v``. |
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1370 |
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1371 target : node |
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1372 If both ``source`` and ``target`` are specified, the similarity |
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1373 value between ``source`` and ``target`` is returned. If |
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1374 ``target`` is specified but ``source`` is not, this argument is |
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1375 ignored. |
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1376 |
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1377 importance_factor : float |
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1378 The relative importance of indirect neighbors with respect to |
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1379 direct neighbors. |
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1380 |
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1381 max_iterations : integer |
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1382 Maximum number of iterations. |
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1383 |
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1384 tolerance : float |
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1385 Error tolerance used to check convergence. When an iteration of |
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1386 the algorithm finds that no similarity value changes more than |
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1387 this amount, the algorithm halts. |
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1388 |
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1389 Returns |
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1390 ------- |
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1391 similarity : numpy matrix, numpy array or float |
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1392 If ``source`` and ``target`` are both ``None``, this returns a |
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1393 Matrix containing SimRank scores of the nodes. |
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1394 |
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1395 If ``source`` is not ``None`` but ``target`` is, this returns an |
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1396 Array containing SimRank scores of ``source`` and that |
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1397 node. |
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1398 |
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1399 If neither ``source`` nor ``target`` is ``None``, this returns |
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1400 the similarity value for the given pair of nodes. |
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1401 |
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1402 Examples |
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1403 -------- |
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1404 >>> from numpy import array |
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1405 >>> G = nx.cycle_graph(4) |
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1406 >>> sim = nx.simrank_similarity_numpy(G) |
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1407 |
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1408 References |
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1409 ---------- |
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1410 .. [1] G. Jeh and J. Widom. |
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1411 "SimRank: a measure of structural-context similarity", |
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1412 In KDD'02: Proceedings of the Eighth ACM SIGKDD |
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1413 International Conference on Knowledge Discovery and Data Mining, |
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1414 pp. 538--543. ACM Press, 2002. |
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1415 """ |
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1416 # This algorithm follows roughly |
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1417 # |
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1418 # S = max{C * (A.T * S * A), I} |
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1419 # |
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1420 # where C is the importance factor, A is the column normalized |
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1421 # adjacency matrix, and I is the identity matrix. |
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1422 import numpy as np |
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1423 |
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1424 adjacency_matrix = nx.to_numpy_array(G) |
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1425 |
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1426 # column-normalize the ``adjacency_matrix`` |
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1427 adjacency_matrix /= adjacency_matrix.sum(axis=0) |
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1428 |
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1429 newsim = np.eye(adjacency_matrix.shape[0], dtype=np.float64) |
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1430 for _ in range(max_iterations): |
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1431 prevsim = np.copy(newsim) |
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1432 newsim = importance_factor * np.matmul( |
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1433 np.matmul(adjacency_matrix.T, prevsim), adjacency_matrix |
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1434 ) |
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1435 np.fill_diagonal(newsim, 1.0) |
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1436 |
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1437 if np.allclose(prevsim, newsim, atol=tolerance): |
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1438 break |
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1439 |
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1440 if source is not None and target is not None: |
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1441 return newsim[source, target] |
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1442 if source is not None: |
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1443 return newsim[source] |
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1444 return newsim |