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

date  Mon, 22 Mar 2021 18:12:50 +0000 
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1 """Functions for computing large cliques.""" 
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2 import networkx as nx 
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3 from networkx.utils import not_implemented_for 
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4 from networkx.algorithms.approximation import ramsey 
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5 
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6 __all__ = ["clique_removal", "max_clique", "large_clique_size"] 
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7 
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8 
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9 def max_clique(G): 
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10 r"""Find the Maximum Clique 
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11 
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12 Finds the $O(V/(logV)^2)$ apx of maximum clique/independent set 
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13 in the worst case. 
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14 
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15 Parameters 
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16  
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17 G : NetworkX graph 
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18 Undirected graph 
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19 
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20 Returns 
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21  
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22 clique : set 
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23 The apxmaximum clique of the graph 
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24 
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25 Notes 
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26  
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27 A clique in an undirected graph G = (V, E) is a subset of the vertex set 
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28 `C \subseteq V` such that for every two vertices in C there exists an edge 
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29 connecting the two. This is equivalent to saying that the subgraph 
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30 induced by C is complete (in some cases, the term clique may also refer 
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31 to the subgraph). 
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32 
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33 A maximum clique is a clique of the largest possible size in a given graph. 
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34 The clique number `\omega(G)` of a graph G is the number of 
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35 vertices in a maximum clique in G. The intersection number of 
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36 G is the smallest number of cliques that together cover all edges of G. 
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37 
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38 https://en.wikipedia.org/wiki/Maximum_clique 
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39 
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40 References 
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41  
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42 .. [1] Boppana, R., & Halldórsson, M. M. (1992). 
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43 Approximating maximum independent sets by excluding subgraphs. 
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44 BIT Numerical Mathematics, 32(2), 180–196. Springer. 
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45 doi:10.1007/BF01994876 
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46 """ 
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47 if G is None: 
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48 raise ValueError("Expected NetworkX graph!") 
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49 
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50 # finding the maximum clique in a graph is equivalent to finding 
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51 # the independent set in the complementary graph 
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52 cgraph = nx.complement(G) 
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53 iset, _ = clique_removal(cgraph) 
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54 return iset 
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55 
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56 
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57 def clique_removal(G): 
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58 r""" Repeatedly remove cliques from the graph. 
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59 
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60 Results in a $O(V/(\log V)^2)$ approximation of maximum clique 
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61 and independent set. Returns the largest independent set found, along 
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62 with found maximal cliques. 
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63 
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64 Parameters 
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65  
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66 G : NetworkX graph 
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67 Undirected graph 
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68 
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69 Returns 
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70  
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71 max_ind_cliques : (set, list) tuple 
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72 2tuple of Maximal Independent Set and list of maximal cliques (sets). 
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73 
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74 References 
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75  
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76 .. [1] Boppana, R., & Halldórsson, M. M. (1992). 
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77 Approximating maximum independent sets by excluding subgraphs. 
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78 BIT Numerical Mathematics, 32(2), 180–196. Springer. 
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79 """ 
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80 graph = G.copy() 
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81 c_i, i_i = ramsey.ramsey_R2(graph) 
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82 cliques = [c_i] 
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83 isets = [i_i] 
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84 while graph: 
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85 graph.remove_nodes_from(c_i) 
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86 c_i, i_i = ramsey.ramsey_R2(graph) 
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87 if c_i: 
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88 cliques.append(c_i) 
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89 if i_i: 
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90 isets.append(i_i) 
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91 # Determine the largest independent set as measured by cardinality. 
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92 maxiset = max(isets, key=len) 
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93 return maxiset, cliques 
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94 
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95 
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96 @not_implemented_for("directed") 
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97 @not_implemented_for("multigraph") 
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98 def large_clique_size(G): 
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99 """Find the size of a large clique in a graph. 
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100 
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101 A *clique* is a subset of nodes in which each pair of nodes is 
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102 adjacent. This function is a heuristic for finding the size of a 
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103 large clique in the graph. 
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104 
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105 Parameters 
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106  
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107 G : NetworkX graph 
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108 
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109 Returns 
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110  
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111 int 
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112 The size of a large clique in the graph. 
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113 
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114 Notes 
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115  
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116 This implementation is from [1]_. Its worst case time complexity is 
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117 :math:`O(n d^2)`, where *n* is the number of nodes in the graph and 
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118 *d* is the maximum degree. 
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119 
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120 This function is a heuristic, which means it may work well in 
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121 practice, but there is no rigorous mathematical guarantee on the 
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122 ratio between the returned number and the actual largest clique size 
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123 in the graph. 
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124 
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125 References 
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126  
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127 .. [1] Pattabiraman, Bharath, et al. 
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128 "Fast Algorithms for the Maximum Clique Problem on Massive Graphs 
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129 with Applications to Overlapping Community Detection." 
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130 *Internet Mathematics* 11.45 (2015): 421448. 
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131 <https://doi.org/10.1080/15427951.2014.986778> 
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132 
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133 See also 
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134  
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135 
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136 :func:`networkx.algorithms.approximation.clique.max_clique` 
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137 A function that returns an approximate maximum clique with a 
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138 guarantee on the approximation ratio. 
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139 
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140 :mod:`networkx.algorithms.clique` 
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141 Functions for finding the exact maximum clique in a graph. 
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142 
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143 """ 
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144 degrees = G.degree 
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145 
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146 def _clique_heuristic(G, U, size, best_size): 
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147 if not U: 
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148 return max(best_size, size) 
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149 u = max(U, key=degrees) 
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150 U.remove(u) 
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151 N_prime = {v for v in G[u] if degrees[v] >= best_size} 
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152 return _clique_heuristic(G, U & N_prime, size + 1, best_size) 
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153 
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154 best_size = 0 
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155 nodes = (u for u in G if degrees[u] >= best_size) 
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156 for u in nodes: 
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157 neighbors = {v for v in G[u] if degrees[v] >= best_size} 
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158 best_size = _clique_heuristic(G, neighbors, 1, best_size) 
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159 return best_size 