Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/bipartite/cluster.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """Functions for computing clustering of pairs |
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2 |
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3 """ |
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4 |
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5 import itertools |
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6 import networkx as nx |
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7 |
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8 __all__ = [ |
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9 "clustering", |
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10 "average_clustering", |
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11 "latapy_clustering", |
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12 "robins_alexander_clustering", |
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13 ] |
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14 |
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15 |
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16 def cc_dot(nu, nv): |
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17 return float(len(nu & nv)) / len(nu | nv) |
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18 |
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19 |
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20 def cc_max(nu, nv): |
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21 return float(len(nu & nv)) / max(len(nu), len(nv)) |
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22 |
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23 |
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24 def cc_min(nu, nv): |
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25 return float(len(nu & nv)) / min(len(nu), len(nv)) |
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26 |
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27 |
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28 modes = {"dot": cc_dot, "min": cc_min, "max": cc_max} |
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29 |
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30 |
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31 def latapy_clustering(G, nodes=None, mode="dot"): |
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32 r"""Compute a bipartite clustering coefficient for nodes. |
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33 |
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34 The bipartie clustering coefficient is a measure of local density |
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35 of connections defined as [1]_: |
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36 |
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37 .. math:: |
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38 |
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39 c_u = \frac{\sum_{v \in N(N(u))} c_{uv} }{|N(N(u))|} |
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40 |
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41 where `N(N(u))` are the second order neighbors of `u` in `G` excluding `u`, |
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42 and `c_{uv}` is the pairwise clustering coefficient between nodes |
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43 `u` and `v`. |
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44 |
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45 The mode selects the function for `c_{uv}` which can be: |
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46 |
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47 `dot`: |
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48 |
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49 .. math:: |
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50 |
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51 c_{uv}=\frac{|N(u)\cap N(v)|}{|N(u) \cup N(v)|} |
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52 |
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53 `min`: |
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54 |
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55 .. math:: |
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56 |
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57 c_{uv}=\frac{|N(u)\cap N(v)|}{min(|N(u)|,|N(v)|)} |
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58 |
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59 `max`: |
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60 |
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61 .. math:: |
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62 |
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63 c_{uv}=\frac{|N(u)\cap N(v)|}{max(|N(u)|,|N(v)|)} |
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64 |
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65 |
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66 Parameters |
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67 ---------- |
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68 G : graph |
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69 A bipartite graph |
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70 |
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71 nodes : list or iterable (optional) |
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72 Compute bipartite clustering for these nodes. The default |
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73 is all nodes in G. |
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74 |
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75 mode : string |
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76 The pariwise bipartite clustering method to be used in the computation. |
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77 It must be "dot", "max", or "min". |
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78 |
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79 Returns |
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80 ------- |
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81 clustering : dictionary |
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82 A dictionary keyed by node with the clustering coefficient value. |
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83 |
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84 |
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85 Examples |
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86 -------- |
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87 >>> from networkx.algorithms import bipartite |
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88 >>> G = nx.path_graph(4) # path graphs are bipartite |
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89 >>> c = bipartite.clustering(G) |
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90 >>> c[0] |
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91 0.5 |
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92 >>> c = bipartite.clustering(G, mode="min") |
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93 >>> c[0] |
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94 1.0 |
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95 |
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96 See Also |
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97 -------- |
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98 robins_alexander_clustering |
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99 square_clustering |
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100 average_clustering |
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101 |
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102 References |
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103 ---------- |
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104 .. [1] Latapy, Matthieu, Clémence Magnien, and Nathalie Del Vecchio (2008). |
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105 Basic notions for the analysis of large two-mode networks. |
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106 Social Networks 30(1), 31--48. |
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107 """ |
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108 if not nx.algorithms.bipartite.is_bipartite(G): |
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109 raise nx.NetworkXError("Graph is not bipartite") |
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110 |
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111 try: |
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112 cc_func = modes[mode] |
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113 except KeyError as e: |
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114 raise nx.NetworkXError( |
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115 "Mode for bipartite clustering must be: dot, min or max" |
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116 ) from e |
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117 |
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118 if nodes is None: |
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119 nodes = G |
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120 ccs = {} |
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121 for v in nodes: |
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122 cc = 0.0 |
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123 nbrs2 = {u for nbr in G[v] for u in G[nbr]} - {v} |
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124 for u in nbrs2: |
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125 cc += cc_func(set(G[u]), set(G[v])) |
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126 if cc > 0.0: # len(nbrs2)>0 |
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127 cc /= len(nbrs2) |
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128 ccs[v] = cc |
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129 return ccs |
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130 |
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131 |
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132 clustering = latapy_clustering |
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133 |
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134 |
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135 def average_clustering(G, nodes=None, mode="dot"): |
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136 r"""Compute the average bipartite clustering coefficient. |
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137 |
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138 A clustering coefficient for the whole graph is the average, |
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139 |
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140 .. math:: |
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141 |
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142 C = \frac{1}{n}\sum_{v \in G} c_v, |
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143 |
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144 where `n` is the number of nodes in `G`. |
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145 |
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146 Similar measures for the two bipartite sets can be defined [1]_ |
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147 |
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148 .. math:: |
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149 |
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150 C_X = \frac{1}{|X|}\sum_{v \in X} c_v, |
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151 |
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152 where `X` is a bipartite set of `G`. |
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153 |
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154 Parameters |
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155 ---------- |
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156 G : graph |
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157 a bipartite graph |
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158 |
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159 nodes : list or iterable, optional |
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160 A container of nodes to use in computing the average. |
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161 The nodes should be either the entire graph (the default) or one of the |
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162 bipartite sets. |
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163 |
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164 mode : string |
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165 The pariwise bipartite clustering method. |
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166 It must be "dot", "max", or "min" |
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167 |
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168 Returns |
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169 ------- |
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170 clustering : float |
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171 The average bipartite clustering for the given set of nodes or the |
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172 entire graph if no nodes are specified. |
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173 |
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174 Examples |
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175 -------- |
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176 >>> from networkx.algorithms import bipartite |
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177 >>> G = nx.star_graph(3) # star graphs are bipartite |
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178 >>> bipartite.average_clustering(G) |
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179 0.75 |
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180 >>> X, Y = bipartite.sets(G) |
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181 >>> bipartite.average_clustering(G, X) |
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182 0.0 |
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183 >>> bipartite.average_clustering(G, Y) |
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184 1.0 |
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185 |
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186 See Also |
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187 -------- |
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188 clustering |
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189 |
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190 Notes |
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191 ----- |
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192 The container of nodes passed to this function must contain all of the nodes |
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193 in one of the bipartite sets ("top" or "bottom") in order to compute |
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194 the correct average bipartite clustering coefficients. |
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195 See :mod:`bipartite documentation <networkx.algorithms.bipartite>` |
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196 for further details on how bipartite graphs are handled in NetworkX. |
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197 |
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198 |
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199 References |
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200 ---------- |
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201 .. [1] Latapy, Matthieu, Clémence Magnien, and Nathalie Del Vecchio (2008). |
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202 Basic notions for the analysis of large two-mode networks. |
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203 Social Networks 30(1), 31--48. |
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204 """ |
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205 if nodes is None: |
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206 nodes = G |
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207 ccs = latapy_clustering(G, nodes=nodes, mode=mode) |
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208 return float(sum(ccs[v] for v in nodes)) / len(nodes) |
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209 |
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210 |
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211 def robins_alexander_clustering(G): |
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212 r"""Compute the bipartite clustering of G. |
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213 |
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214 Robins and Alexander [1]_ defined bipartite clustering coefficient as |
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215 four times the number of four cycles `C_4` divided by the number of |
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216 three paths `L_3` in a bipartite graph: |
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217 |
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218 .. math:: |
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219 |
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220 CC_4 = \frac{4 * C_4}{L_3} |
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221 |
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222 Parameters |
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223 ---------- |
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224 G : graph |
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225 a bipartite graph |
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226 |
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227 Returns |
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228 ------- |
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229 clustering : float |
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230 The Robins and Alexander bipartite clustering for the input graph. |
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231 |
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232 Examples |
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233 -------- |
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234 >>> from networkx.algorithms import bipartite |
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235 >>> G = nx.davis_southern_women_graph() |
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236 >>> print(round(bipartite.robins_alexander_clustering(G), 3)) |
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237 0.468 |
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238 |
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239 See Also |
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240 -------- |
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241 latapy_clustering |
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242 square_clustering |
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243 |
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244 References |
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245 ---------- |
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246 .. [1] Robins, G. and M. Alexander (2004). Small worlds among interlocking |
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247 directors: Network structure and distance in bipartite graphs. |
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248 Computational & Mathematical Organization Theory 10(1), 69–94. |
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249 |
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250 """ |
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251 if G.order() < 4 or G.size() < 3: |
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252 return 0 |
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253 L_3 = _threepaths(G) |
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254 if L_3 == 0: |
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255 return 0 |
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256 C_4 = _four_cycles(G) |
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257 return (4.0 * C_4) / L_3 |
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258 |
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259 |
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260 def _four_cycles(G): |
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261 cycles = 0 |
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262 for v in G: |
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263 for u, w in itertools.combinations(G[v], 2): |
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264 cycles += len((set(G[u]) & set(G[w])) - {v}) |
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265 return cycles / 4 |
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266 |
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267 |
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268 def _threepaths(G): |
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269 paths = 0 |
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270 for v in G: |
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271 for u in G[v]: |
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272 for w in set(G[u]) - {v}: |
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273 paths += len(set(G[w]) - {v, u}) |
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274 # Divide by two because we count each three path twice |
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275 # one for each possible starting point |
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276 return paths / 2 |