Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/sitepackages/networkx/algorithms/bipartite/centrality.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 import networkx as nx 
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2 
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3 __all__ = ["degree_centrality", "betweenness_centrality", "closeness_centrality"] 
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4 
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5 
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6 def degree_centrality(G, nodes): 
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7 r"""Compute the degree centrality for nodes in a bipartite network. 
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8 
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9 The degree centrality for a node `v` is the fraction of nodes 
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10 connected to it. 
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11 
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12 Parameters 
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13  
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14 G : graph 
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15 A bipartite network 
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16 
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17 nodes : list or container 
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18 Container with all nodes in one bipartite node set. 
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19 
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20 Returns 
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21  
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22 centrality : dictionary 
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23 Dictionary keyed by node with bipartite degree centrality as the value. 
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24 
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25 See Also 
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26  
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27 betweenness_centrality, 
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28 closeness_centrality, 
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29 sets, 
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30 is_bipartite 
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31 
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32 Notes 
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33  
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34 The nodes input parameter must contain all nodes in one bipartite node set, 
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35 but the dictionary returned contains all nodes from both bipartite node 
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36 sets. See :mod:`bipartite documentation <networkx.algorithms.bipartite>` 
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37 for further details on how bipartite graphs are handled in NetworkX. 
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38 
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39 For unipartite networks, the degree centrality values are 
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40 normalized by dividing by the maximum possible degree (which is 
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41 `n1` where `n` is the number of nodes in G). 
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42 
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43 In the bipartite case, the maximum possible degree of a node in a 
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44 bipartite node set is the number of nodes in the opposite node set 
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45 [1]_. The degree centrality for a node `v` in the bipartite 
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46 sets `U` with `n` nodes and `V` with `m` nodes is 
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47 
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48 .. math:: 
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49 
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50 d_{v} = \frac{deg(v)}{m}, \mbox{for} v \in U , 
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51 
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52 d_{v} = \frac{deg(v)}{n}, \mbox{for} v \in V , 
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53 
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54 
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55 where `deg(v)` is the degree of node `v`. 
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56 
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57 References 
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58  
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59 .. [1] Borgatti, S.P. and Halgin, D. In press. "Analyzing Affiliation 
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60 Networks". In Carrington, P. and Scott, J. (eds) The Sage Handbook 
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61 of Social Network Analysis. Sage Publications. 
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62 http://www.steveborgatti.com/research/publications/bhaffiliations.pdf 
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63 """ 
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64 top = set(nodes) 
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65 bottom = set(G)  top 
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66 s = 1.0 / len(bottom) 
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67 centrality = {n: d * s for n, d in G.degree(top)} 
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68 s = 1.0 / len(top) 
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69 centrality.update({n: d * s for n, d in G.degree(bottom)}) 
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70 return centrality 
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71 
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72 
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73 def betweenness_centrality(G, nodes): 
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74 r"""Compute betweenness centrality for nodes in a bipartite network. 
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75 
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76 Betweenness centrality of a node `v` is the sum of the 
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77 fraction of allpairs shortest paths that pass through `v`. 
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78 
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79 Values of betweenness are normalized by the maximum possible 
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80 value which for bipartite graphs is limited by the relative size 
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81 of the two node sets [1]_. 
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82 
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83 Let `n` be the number of nodes in the node set `U` and 
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84 `m` be the number of nodes in the node set `V`, then 
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85 nodes in `U` are normalized by dividing by 
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86 
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87 .. math:: 
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88 
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89 \frac{1}{2} [m^2 (s + 1)^2 + m (s + 1)(2t  s  1)  t (2s  t + 3)] , 
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90 
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91 where 
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92 
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93 .. math:: 
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94 
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95 s = (n  1) \div m , t = (n  1) \mod m , 
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96 
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97 and nodes in `V` are normalized by dividing by 
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98 
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99 .. math:: 
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100 
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101 \frac{1}{2} [n^2 (p + 1)^2 + n (p + 1)(2r  p  1)  r (2p  r + 3)] , 
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102 
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103 where, 
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104 
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105 .. math:: 
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106 
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107 p = (m  1) \div n , r = (m  1) \mod n . 
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108 
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109 Parameters 
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110  
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111 G : graph 
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112 A bipartite graph 
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113 
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114 nodes : list or container 
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115 Container with all nodes in one bipartite node set. 
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116 
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117 Returns 
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118  
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119 betweenness : dictionary 
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120 Dictionary keyed by node with bipartite betweenness centrality 
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121 as the value. 
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122 
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123 See Also 
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124  
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125 degree_centrality, 
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126 closeness_centrality, 
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127 sets, 
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128 is_bipartite 
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129 
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130 Notes 
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131  
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132 The nodes input parameter must contain all nodes in one bipartite node set, 
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133 but the dictionary returned contains all nodes from both node sets. 
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134 See :mod:`bipartite documentation <networkx.algorithms.bipartite>` 
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135 for further details on how bipartite graphs are handled in NetworkX. 
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136 
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137 
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138 References 
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139  
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140 .. [1] Borgatti, S.P. and Halgin, D. In press. "Analyzing Affiliation 
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141 Networks". In Carrington, P. and Scott, J. (eds) The Sage Handbook 
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142 of Social Network Analysis. Sage Publications. 
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143 http://www.steveborgatti.com/research/publications/bhaffiliations.pdf 
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144 """ 
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145 top = set(nodes) 
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146 bottom = set(G)  top 
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147 n = float(len(top)) 
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148 m = float(len(bottom)) 
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149 s = (n  1) // m 
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150 t = (n  1) % m 
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151 bet_max_top = ( 
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152 ((m ** 2) * ((s + 1) ** 2)) 
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153 + (m * (s + 1) * (2 * t  s  1)) 
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154  (t * ((2 * s)  t + 3)) 
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155 ) / 2.0 
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156 p = (m  1) // n 
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157 r = (m  1) % n 
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158 bet_max_bot = ( 
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159 ((n ** 2) * ((p + 1) ** 2)) 
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160 + (n * (p + 1) * (2 * r  p  1)) 
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161  (r * ((2 * p)  r + 3)) 
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162 ) / 2.0 
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163 betweenness = nx.betweenness_centrality(G, normalized=False, weight=None) 
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164 for node in top: 
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165 betweenness[node] /= bet_max_top 
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166 for node in bottom: 
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167 betweenness[node] /= bet_max_bot 
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168 return betweenness 
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169 
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170 
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171 def closeness_centrality(G, nodes, normalized=True): 
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172 r"""Compute the closeness centrality for nodes in a bipartite network. 
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173 
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174 The closeness of a node is the distance to all other nodes in the 
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175 graph or in the case that the graph is not connected to all other nodes 
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176 in the connected component containing that node. 
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177 
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178 Parameters 
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179  
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180 G : graph 
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181 A bipartite network 
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182 
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183 nodes : list or container 
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184 Container with all nodes in one bipartite node set. 
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185 
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186 normalized : bool, optional 
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187 If True (default) normalize by connected component size. 
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188 
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189 Returns 
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190  
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191 closeness : dictionary 
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192 Dictionary keyed by node with bipartite closeness centrality 
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193 as the value. 
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194 
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195 See Also 
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196  
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197 betweenness_centrality, 
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198 degree_centrality 
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199 sets, 
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200 is_bipartite 
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201 
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202 Notes 
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203  
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204 The nodes input parameter must contain all nodes in one bipartite node set, 
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205 but the dictionary returned contains all nodes from both node sets. 
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206 See :mod:`bipartite documentation <networkx.algorithms.bipartite>` 
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207 for further details on how bipartite graphs are handled in NetworkX. 
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208 
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209 
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210 Closeness centrality is normalized by the minimum distance possible. 
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211 In the bipartite case the minimum distance for a node in one bipartite 
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212 node set is 1 from all nodes in the other node set and 2 from all 
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213 other nodes in its own set [1]_. Thus the closeness centrality 
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214 for node `v` in the two bipartite sets `U` with 
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215 `n` nodes and `V` with `m` nodes is 
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216 
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217 .. math:: 
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218 
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219 c_{v} = \frac{m + 2(n  1)}{d}, \mbox{for} v \in U, 
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220 
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221 c_{v} = \frac{n + 2(m  1)}{d}, \mbox{for} v \in V, 
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222 
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223 where `d` is the sum of the distances from `v` to all 
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224 other nodes. 
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225 
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226 Higher values of closeness indicate higher centrality. 
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227 
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228 As in the unipartite case, setting normalized=True causes the 
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229 values to normalized further to n1 / size(G)1 where n is the 
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230 number of nodes in the connected part of graph containing the 
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231 node. If the graph is not completely connected, this algorithm 
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232 computes the closeness centrality for each connected part 
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233 separately. 
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234 
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235 References 
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236  
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237 .. [1] Borgatti, S.P. and Halgin, D. In press. "Analyzing Affiliation 
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238 Networks". In Carrington, P. and Scott, J. (eds) The Sage Handbook 
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239 of Social Network Analysis. Sage Publications. 
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240 http://www.steveborgatti.com/research/publications/bhaffiliations.pdf 
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241 """ 
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242 closeness = {} 
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243 path_length = nx.single_source_shortest_path_length 
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244 top = set(nodes) 
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245 bottom = set(G)  top 
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246 n = float(len(top)) 
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247 m = float(len(bottom)) 
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248 for node in top: 
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249 sp = dict(path_length(G, node)) 
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250 totsp = sum(sp.values()) 
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251 if totsp > 0.0 and len(G) > 1: 
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252 closeness[node] = (m + 2 * (n  1)) / totsp 
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253 if normalized: 
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254 s = (len(sp)  1.0) / (len(G)  1) 
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255 closeness[node] *= s 
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256 else: 
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257 closeness[n] = 0.0 
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258 for node in bottom: 
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259 sp = dict(path_length(G, node)) 
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260 totsp = sum(sp.values()) 
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261 if totsp > 0.0 and len(G) > 1: 
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262 closeness[node] = (n + 2 * (m  1)) / totsp 
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263 if normalized: 
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264 s = (len(sp)  1.0) / (len(G)  1) 
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265 closeness[node] *= s 
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266 else: 
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267 closeness[n] = 0.0 
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268 return closeness 