Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/sitepackages/networkx/algorithms/centrality/betweenness.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 """Betweenness centrality measures.""" 
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2 from heapq import heappush, heappop 
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3 from itertools import count 
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4 import warnings 
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5 
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6 from networkx.utils import py_random_state 
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7 from networkx.utils.decorators import not_implemented_for 
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8 
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9 __all__ = ["betweenness_centrality", "edge_betweenness_centrality", "edge_betweenness"] 
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10 
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11 
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12 @py_random_state(5) 
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13 @not_implemented_for("multigraph") 
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14 def betweenness_centrality( 
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15 G, k=None, normalized=True, weight=None, endpoints=False, seed=None 
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16 ): 
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17 r"""Compute the shortestpath betweenness centrality for nodes. 
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18 
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19 Betweenness centrality of a node $v$ is the sum of the 
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20 fraction of allpairs shortest paths that pass through $v$ 
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21 
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22 .. math:: 
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23 
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24 c_B(v) =\sum_{s,t \in V} \frac{\sigma(s, tv)}{\sigma(s, t)} 
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25 
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26 where $V$ is the set of nodes, $\sigma(s, t)$ is the number of 
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27 shortest $(s, t)$paths, and $\sigma(s, tv)$ is the number of 
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28 those paths passing through some node $v$ other than $s, t$. 
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29 If $s = t$, $\sigma(s, t) = 1$, and if $v \in {s, t}$, 
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30 $\sigma(s, tv) = 0$ [2]_. 
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31 
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32 Parameters 
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33  
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34 G : graph 
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35 A NetworkX graph. 
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36 
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37 k : int, optional (default=None) 
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38 If k is not None use k node samples to estimate betweenness. 
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39 The value of k <= n where n is the number of nodes in the graph. 
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40 Higher values give better approximation. 
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41 
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42 normalized : bool, optional 
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43 If True the betweenness values are normalized by `2/((n1)(n2))` 
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44 for graphs, and `1/((n1)(n2))` for directed graphs where `n` 
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45 is the number of nodes in G. 
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46 
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47 weight : None or string, optional (default=None) 
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48 If None, all edge weights are considered equal. 
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49 Otherwise holds the name of the edge attribute used as weight. 
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50 
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51 endpoints : bool, optional 
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52 If True include the endpoints in the shortest path counts. 
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53 
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54 seed : integer, random_state, or None (default) 
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55 Indicator of random number generation state. 
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56 See :ref:`Randomness<randomness>`. 
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57 Note that this is only used if k is not None. 
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58 
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59 Returns 
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60  
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61 nodes : dictionary 
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62 Dictionary of nodes with betweenness centrality as the value. 
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63 
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64 See Also 
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65  
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66 edge_betweenness_centrality 
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67 load_centrality 
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68 
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69 Notes 
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70  
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71 The algorithm is from Ulrik Brandes [1]_. 
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72 See [4]_ for the original first published version and [2]_ for details on 
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73 algorithms for variations and related metrics. 
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74 
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75 For approximate betweenness calculations set k=#samples to use 
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76 k nodes ("pivots") to estimate the betweenness values. For an estimate 
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77 of the number of pivots needed see [3]_. 
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78 
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79 For weighted graphs the edge weights must be greater than zero. 
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80 Zero edge weights can produce an infinite number of equal length 
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81 paths between pairs of nodes. 
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82 
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83 The total number of paths between source and target is counted 
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84 differently for directed and undirected graphs. Directed paths 
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85 are easy to count. Undirected paths are tricky: should a path 
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86 from "u" to "v" count as 1 undirected path or as 2 directed paths? 
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87 
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88 For betweenness_centrality we report the number of undirected 
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89 paths when G is undirected. 
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90 
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91 For betweenness_centrality_subset the reporting is different. 
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92 If the source and target subsets are the same, then we want 
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93 to count undirected paths. But if the source and target subsets 
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94 differ  for example, if sources is {0} and targets is {1}, 
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95 then we are only counting the paths in one direction. They are 
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96 undirected paths but we are counting them in a directed way. 
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97 To count them as undirected paths, each should count as half a path. 
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98 
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99 References 
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100  
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101 .. [1] Ulrik Brandes: 
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102 A Faster Algorithm for Betweenness Centrality. 
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103 Journal of Mathematical Sociology 25(2):163177, 2001. 
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104 http://www.inf.unikonstanz.de/algo/publications/bfabc01.pdf 
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105 .. [2] Ulrik Brandes: 
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106 On Variants of ShortestPath Betweenness 
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107 Centrality and their Generic Computation. 
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108 Social Networks 30(2):136145, 2008. 
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109 http://www.inf.unikonstanz.de/algo/publications/bvspbc08.pdf 
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110 .. [3] Ulrik Brandes and Christian Pich: 
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111 Centrality Estimation in Large Networks. 
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112 International Journal of Bifurcation and Chaos 17(7):23032318, 2007. 
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113 http://www.inf.unikonstanz.de/algo/publications/bpceln06.pdf 
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114 .. [4] Linton C. Freeman: 
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115 A set of measures of centrality based on betweenness. 
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116 Sociometry 40: 35â€“41, 1977 
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117 http://moreno.ss.uci.edu/23.pdf 
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118 """ 
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119 betweenness = dict.fromkeys(G, 0.0) # b[v]=0 for v in G 
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120 if k is None: 
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121 nodes = G 
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122 else: 
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123 nodes = seed.sample(G.nodes(), k) 
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124 for s in nodes: 
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125 # single source shortest paths 
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126 if weight is None: # use BFS 
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127 S, P, sigma = _single_source_shortest_path_basic(G, s) 
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128 else: # use Dijkstra's algorithm 
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129 S, P, sigma = _single_source_dijkstra_path_basic(G, s, weight) 
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130 # accumulation 
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131 if endpoints: 
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132 betweenness = _accumulate_endpoints(betweenness, S, P, sigma, s) 
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133 else: 
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134 betweenness = _accumulate_basic(betweenness, S, P, sigma, s) 
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135 # rescaling 
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136 betweenness = _rescale( 
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137 betweenness, 
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138 len(G), 
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139 normalized=normalized, 
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140 directed=G.is_directed(), 
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141 k=k, 
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142 endpoints=endpoints, 
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143 ) 
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144 return betweenness 
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145 
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146 
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147 @py_random_state(4) 
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148 def edge_betweenness_centrality(G, k=None, normalized=True, weight=None, seed=None): 
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149 r"""Compute betweenness centrality for edges. 
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150 
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151 Betweenness centrality of an edge $e$ is the sum of the 
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152 fraction of allpairs shortest paths that pass through $e$ 
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153 
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154 .. math:: 
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155 
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156 c_B(e) =\sum_{s,t \in V} \frac{\sigma(s, te)}{\sigma(s, t)} 
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157 
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158 where $V$ is the set of nodes, $\sigma(s, t)$ is the number of 
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159 shortest $(s, t)$paths, and $\sigma(s, te)$ is the number of 
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160 those paths passing through edge $e$ [2]_. 
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161 
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162 Parameters 
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163  
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164 G : graph 
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165 A NetworkX graph. 
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166 
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167 k : int, optional (default=None) 
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168 If k is not None use k node samples to estimate betweenness. 
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169 The value of k <= n where n is the number of nodes in the graph. 
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170 Higher values give better approximation. 
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171 
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172 normalized : bool, optional 
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173 If True the betweenness values are normalized by $2/(n(n1))$ 
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174 for graphs, and $1/(n(n1))$ for directed graphs where $n$ 
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175 is the number of nodes in G. 
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176 
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177 weight : None or string, optional (default=None) 
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178 If None, all edge weights are considered equal. 
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179 Otherwise holds the name of the edge attribute used as weight. 
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180 
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181 seed : integer, random_state, or None (default) 
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182 Indicator of random number generation state. 
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183 See :ref:`Randomness<randomness>`. 
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184 Note that this is only used if k is not None. 
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185 
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186 Returns 
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187  
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188 edges : dictionary 
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189 Dictionary of edges with betweenness centrality as the value. 
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190 
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191 See Also 
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192  
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193 betweenness_centrality 
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194 edge_load 
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195 
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196 Notes 
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197  
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198 The algorithm is from Ulrik Brandes [1]_. 
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199 
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200 For weighted graphs the edge weights must be greater than zero. 
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201 Zero edge weights can produce an infinite number of equal length 
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202 paths between pairs of nodes. 
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203 
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204 References 
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205  
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206 .. [1] A Faster Algorithm for Betweenness Centrality. Ulrik Brandes, 
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207 Journal of Mathematical Sociology 25(2):163177, 2001. 
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208 http://www.inf.unikonstanz.de/algo/publications/bfabc01.pdf 
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209 .. [2] Ulrik Brandes: On Variants of ShortestPath Betweenness 
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210 Centrality and their Generic Computation. 
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211 Social Networks 30(2):136145, 2008. 
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212 http://www.inf.unikonstanz.de/algo/publications/bvspbc08.pdf 
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213 """ 
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214 betweenness = dict.fromkeys(G, 0.0) # b[v]=0 for v in G 
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215 # b[e]=0 for e in G.edges() 
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216 betweenness.update(dict.fromkeys(G.edges(), 0.0)) 
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217 if k is None: 
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218 nodes = G 
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219 else: 
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220 nodes = seed.sample(G.nodes(), k) 
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221 for s in nodes: 
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222 # single source shortest paths 
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223 if weight is None: # use BFS 
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224 S, P, sigma = _single_source_shortest_path_basic(G, s) 
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225 else: # use Dijkstra's algorithm 
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226 S, P, sigma = _single_source_dijkstra_path_basic(G, s, weight) 
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227 # accumulation 
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228 betweenness = _accumulate_edges(betweenness, S, P, sigma, s) 
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229 # rescaling 
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230 for n in G: # remove nodes to only return edges 
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231 del betweenness[n] 
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232 betweenness = _rescale_e( 
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233 betweenness, len(G), normalized=normalized, directed=G.is_directed() 
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234 ) 
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235 return betweenness 
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236 
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237 
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238 # obsolete name 
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239 def edge_betweenness(G, k=None, normalized=True, weight=None, seed=None): 
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240 warnings.warn( 
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241 "edge_betweeness is replaced by edge_betweenness_centrality", DeprecationWarning 
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242 ) 
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243 return edge_betweenness_centrality(G, k, normalized, weight, seed) 
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244 
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245 
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246 # helpers for betweenness centrality 
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247 
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248 
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249 def _single_source_shortest_path_basic(G, s): 
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250 S = [] 
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251 P = {} 
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252 for v in G: 
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253 P[v] = [] 
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254 sigma = dict.fromkeys(G, 0.0) # sigma[v]=0 for v in G 
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255 D = {} 
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256 sigma[s] = 1.0 
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257 D[s] = 0 
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258 Q = [s] 
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259 while Q: # use BFS to find shortest paths 
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260 v = Q.pop(0) 
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261 S.append(v) 
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262 Dv = D[v] 
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263 sigmav = sigma[v] 
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264 for w in G[v]: 
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265 if w not in D: 
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266 Q.append(w) 
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267 D[w] = Dv + 1 
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268 if D[w] == Dv + 1: # this is a shortest path, count paths 
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269 sigma[w] += sigmav 
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270 P[w].append(v) # predecessors 
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271 return S, P, sigma 
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272 
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273 
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274 def _single_source_dijkstra_path_basic(G, s, weight): 
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275 # modified from Eppstein 
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276 S = [] 
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277 P = {} 
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278 for v in G: 
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279 P[v] = [] 
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280 sigma = dict.fromkeys(G, 0.0) # sigma[v]=0 for v in G 
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281 D = {} 
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282 sigma[s] = 1.0 
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283 push = heappush 
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284 pop = heappop 
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285 seen = {s: 0} 
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286 c = count() 
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287 Q = [] # use Q as heap with (distance,node id) tuples 
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288 push(Q, (0, next(c), s, s)) 
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289 while Q: 
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290 (dist, _, pred, v) = pop(Q) 
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291 if v in D: 
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292 continue # already searched this node. 
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293 sigma[v] += sigma[pred] # count paths 
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294 S.append(v) 
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295 D[v] = dist 
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296 for w, edgedata in G[v].items(): 
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297 vw_dist = dist + edgedata.get(weight, 1) 
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298 if w not in D and (w not in seen or vw_dist < seen[w]): 
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299 seen[w] = vw_dist 
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300 push(Q, (vw_dist, next(c), v, w)) 
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301 sigma[w] = 0.0 
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302 P[w] = [v] 
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303 elif vw_dist == seen[w]: # handle equal paths 
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304 sigma[w] += sigma[v] 
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305 P[w].append(v) 
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306 return S, P, sigma 
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307 
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308 
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309 def _accumulate_basic(betweenness, S, P, sigma, s): 
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310 delta = dict.fromkeys(S, 0) 
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311 while S: 
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312 w = S.pop() 
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313 coeff = (1 + delta[w]) / sigma[w] 
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314 for v in P[w]: 
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315 delta[v] += sigma[v] * coeff 
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316 if w != s: 
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317 betweenness[w] += delta[w] 
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318 return betweenness 
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319 
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320 
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321 def _accumulate_endpoints(betweenness, S, P, sigma, s): 
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322 betweenness[s] += len(S)  1 
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323 delta = dict.fromkeys(S, 0) 
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324 while S: 
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325 w = S.pop() 
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326 coeff = (1 + delta[w]) / sigma[w] 
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327 for v in P[w]: 
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328 delta[v] += sigma[v] * coeff 
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329 if w != s: 
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330 betweenness[w] += delta[w] + 1 
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331 return betweenness 
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332 
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333 
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334 def _accumulate_edges(betweenness, S, P, sigma, s): 
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335 delta = dict.fromkeys(S, 0) 
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336 while S: 
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337 w = S.pop() 
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338 coeff = (1 + delta[w]) / sigma[w] 
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339 for v in P[w]: 
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340 c = sigma[v] * coeff 
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341 if (v, w) not in betweenness: 
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342 betweenness[(w, v)] += c 
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343 else: 
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344 betweenness[(v, w)] += c 
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345 delta[v] += c 
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346 if w != s: 
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347 betweenness[w] += delta[w] 
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348 return betweenness 
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349 
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350 
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351 def _rescale(betweenness, n, normalized, directed=False, k=None, endpoints=False): 
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352 if normalized: 
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353 if endpoints: 
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354 if n < 2: 
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355 scale = None # no normalization 
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356 else: 
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357 # Scale factor should include endpoint nodes 
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358 scale = 1 / (n * (n  1)) 
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359 elif n <= 2: 
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360 scale = None # no normalization b=0 for all nodes 
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361 else: 
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362 scale = 1 / ((n  1) * (n  2)) 
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363 else: # rescale by 2 for undirected graphs 
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364 if not directed: 
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365 scale = 0.5 
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366 else: 
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367 scale = None 
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368 if scale is not None: 
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369 if k is not None: 
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370 scale = scale * n / k 
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371 for v in betweenness: 
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372 betweenness[v] *= scale 
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373 return betweenness 
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374 
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375 
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376 def _rescale_e(betweenness, n, normalized, directed=False, k=None): 
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377 if normalized: 
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378 if n <= 1: 
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379 scale = None # no normalization b=0 for all nodes 
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380 else: 
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381 scale = 1 / (n * (n  1)) 
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382 else: # rescale by 2 for undirected graphs 
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383 if not directed: 
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384 scale = 0.5 
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385 else: 
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386 scale = None 
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387 if scale is not None: 
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388 if k is not None: 
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389 scale = scale * n / k 
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390 for v in betweenness: 
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391 betweenness[v] *= scale 
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392 return betweenness 