Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/sitepackages/networkx/algorithms/centrality/closeness.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 """ 
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2 Closeness centrality measures. 
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3 """ 
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4 import functools 
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5 import networkx as nx 
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6 from networkx.exception import NetworkXError 
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7 from networkx.utils.decorators import not_implemented_for 
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8 
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9 __all__ = ["closeness_centrality", "incremental_closeness_centrality"] 
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10 
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11 
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12 def closeness_centrality(G, u=None, distance=None, wf_improved=True): 
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13 r"""Compute closeness centrality for nodes. 
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14 
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15 Closeness centrality [1]_ of a node `u` is the reciprocal of the 
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16 average shortest path distance to `u` over all `n1` reachable nodes. 
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17 
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18 .. math:: 
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19 
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20 C(u) = \frac{n  1}{\sum_{v=1}^{n1} d(v, u)}, 
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21 
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22 where `d(v, u)` is the shortestpath distance between `v` and `u`, 
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23 and `n` is the number of nodes that can reach `u`. Notice that the 
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24 closeness distance function computes the incoming distance to `u` 
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25 for directed graphs. To use outward distance, act on `G.reverse()`. 
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26 
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27 Notice that higher values of closeness indicate higher centrality. 
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28 
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29 Wasserman and Faust propose an improved formula for graphs with 
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30 more than one connected component. The result is "a ratio of the 
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31 fraction of actors in the group who are reachable, to the average 
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32 distance" from the reachable actors [2]_. You might think this 
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33 scale factor is inverted but it is not. As is, nodes from small 
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34 components receive a smaller closeness value. Letting `N` denote 
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35 the number of nodes in the graph, 
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36 
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37 .. math:: 
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38 
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39 C_{WF}(u) = \frac{n1}{N1} \frac{n  1}{\sum_{v=1}^{n1} d(v, u)}, 
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40 
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41 Parameters 
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42  
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43 G : graph 
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44 A NetworkX graph 
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45 
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46 u : node, optional 
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47 Return only the value for node u 
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48 
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49 distance : edge attribute key, optional (default=None) 
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50 Use the specified edge attribute as the edge distance in shortest 
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51 path calculations 
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52 
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53 wf_improved : bool, optional (default=True) 
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54 If True, scale by the fraction of nodes reachable. This gives the 
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55 Wasserman and Faust improved formula. For single component graphs 
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56 it is the same as the original formula. 
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57 
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58 Returns 
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59  
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60 nodes : dictionary 
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61 Dictionary of nodes with closeness centrality as the value. 
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62 
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63 See Also 
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64  
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65 betweenness_centrality, load_centrality, eigenvector_centrality, 
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66 degree_centrality, incremental_closeness_centrality 
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67 
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68 Notes 
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69  
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70 The closeness centrality is normalized to `(n1)/(G1)` where 
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71 `n` is the number of nodes in the connected part of graph 
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72 containing the node. If the graph is not completely connected, 
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73 this algorithm computes the closeness centrality for each 
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74 connected part separately scaled by that parts size. 
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75 
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76 If the 'distance' keyword is set to an edge attribute key then the 
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77 shortestpath length will be computed using Dijkstra's algorithm with 
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78 that edge attribute as the edge weight. 
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79 
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80 The closeness centrality uses *inward* distance to a node, not outward. 
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81 If you want to use outword distances apply the function to `G.reverse()` 
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82 
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83 In NetworkX 2.2 and earlier a bug caused Dijkstra's algorithm to use the 
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84 outward distance rather than the inward distance. If you use a 'distance' 
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85 keyword and a DiGraph, your results will change between v2.2 and v2.3. 
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86 
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87 References 
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88  
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89 .. [1] Linton C. Freeman: Centrality in networks: I. 
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90 Conceptual clarification. Social Networks 1:215239, 1979. 
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91 http://leonidzhukov.ru/hse/2013/socialnetworks/papers/freeman79centrality.pdf 
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92 .. [2] pg. 201 of Wasserman, S. and Faust, K., 
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93 Social Network Analysis: Methods and Applications, 1994, 
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94 Cambridge University Press. 
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95 """ 
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96 if G.is_directed(): 
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97 G = G.reverse() # create a reversed graph view 
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98 
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99 if distance is not None: 
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100 # use Dijkstra's algorithm with specified attribute as edge weight 
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101 path_length = functools.partial( 
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102 nx.single_source_dijkstra_path_length, weight=distance 
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103 ) 
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104 else: 
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105 path_length = nx.single_source_shortest_path_length 
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106 
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107 if u is None: 
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108 nodes = G.nodes 
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109 else: 
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110 nodes = [u] 
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111 closeness_centrality = {} 
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112 for n in nodes: 
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113 sp = path_length(G, n) 
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114 totsp = sum(sp.values()) 
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115 len_G = len(G) 
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116 _closeness_centrality = 0.0 
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117 if totsp > 0.0 and len_G > 1: 
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118 _closeness_centrality = (len(sp)  1.0) / totsp 
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119 # normalize to number of nodes1 in connected part 
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120 if wf_improved: 
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121 s = (len(sp)  1.0) / (len_G  1) 
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122 _closeness_centrality *= s 
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123 closeness_centrality[n] = _closeness_centrality 
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124 if u is not None: 
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125 return closeness_centrality[u] 
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126 else: 
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127 return closeness_centrality 
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128 
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129 
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130 @not_implemented_for("directed") 
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131 def incremental_closeness_centrality( 
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132 G, edge, prev_cc=None, insertion=True, wf_improved=True 
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133 ): 
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134 r"""Incremental closeness centrality for nodes. 
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135 
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136 Compute closeness centrality for nodes using levelbased work filtering 
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137 as described in Incremental Algorithms for Closeness Centrality by Sariyuce et al. 
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138 
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139 Levelbased work filtering detects unnecessary updates to the closeness 
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140 centrality and filters them out. 
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141 
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142  
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143 From "Incremental Algorithms for Closeness Centrality": 
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144 
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145 Theorem 1: Let :math:`G = (V, E)` be a graph and u and v be two vertices in V 
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146 such that there is no edge (u, v) in E. Let :math:`G' = (V, E \cup uv)` 
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147 Then :math:`cc[s] = cc'[s]` if and only if :math:`\leftdG(s, u)  dG(s, v)\right \leq 1`. 
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148 
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149 Where :math:`dG(u, v)` denotes the length of the shortest path between 
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150 two vertices u, v in a graph G, cc[s] is the closeness centrality for a 
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151 vertex s in V, and cc'[s] is the closeness centrality for a 
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152 vertex s in V, with the (u, v) edge added. 
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153  
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154 
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155 We use Theorem 1 to filter out updates when adding or removing an edge. 
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156 When adding an edge (u, v), we compute the shortest path lengths from all 
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157 other nodes to u and to v before the node is added. When removing an edge, 
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158 we compute the shortest path lengths after the edge is removed. Then we 
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159 apply Theorem 1 to use previously computed closeness centrality for nodes 
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160 where :math:`\leftdG(s, u)  dG(s, v)\right \leq 1`. This works only for 
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161 undirected, unweighted graphs; the distance argument is not supported. 
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162 
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163 Closeness centrality [1]_ of a node `u` is the reciprocal of the 
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164 sum of the shortest path distances from `u` to all `n1` other nodes. 
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165 Since the sum of distances depends on the number of nodes in the 
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166 graph, closeness is normalized by the sum of minimum possible 
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167 distances `n1`. 
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168 
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169 .. math:: 
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170 
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171 C(u) = \frac{n  1}{\sum_{v=1}^{n1} d(v, u)}, 
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172 
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173 where `d(v, u)` is the shortestpath distance between `v` and `u`, 
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174 and `n` is the number of nodes in the graph. 
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175 
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176 Notice that higher values of closeness indicate higher centrality. 
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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 NetworkX graph 
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182 
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183 edge : tuple 
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184 The modified edge (u, v) in the graph. 
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185 
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186 prev_cc : dictionary 
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187 The previous closeness centrality for all nodes in the graph. 
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188 
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189 insertion : bool, optional 
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190 If True (default) the edge was inserted, otherwise it was deleted from the graph. 
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191 
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192 wf_improved : bool, optional (default=True) 
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193 If True, scale by the fraction of nodes reachable. This gives the 
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194 Wasserman and Faust improved formula. For single component graphs 
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195 it is the same as the original formula. 
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196 
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197 Returns 
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198  
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199 nodes : dictionary 
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200 Dictionary of nodes with closeness centrality as the value. 
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201 
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202 See Also 
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203  
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204 betweenness_centrality, load_centrality, eigenvector_centrality, 
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205 degree_centrality, closeness_centrality 
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206 
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207 Notes 
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208  
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209 The closeness centrality is normalized to `(n1)/(G1)` where 
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210 `n` is the number of nodes in the connected part of graph 
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211 containing the node. If the graph is not completely connected, 
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212 this algorithm computes the closeness centrality for each 
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213 connected part separately. 
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214 
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215 References 
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216  
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217 .. [1] Freeman, L.C., 1979. Centrality in networks: I. 
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218 Conceptual clarification. Social Networks 1, 215239. 
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219 http://www.soc.ucsb.edu/faculty/friedkin/Syllabi/Soc146/Freeman78.PDF 
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220 .. [2] Sariyuce, A.E. ; Kaya, K. ; Saule, E. ; Catalyiirek, U.V. Incremental 
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221 Algorithms for Closeness Centrality. 2013 IEEE International Conference on Big Data 
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222 http://sariyuce.com/papers/bigdata13.pdf 
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223 """ 
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224 if prev_cc is not None and set(prev_cc.keys()) != set(G.nodes()): 
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225 raise NetworkXError("prev_cc and G do not have the same nodes") 
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226 
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227 # Unpack edge 
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228 (u, v) = edge 
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229 path_length = nx.single_source_shortest_path_length 
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230 
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231 if insertion: 
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232 # For edge insertion, we want shortest paths before the edge is inserted 
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233 du = path_length(G, u) 
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234 dv = path_length(G, v) 
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235 
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236 G.add_edge(u, v) 
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237 else: 
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238 G.remove_edge(u, v) 
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239 
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240 # For edge removal, we want shortest paths after the edge is removed 
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241 du = path_length(G, u) 
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242 dv = path_length(G, v) 
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243 
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244 if prev_cc is None: 
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245 return nx.closeness_centrality(G) 
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246 
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247 nodes = G.nodes() 
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248 closeness_centrality = {} 
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249 for n in nodes: 
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250 if n in du and n in dv and abs(du[n]  dv[n]) <= 1: 
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251 closeness_centrality[n] = prev_cc[n] 
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252 else: 
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253 sp = path_length(G, n) 
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254 totsp = sum(sp.values()) 
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255 len_G = len(G) 
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256 _closeness_centrality = 0.0 
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257 if totsp > 0.0 and len_G > 1: 
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258 _closeness_centrality = (len(sp)  1.0) / totsp 
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259 # normalize to number of nodes1 in connected part 
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260 if wf_improved: 
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261 s = (len(sp)  1.0) / (len_G  1) 
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262 _closeness_centrality *= s 
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263 closeness_centrality[n] = _closeness_centrality 
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264 
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265 # Leave the graph as we found it 
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266 if insertion: 
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267 G.remove_edge(u, v) 
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268 else: 
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269 G.add_edge(u, v) 
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270 
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271 return closeness_centrality 