Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/centrality/group.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 """Group centrality measures.""" |
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2 from itertools import combinations |
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3 |
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4 |
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5 import networkx as nx |
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6 from networkx.utils.decorators import not_implemented_for |
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7 |
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8 |
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9 __all__ = [ |
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10 "group_betweenness_centrality", |
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11 "group_closeness_centrality", |
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12 "group_degree_centrality", |
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13 "group_in_degree_centrality", |
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14 "group_out_degree_centrality", |
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15 ] |
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16 |
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17 |
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18 def group_betweenness_centrality(G, C, normalized=True, weight=None): |
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19 r"""Compute the group betweenness centrality for a group of nodes. |
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20 |
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21 Group betweenness centrality of a group of nodes $C$ is the sum of the |
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22 fraction of all-pairs shortest paths that pass through any vertex in $C$ |
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23 |
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24 .. math:: |
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25 |
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26 c_B(C) =\sum_{s,t \in V-C; s<t} \frac{\sigma(s, t|C)}{\sigma(s, t)} |
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27 |
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28 where $V$ is the set of nodes, $\sigma(s, t)$ is the number of |
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29 shortest $(s, t)$-paths, and $\sigma(s, t|C)$ is the number of |
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30 those paths passing through some node in group $C$. Note that |
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31 $(s, t)$ are not members of the group ($V-C$ is the set of nodes |
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32 in $V$ that are not in $C$). |
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33 |
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34 Parameters |
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35 ---------- |
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36 G : graph |
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37 A NetworkX graph. |
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38 |
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39 C : list or set |
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40 C is a group of nodes which belong to G, for which group betweenness |
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41 centrality is to be calculated. |
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42 |
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43 normalized : bool, optional |
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44 If True, group betweenness is normalized by `2/((|V|-|C|)(|V|-|C|-1))` |
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45 for graphs and `1/((|V|-|C|)(|V|-|C|-1))` for directed graphs where `|V|` |
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46 is the number of nodes in G and `|C|` is the number of nodes in C. |
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47 |
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48 weight : None or string, optional (default=None) |
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49 If None, all edge weights are considered equal. |
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50 Otherwise holds the name of the edge attribute used as weight. |
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51 |
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52 Raises |
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53 ------ |
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54 NodeNotFound |
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55 If node(s) in C are not present in G. |
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56 |
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57 Returns |
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58 ------- |
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59 betweenness : float |
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60 Group betweenness centrality of the group C. |
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61 |
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62 See Also |
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63 -------- |
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64 betweenness_centrality |
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65 |
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66 Notes |
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67 ----- |
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68 The measure is described in [1]_. |
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69 The algorithm is an extension of the one proposed by Ulrik Brandes for |
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70 betweenness centrality of nodes. Group betweenness is also mentioned in |
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71 his paper [2]_ along with the algorithm. The importance of the measure is |
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72 discussed in [3]_. |
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73 |
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74 The number of nodes in the group must be a maximum of n - 2 where `n` |
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75 is the total number of nodes in the graph. |
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76 |
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77 For weighted graphs the edge weights must be greater than zero. |
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78 Zero edge weights can produce an infinite number of equal length |
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79 paths between pairs of nodes. |
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80 |
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81 References |
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82 ---------- |
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83 .. [1] M G Everett and S P Borgatti: |
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84 The Centrality of Groups and Classes. |
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85 Journal of Mathematical Sociology. 23(3): 181-201. 1999. |
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86 http://www.analytictech.com/borgatti/group_centrality.htm |
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87 .. [2] Ulrik Brandes: |
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88 On Variants of Shortest-Path Betweenness |
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89 Centrality and their Generic Computation. |
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90 Social Networks 30(2):136-145, 2008. |
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91 http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.72.9610&rep=rep1&type=pdf |
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92 .. [3] Sourav Medya et. al.: |
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93 Group Centrality Maximization via Network Design. |
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94 SIAM International Conference on Data Mining, SDM 2018, 126–134. |
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95 https://sites.cs.ucsb.edu/~arlei/pubs/sdm18.pdf |
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96 """ |
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97 betweenness = 0 # initialize betweenness to 0 |
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98 V = set(G) # set of nodes in G |
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99 C = set(C) # set of nodes in C (group) |
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100 if len(C - V) != 0: # element(s) of C not in V |
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101 raise nx.NodeNotFound( |
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102 "The node(s) " + str(list(C - V)) + " are not " "in the graph." |
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103 ) |
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104 V_C = V - C # set of nodes in V but not in C |
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105 # accumulation |
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106 for pair in combinations(V_C, 2): # (s, t) pairs of V_C |
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107 try: |
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108 paths = 0 |
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109 paths_through_C = 0 |
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110 for path in nx.all_shortest_paths( |
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111 G, source=pair[0], target=pair[1], weight=weight |
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112 ): |
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113 if set(path) & C: |
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114 paths_through_C += 1 |
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115 paths += 1 |
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116 betweenness += paths_through_C / paths |
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117 except nx.exception.NetworkXNoPath: |
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118 betweenness += 0 |
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119 # rescaling |
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120 v, c = len(G), len(C) |
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121 if normalized: |
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122 scale = 1 / ((v - c) * (v - c - 1)) |
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123 if not G.is_directed(): |
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124 scale *= 2 |
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125 else: |
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126 scale = None |
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127 if scale is not None: |
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128 betweenness *= scale |
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129 return betweenness |
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130 |
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131 |
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132 def group_closeness_centrality(G, S, weight=None): |
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133 r"""Compute the group closeness centrality for a group of nodes. |
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134 |
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135 Group closeness centrality of a group of nodes $S$ is a measure |
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136 of how close the group is to the other nodes in the graph. |
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137 |
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138 .. math:: |
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139 |
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140 c_{close}(S) = \frac{|V-S|}{\sum_{v \in V-S} d_{S, v}} |
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141 |
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142 d_{S, v} = min_{u \in S} (d_{u, v}) |
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143 |
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144 where $V$ is the set of nodes, $d_{S, v}$ is the distance of |
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145 the group $S$ from $v$ defined as above. ($V-S$ is the set of nodes |
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146 in $V$ that are not in $S$). |
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147 |
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148 Parameters |
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149 ---------- |
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150 G : graph |
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151 A NetworkX graph. |
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152 |
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153 S : list or set |
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154 S is a group of nodes which belong to G, for which group closeness |
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155 centrality is to be calculated. |
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156 |
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157 weight : None or string, optional (default=None) |
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158 If None, all edge weights are considered equal. |
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159 Otherwise holds the name of the edge attribute used as weight. |
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160 |
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161 Raises |
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162 ------ |
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163 NodeNotFound |
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164 If node(s) in S are not present in G. |
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165 |
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166 Returns |
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167 ------- |
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168 closeness : float |
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169 Group closeness centrality of the group S. |
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170 |
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171 See Also |
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172 -------- |
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173 closeness_centrality |
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174 |
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175 Notes |
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176 ----- |
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177 The measure was introduced in [1]_. |
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178 The formula implemented here is described in [2]_. |
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179 |
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180 Higher values of closeness indicate greater centrality. |
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181 |
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182 It is assumed that 1 / 0 is 0 (required in the case of directed graphs, |
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183 or when a shortest path length is 0). |
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184 |
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185 The number of nodes in the group must be a maximum of n - 1 where `n` |
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186 is the total number of nodes in the graph. |
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187 |
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188 For directed graphs, the incoming distance is utilized here. To use the |
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189 outward distance, act on `G.reverse()`. |
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190 |
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191 For weighted graphs the edge weights must be greater than zero. |
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192 Zero edge weights can produce an infinite number of equal length |
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193 paths between pairs of nodes. |
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194 |
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195 References |
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196 ---------- |
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197 .. [1] M G Everett and S P Borgatti: |
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198 The Centrality of Groups and Classes. |
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199 Journal of Mathematical Sociology. 23(3): 181-201. 1999. |
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200 http://www.analytictech.com/borgatti/group_centrality.htm |
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201 .. [2] J. Zhao et. al.: |
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202 Measuring and Maximizing Group Closeness Centrality over |
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203 Disk Resident Graphs. |
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204 WWWConference Proceedings, 2014. 689-694. |
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205 http://wwwconference.org/proceedings/www2014/companion/p689.pdf |
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206 """ |
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207 if G.is_directed(): |
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208 G = G.reverse() # reverse view |
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209 closeness = 0 # initialize to 0 |
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210 V = set(G) # set of nodes in G |
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211 S = set(S) # set of nodes in group S |
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212 V_S = V - S # set of nodes in V but not S |
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213 shortest_path_lengths = nx.multi_source_dijkstra_path_length(G, S, weight=weight) |
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214 # accumulation |
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215 for v in V_S: |
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216 try: |
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217 closeness += shortest_path_lengths[v] |
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218 except KeyError: # no path exists |
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219 closeness += 0 |
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220 try: |
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221 closeness = len(V_S) / closeness |
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222 except ZeroDivisionError: # 1 / 0 assumed as 0 |
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223 closeness = 0 |
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224 return closeness |
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225 |
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226 |
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227 def group_degree_centrality(G, S): |
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228 """Compute the group degree centrality for a group of nodes. |
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229 |
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230 Group degree centrality of a group of nodes $S$ is the fraction |
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231 of non-group members connected to group members. |
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232 |
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233 Parameters |
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234 ---------- |
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235 G : graph |
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236 A NetworkX graph. |
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237 |
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238 S : list or set |
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239 S is a group of nodes which belong to G, for which group degree |
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240 centrality is to be calculated. |
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241 |
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242 Raises |
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243 ------ |
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244 NetworkXError |
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245 If node(s) in S are not in G. |
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246 |
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247 Returns |
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248 ------- |
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249 centrality : float |
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250 Group degree centrality of the group S. |
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251 |
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252 See Also |
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253 -------- |
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254 degree_centrality |
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255 group_in_degree_centrality |
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256 group_out_degree_centrality |
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257 |
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258 Notes |
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259 ----- |
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260 The measure was introduced in [1]_. |
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261 |
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262 The number of nodes in the group must be a maximum of n - 1 where `n` |
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263 is the total number of nodes in the graph. |
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264 |
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265 References |
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266 ---------- |
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267 .. [1] M G Everett and S P Borgatti: |
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268 The Centrality of Groups and Classes. |
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269 Journal of Mathematical Sociology. 23(3): 181-201. 1999. |
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270 http://www.analytictech.com/borgatti/group_centrality.htm |
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271 """ |
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272 centrality = len(set().union(*list(set(G.neighbors(i)) for i in S)) - set(S)) |
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273 centrality /= len(G.nodes()) - len(S) |
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274 return centrality |
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275 |
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276 |
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277 @not_implemented_for("undirected") |
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278 def group_in_degree_centrality(G, S): |
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279 """Compute the group in-degree centrality for a group of nodes. |
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280 |
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281 Group in-degree centrality of a group of nodes $S$ is the fraction |
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282 of non-group members connected to group members by incoming edges. |
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283 |
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284 Parameters |
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285 ---------- |
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286 G : graph |
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287 A NetworkX graph. |
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288 |
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289 S : list or set |
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290 S is a group of nodes which belong to G, for which group in-degree |
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291 centrality is to be calculated. |
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292 |
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293 Returns |
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294 ------- |
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295 centrality : float |
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296 Group in-degree centrality of the group S. |
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297 |
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298 Raises |
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299 ------ |
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300 NetworkXNotImplemented |
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301 If G is undirected. |
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302 |
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303 NodeNotFound |
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304 If node(s) in S are not in G. |
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305 |
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306 See Also |
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307 -------- |
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308 degree_centrality |
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309 group_degree_centrality |
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310 group_out_degree_centrality |
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311 |
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312 Notes |
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313 ----- |
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314 The number of nodes in the group must be a maximum of n - 1 where `n` |
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315 is the total number of nodes in the graph. |
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316 |
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317 `G.neighbors(i)` gives nodes with an outward edge from i, in a DiGraph, |
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318 so for group in-degree centrality, the reverse graph is used. |
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319 """ |
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320 return group_degree_centrality(G.reverse(), S) |
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321 |
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322 |
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323 @not_implemented_for("undirected") |
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324 def group_out_degree_centrality(G, S): |
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325 """Compute the group out-degree centrality for a group of nodes. |
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326 |
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327 Group out-degree centrality of a group of nodes $S$ is the fraction |
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328 of non-group members connected to group members by outgoing edges. |
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329 |
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330 Parameters |
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331 ---------- |
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332 G : graph |
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333 A NetworkX graph. |
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334 |
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335 S : list or set |
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336 S is a group of nodes which belong to G, for which group in-degree |
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337 centrality is to be calculated. |
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338 |
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339 Returns |
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340 ------- |
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341 centrality : float |
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342 Group out-degree centrality of the group S. |
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343 |
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344 Raises |
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345 ------ |
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346 NetworkXNotImplemented |
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347 If G is undirected. |
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348 |
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349 NodeNotFound |
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350 If node(s) in S are not in G. |
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351 |
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352 See Also |
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353 -------- |
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354 degree_centrality |
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355 group_degree_centrality |
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356 group_in_degree_centrality |
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357 |
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358 Notes |
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359 ----- |
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360 The number of nodes in the group must be a maximum of n - 1 where `n` |
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361 is the total number of nodes in the graph. |
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362 |
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363 `G.neighbors(i)` gives nodes with an outward edge from i, in a DiGraph, |
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364 so for group out-degree centrality, the graph itself is used. |
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365 """ |
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366 return group_degree_centrality(G, S) |