Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/bipartite/centrality.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 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 `n-1` 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 all-pairs 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 n-1 / 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 |