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annotate planemo/lib/python3.7/site-packages/networkx/algorithms/cluster.py @ 1:56ad4e20f292 draft

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"planemo upload commit 6eee67778febed82ddd413c3ca40b3183a3898f1"
author guerler
date Fri, 31 Jul 2020 00:32:28 -0400
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1 # -*- coding: utf-8 -*-
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2 #
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3 # Copyright (C) 2004-2019 by
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4 # Aric Hagberg <hagberg@lanl.gov>
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5 # Dan Schult <dschult@colgate.edu>
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6 # Pieter Swart <swart@lanl.gov>
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7 # All rights reserved.
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8 # BSD license.
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9 """Algorithms to characterize the number of triangles in a graph."""
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10
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11 from itertools import chain
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12 from itertools import combinations
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13 from collections import Counter
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14
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15 import networkx as nx
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16 from networkx.utils import not_implemented_for
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17
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18 __author__ = """\n""".join(['Aric Hagberg <aric.hagberg@gmail.com>',
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19 'Dan Schult (dschult@colgate.edu)',
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20 'Pieter Swart (swart@lanl.gov)',
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21 'Jordi Torrents <jtorrents@milnou.net>'])
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22
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23 __all__ = ['triangles', 'average_clustering', 'clustering', 'transitivity',
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24 'square_clustering', 'generalized_degree']
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25
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26
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27 @not_implemented_for('directed')
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28 def triangles(G, nodes=None):
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29 """Compute the number of triangles.
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30
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31 Finds the number of triangles that include a node as one vertex.
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32
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33 Parameters
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34 ----------
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35 G : graph
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36 A networkx graph
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37 nodes : container of nodes, optional (default= all nodes in G)
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38 Compute triangles for nodes in this container.
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39
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40 Returns
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41 -------
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42 out : dictionary
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43 Number of triangles keyed by node label.
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44
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45 Examples
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46 --------
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47 >>> G=nx.complete_graph(5)
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48 >>> print(nx.triangles(G,0))
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49 6
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50 >>> print(nx.triangles(G))
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51 {0: 6, 1: 6, 2: 6, 3: 6, 4: 6}
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52 >>> print(list(nx.triangles(G,(0,1)).values()))
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53 [6, 6]
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54
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55 Notes
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56 -----
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57 When computing triangles for the entire graph each triangle is counted
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58 three times, once at each node. Self loops are ignored.
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59
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60 """
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61 # If `nodes` represents a single node in the graph, return only its number
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62 # of triangles.
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63 if nodes in G:
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64 return next(_triangles_and_degree_iter(G, nodes))[2] // 2
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65 # Otherwise, `nodes` represents an iterable of nodes, so return a
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66 # dictionary mapping node to number of triangles.
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67 return {v: t // 2 for v, d, t, _ in _triangles_and_degree_iter(G, nodes)}
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68
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69
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70 @not_implemented_for('multigraph')
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71 def _triangles_and_degree_iter(G, nodes=None):
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72 """ Return an iterator of (node, degree, triangles, generalized degree).
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73
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74 This double counts triangles so you may want to divide by 2.
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75 See degree(), triangles() and generalized_degree() for definitions
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76 and details.
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77
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78 """
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79 if nodes is None:
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80 nodes_nbrs = G.adj.items()
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81 else:
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82 nodes_nbrs = ((n, G[n]) for n in G.nbunch_iter(nodes))
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83
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84 for v, v_nbrs in nodes_nbrs:
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85 vs = set(v_nbrs) - {v}
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86 gen_degree = Counter(len(vs & (set(G[w]) - {w})) for w in vs)
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87 ntriangles = sum(k * val for k, val in gen_degree.items())
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88 yield (v, len(vs), ntriangles, gen_degree)
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89
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90
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91 @not_implemented_for('multigraph')
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92 def _weighted_triangles_and_degree_iter(G, nodes=None, weight='weight'):
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93 """ Return an iterator of (node, degree, weighted_triangles).
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94
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95 Used for weighted clustering.
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96
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97 """
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98 if weight is None or G.number_of_edges() == 0:
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99 max_weight = 1
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100 else:
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101 max_weight = max(d.get(weight, 1) for u, v, d in G.edges(data=True))
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102 if nodes is None:
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103 nodes_nbrs = G.adj.items()
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104 else:
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105 nodes_nbrs = ((n, G[n]) for n in G.nbunch_iter(nodes))
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106
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107 def wt(u, v):
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108 return G[u][v].get(weight, 1) / max_weight
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109
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110 for i, nbrs in nodes_nbrs:
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111 inbrs = set(nbrs) - {i}
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112 weighted_triangles = 0
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113 seen = set()
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114 for j in inbrs:
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115 seen.add(j)
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116 # This prevents double counting.
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117 jnbrs = set(G[j]) - seen
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118 # Only compute the edge weight once, before the inner inner
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119 # loop.
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120 wij = wt(i, j)
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121 weighted_triangles += sum((wij * wt(j, k) * wt(k, i)) ** (1 / 3)
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122 for k in inbrs & jnbrs)
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123 yield (i, len(inbrs), 2 * weighted_triangles)
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124
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125
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126 @not_implemented_for('multigraph')
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127 def _directed_triangles_and_degree_iter(G, nodes=None):
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128 """ Return an iterator of
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129 (node, total_degree, reciprocal_degree, directed_triangles).
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130
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131 Used for directed clustering.
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132
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133 """
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134 nodes_nbrs = ((n, G._pred[n], G._succ[n]) for n in G.nbunch_iter(nodes))
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135
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136 for i, preds, succs in nodes_nbrs:
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137 ipreds = set(preds) - {i}
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138 isuccs = set(succs) - {i}
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139
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140 directed_triangles = 0
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141 for j in chain(ipreds, isuccs):
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142 jpreds = set(G._pred[j]) - {j}
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143 jsuccs = set(G._succ[j]) - {j}
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144 directed_triangles += sum((1 for k in
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145 chain((ipreds & jpreds),
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146 (ipreds & jsuccs),
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147 (isuccs & jpreds),
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148 (isuccs & jsuccs))))
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149 dtotal = len(ipreds) + len(isuccs)
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150 dbidirectional = len(ipreds & isuccs)
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151 yield (i, dtotal, dbidirectional, directed_triangles)
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152
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153
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154 @not_implemented_for('multigraph')
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155 def _directed_weighted_triangles_and_degree_iter(G, nodes=None, weight = 'weight'):
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156 """ Return an iterator of
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157 (node, total_degree, reciprocal_degree, directed_weighted_triangles).
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158
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159 Used for directed weighted clustering.
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160
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161 """
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162 if weight is None or G.number_of_edges() == 0:
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163 max_weight = 1
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164 else:
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165 max_weight = max(d.get(weight, 1) for u, v, d in G.edges(data=True))
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166
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167 nodes_nbrs = ((n, G._pred[n], G._succ[n]) for n in G.nbunch_iter(nodes))
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168
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169 def wt(u, v):
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170 return G[u][v].get(weight, 1) / max_weight
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171
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172 for i, preds, succs in nodes_nbrs:
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173 ipreds = set(preds) - {i}
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174 isuccs = set(succs) - {i}
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175
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176 directed_triangles = 0
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177 for j in ipreds:
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178 jpreds = set(G._pred[j]) - {j}
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179 jsuccs = set(G._succ[j]) - {j}
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180 directed_triangles += sum((wt(j, i) * wt(k, i) * wt(k, j))**(1 / 3)
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181 for k in ipreds & jpreds)
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182 directed_triangles += sum((wt(j, i) * wt(k, i) * wt(j, k))**(1 / 3)
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183 for k in ipreds & jsuccs)
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184 directed_triangles += sum((wt(j, i) * wt(i, k) * wt(k, j))**(1 / 3)
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185 for k in isuccs & jpreds)
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186 directed_triangles += sum((wt(j, i) * wt(i, k) * wt(j, k))**(1 / 3)
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187 for k in isuccs & jsuccs)
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188
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189 for j in isuccs:
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190 jpreds = set(G._pred[j]) - {j}
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191 jsuccs = set(G._succ[j]) - {j}
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192 directed_triangles += sum((wt(i, j) * wt(k, i) * wt(k, j))**(1 / 3)
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193 for k in ipreds & jpreds)
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194 directed_triangles += sum((wt(i, j) * wt(k, i) * wt(j, k))**(1 / 3)
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195 for k in ipreds & jsuccs)
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196 directed_triangles += sum((wt(i, j) * wt(i, k) * wt(k, j))**(1 / 3)
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197 for k in isuccs & jpreds)
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198 directed_triangles += sum((wt(i, j) * wt(i, k) * wt(j, k))**(1 / 3)
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199 for k in isuccs & jsuccs)
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200
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201 dtotal = len(ipreds) + len(isuccs)
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202 dbidirectional = len(ipreds & isuccs)
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203 yield (i, dtotal, dbidirectional, directed_triangles)
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204
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205
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206 def average_clustering(G, nodes=None, weight=None, count_zeros=True):
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207 r"""Compute the average clustering coefficient for the graph G.
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208
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209 The clustering coefficient for the graph is the average,
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210
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211 .. math::
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212
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213 C = \frac{1}{n}\sum_{v \in G} c_v,
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214
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215 where :math:`n` is the number of nodes in `G`.
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216
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217 Parameters
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218 ----------
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219 G : graph
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220
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221 nodes : container of nodes, optional (default=all nodes in G)
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222 Compute average clustering for nodes in this container.
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223
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224 weight : string or None, optional (default=None)
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225 The edge attribute that holds the numerical value used as a weight.
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226 If None, then each edge has weight 1.
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227
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228 count_zeros : bool
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229 If False include only the nodes with nonzero clustering in the average.
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230
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231 Returns
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232 -------
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233 avg : float
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234 Average clustering
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235
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236 Examples
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237 --------
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238 >>> G=nx.complete_graph(5)
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239 >>> print(nx.average_clustering(G))
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240 1.0
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241
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242 Notes
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243 -----
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244 This is a space saving routine; it might be faster
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245 to use the clustering function to get a list and then take the average.
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246
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247 Self loops are ignored.
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248
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249 References
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250 ----------
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251 .. [1] Generalizations of the clustering coefficient to weighted
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252 complex networks by J. Saramäki, M. Kivelä, J.-P. Onnela,
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253 K. Kaski, and J. Kertész, Physical Review E, 75 027105 (2007).
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254 http://jponnela.com/web_documents/a9.pdf
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255 .. [2] Marcus Kaiser, Mean clustering coefficients: the role of isolated
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256 nodes and leafs on clustering measures for small-world networks.
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257 https://arxiv.org/abs/0802.2512
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258 """
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259 c = clustering(G, nodes, weight=weight).values()
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260 if not count_zeros:
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261 c = [v for v in c if v > 0]
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262 return sum(c) / len(c)
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263
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264
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265 def clustering(G, nodes=None, weight=None):
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266 r"""Compute the clustering coefficient for nodes.
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267
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268 For unweighted graphs, the clustering of a node :math:`u`
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269 is the fraction of possible triangles through that node that exist,
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270
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271 .. math::
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272
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273 c_u = \frac{2 T(u)}{deg(u)(deg(u)-1)},
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274
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275 where :math:`T(u)` is the number of triangles through node :math:`u` and
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276 :math:`deg(u)` is the degree of :math:`u`.
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277
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278 For weighted graphs, there are several ways to define clustering [1]_.
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279 the one used here is defined
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280 as the geometric average of the subgraph edge weights [2]_,
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281
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282 .. math::
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283
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284 c_u = \frac{1}{deg(u)(deg(u)-1))}
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285 \sum_{vw} (\hat{w}_{uv} \hat{w}_{uw} \hat{w}_{vw})^{1/3}.
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286
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287 The edge weights :math:`\hat{w}_{uv}` are normalized by the maximum weight
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288 in the network :math:`\hat{w}_{uv} = w_{uv}/\max(w)`.
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289
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290 The value of :math:`c_u` is assigned to 0 if :math:`deg(u) < 2`.
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291
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292 For directed graphs, the clustering is similarly defined as the fraction
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293 of all possible directed triangles or geometric average of the subgraph
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294 edge weights for unweighted and weighted directed graph respectively [3]_.
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295
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296 .. math::
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297
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298 c_u = \frac{1}{deg^{tot}(u)(deg^{tot}(u)-1) - 2deg^{\leftrightarrow}(u)}
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299 T(u),
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300
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301 where :math:`T(u)` is the number of directed triangles through node
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302 :math:`u`, :math:`deg^{tot}(u)` is the sum of in degree and out degree of
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303 :math:`u` and :math:`deg^{\leftrightarrow}(u)` is the reciprocal degree of
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304 :math:`u`.
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305
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306 Parameters
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307 ----------
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308 G : graph
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309
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310 nodes : container of nodes, optional (default=all nodes in G)
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311 Compute clustering for nodes in this container.
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312
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313 weight : string or None, optional (default=None)
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314 The edge attribute that holds the numerical value used as a weight.
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315 If None, then each edge has weight 1.
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316
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317 Returns
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318 -------
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319 out : float, or dictionary
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320 Clustering coefficient at specified nodes
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321
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322 Examples
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323 --------
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324 >>> G=nx.complete_graph(5)
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325 >>> print(nx.clustering(G,0))
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326 1.0
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327 >>> print(nx.clustering(G))
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328 {0: 1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0}
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329
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330 Notes
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331 -----
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332 Self loops are ignored.
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333
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334 References
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335 ----------
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336 .. [1] Generalizations of the clustering coefficient to weighted
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337 complex networks by J. Saramäki, M. Kivelä, J.-P. Onnela,
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338 K. Kaski, and J. Kertész, Physical Review E, 75 027105 (2007).
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339 http://jponnela.com/web_documents/a9.pdf
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340 .. [2] Intensity and coherence of motifs in weighted complex
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341 networks by J. P. Onnela, J. Saramäki, J. Kertész, and K. Kaski,
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342 Physical Review E, 71(6), 065103 (2005).
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343 .. [3] Clustering in complex directed networks by G. Fagiolo,
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344 Physical Review E, 76(2), 026107 (2007).
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345 """
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346 if G.is_directed():
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347 if weight is not None:
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348 td_iter = _directed_weighted_triangles_and_degree_iter(
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349 G, nodes, weight)
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350 clusterc = {v: 0 if t == 0 else t / ((dt * (dt - 1) - 2 * db) * 2)
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351 for v, dt, db, t in td_iter}
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352 else:
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353 td_iter = _directed_triangles_and_degree_iter(G, nodes)
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354 clusterc = {v: 0 if t == 0 else t / ((dt * (dt - 1) - 2 * db) * 2)
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355 for v, dt, db, t in td_iter}
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356 else:
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357 if weight is not None:
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358 td_iter = _weighted_triangles_and_degree_iter(G, nodes, weight)
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359 clusterc = {v: 0 if t == 0 else t / (d * (d - 1)) for
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360 v, d, t in td_iter}
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361 else:
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362 td_iter = _triangles_and_degree_iter(G, nodes)
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363 clusterc = {v: 0 if t == 0 else t / (d * (d - 1)) for
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364 v, d, t, _ in td_iter}
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365 if nodes in G:
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366 # Return the value of the sole entry in the dictionary.
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367 return clusterc[nodes]
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368 return clusterc
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369
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370
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371 def transitivity(G):
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372 r"""Compute graph transitivity, the fraction of all possible triangles
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373 present in G.
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374
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375 Possible triangles are identified by the number of "triads"
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376 (two edges with a shared vertex).
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377
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378 The transitivity is
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379
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380 .. math::
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381
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382 T = 3\frac{\#triangles}{\#triads}.
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383
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384 Parameters
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385 ----------
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386 G : graph
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387
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388 Returns
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389 -------
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390 out : float
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391 Transitivity
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392
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393 Examples
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394 --------
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395 >>> G = nx.complete_graph(5)
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396 >>> print(nx.transitivity(G))
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397 1.0
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398 """
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399 triangles = sum(t for v, d, t, _ in _triangles_and_degree_iter(G))
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400 contri = sum(d * (d - 1) for v, d, t, _ in _triangles_and_degree_iter(G))
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401 return 0 if triangles == 0 else triangles / contri
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402
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403
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404 def square_clustering(G, nodes=None):
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405 r""" Compute the squares clustering coefficient for nodes.
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406
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407 For each node return the fraction of possible squares that exist at
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408 the node [1]_
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409
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410 .. math::
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411 C_4(v) = \frac{ \sum_{u=1}^{k_v}
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412 \sum_{w=u+1}^{k_v} q_v(u,w) }{ \sum_{u=1}^{k_v}
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413 \sum_{w=u+1}^{k_v} [a_v(u,w) + q_v(u,w)]},
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414
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415 where :math:`q_v(u,w)` are the number of common neighbors of :math:`u` and
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416 :math:`w` other than :math:`v` (ie squares), and :math:`a_v(u,w) = (k_u -
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417 (1+q_v(u,w)+\theta_{uv}))(k_w - (1+q_v(u,w)+\theta_{uw}))`, where
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418 :math:`\theta_{uw} = 1` if :math:`u` and :math:`w` are connected and 0
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419 otherwise.
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420
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421 Parameters
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422 ----------
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423 G : graph
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424
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425 nodes : container of nodes, optional (default=all nodes in G)
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426 Compute clustering for nodes in this container.
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427
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428 Returns
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429 -------
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430 c4 : dictionary
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431 A dictionary keyed by node with the square clustering coefficient value.
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432
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433 Examples
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434 --------
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435 >>> G=nx.complete_graph(5)
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436 >>> print(nx.square_clustering(G,0))
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437 1.0
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438 >>> print(nx.square_clustering(G))
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439 {0: 1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0}
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440
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441 Notes
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442 -----
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443 While :math:`C_3(v)` (triangle clustering) gives the probability that
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444 two neighbors of node v are connected with each other, :math:`C_4(v)` is
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445 the probability that two neighbors of node v share a common
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446 neighbor different from v. This algorithm can be applied to both
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447 bipartite and unipartite networks.
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448
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449 References
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450 ----------
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451 .. [1] Pedro G. Lind, Marta C. González, and Hans J. Herrmann. 2005
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452 Cycles and clustering in bipartite networks.
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453 Physical Review E (72) 056127.
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454 """
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455 if nodes is None:
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456 node_iter = G
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457 else:
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458 node_iter = G.nbunch_iter(nodes)
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459 clustering = {}
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460 for v in node_iter:
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461 clustering[v] = 0
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462 potential = 0
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463 for u, w in combinations(G[v], 2):
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464 squares = len((set(G[u]) & set(G[w])) - set([v]))
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465 clustering[v] += squares
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466 degm = squares + 1
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467 if w in G[u]:
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468 degm += 1
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469 potential += (len(G[u]) - degm) * (len(G[w]) - degm) + squares
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470 if potential > 0:
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471 clustering[v] /= potential
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472 if nodes in G:
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473 # Return the value of the sole entry in the dictionary.
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474 return clustering[nodes]
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475 return clustering
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476
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477
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478 @not_implemented_for('directed')
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479 def generalized_degree(G, nodes=None):
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480 r""" Compute the generalized degree for nodes.
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481
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482 For each node, the generalized degree shows how many edges of given
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483 triangle multiplicity the node is connected to. The triangle multiplicity
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484 of an edge is the number of triangles an edge participates in. The
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485 generalized degree of node :math:`i` can be written as a vector
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486 :math:`\mathbf{k}_i=(k_i^{(0)}, \dotsc, k_i^{(N-2)})` where
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487 :math:`k_i^{(j)}` is the number of edges attached to node :math:`i` that
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488 participate in :math:`j` triangles.
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489
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490 Parameters
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491 ----------
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492 G : graph
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493
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494 nodes : container of nodes, optional (default=all nodes in G)
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495 Compute the generalized degree for nodes in this container.
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496
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497 Returns
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498 -------
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499 out : Counter, or dictionary of Counters
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500 Generalized degree of specified nodes. The Counter is keyed by edge
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501 triangle multiplicity.
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502
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503 Examples
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504 --------
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505 >>> G=nx.complete_graph(5)
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506 >>> print(nx.generalized_degree(G,0))
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507 Counter({3: 4})
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508 >>> print(nx.generalized_degree(G))
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509 {0: Counter({3: 4}), 1: Counter({3: 4}), 2: Counter({3: 4}), 3: Counter({3: 4}), 4: Counter({3: 4})}
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510
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511 To recover the number of triangles attached to a node:
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512
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513 >>> k1 = nx.generalized_degree(G,0)
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514 >>> sum([k*v for k,v in k1.items()])/2 == nx.triangles(G,0)
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515 True
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516
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517 Notes
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518 -----
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519 In a network of N nodes, the highest triangle multiplicty an edge can have
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520 is N-2.
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521
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522 The return value does not include a `zero` entry if no edges of a
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523 particular triangle multiplicity are present.
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524
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525 The number of triangles node :math:`i` is attached to can be recovered from
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526 the generalized degree :math:`\mathbf{k}_i=(k_i^{(0)}, \dotsc,
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527 k_i^{(N-2)})` by :math:`(k_i^{(1)}+2k_i^{(2)}+\dotsc +(N-2)k_i^{(N-2)})/2`.
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528
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529 References
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530 ----------
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531 .. [1] Networks with arbitrary edge multiplicities by V. Zlatić,
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532 D. Garlaschelli and G. Caldarelli, EPL (Europhysics Letters),
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533 Volume 97, Number 2 (2012).
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534 https://iopscience.iop.org/article/10.1209/0295-5075/97/28005
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535 """
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536 if nodes in G:
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537 return next(_triangles_and_degree_iter(G, nodes))[3]
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538 return {v: gd for v, d, t, gd in _triangles_and_degree_iter(G, nodes)}