Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/cluster.py @ 0:4f3585e2f14b draft default tip
"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """Algorithms to characterize the number of triangles in a graph.""" |
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2 |
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3 from itertools import chain |
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4 from itertools import combinations |
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5 from collections import Counter |
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6 |
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7 from networkx.utils import not_implemented_for |
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8 |
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9 __all__ = [ |
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10 "triangles", |
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11 "average_clustering", |
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12 "clustering", |
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13 "transitivity", |
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14 "square_clustering", |
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15 "generalized_degree", |
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16 ] |
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17 |
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18 |
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19 @not_implemented_for("directed") |
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20 def triangles(G, nodes=None): |
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21 """Compute the number of triangles. |
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22 |
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23 Finds the number of triangles that include a node as one vertex. |
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24 |
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25 Parameters |
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26 ---------- |
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27 G : graph |
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28 A networkx graph |
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29 nodes : container of nodes, optional (default= all nodes in G) |
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30 Compute triangles for nodes in this container. |
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31 |
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32 Returns |
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33 ------- |
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34 out : dictionary |
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35 Number of triangles keyed by node label. |
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36 |
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37 Examples |
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38 -------- |
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39 >>> G = nx.complete_graph(5) |
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40 >>> print(nx.triangles(G, 0)) |
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41 6 |
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42 >>> print(nx.triangles(G)) |
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43 {0: 6, 1: 6, 2: 6, 3: 6, 4: 6} |
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44 >>> print(list(nx.triangles(G, (0, 1)).values())) |
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45 [6, 6] |
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46 |
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47 Notes |
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48 ----- |
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49 When computing triangles for the entire graph each triangle is counted |
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50 three times, once at each node. Self loops are ignored. |
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51 |
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52 """ |
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53 # If `nodes` represents a single node in the graph, return only its number |
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54 # of triangles. |
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55 if nodes in G: |
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56 return next(_triangles_and_degree_iter(G, nodes))[2] // 2 |
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57 # Otherwise, `nodes` represents an iterable of nodes, so return a |
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58 # dictionary mapping node to number of triangles. |
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59 return {v: t // 2 for v, d, t, _ in _triangles_and_degree_iter(G, nodes)} |
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60 |
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61 |
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62 @not_implemented_for("multigraph") |
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63 def _triangles_and_degree_iter(G, nodes=None): |
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64 """ Return an iterator of (node, degree, triangles, generalized degree). |
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65 |
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66 This double counts triangles so you may want to divide by 2. |
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67 See degree(), triangles() and generalized_degree() for definitions |
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68 and details. |
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69 |
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70 """ |
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71 if nodes is None: |
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72 nodes_nbrs = G.adj.items() |
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73 else: |
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74 nodes_nbrs = ((n, G[n]) for n in G.nbunch_iter(nodes)) |
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75 |
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76 for v, v_nbrs in nodes_nbrs: |
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77 vs = set(v_nbrs) - {v} |
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78 gen_degree = Counter(len(vs & (set(G[w]) - {w})) for w in vs) |
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79 ntriangles = sum(k * val for k, val in gen_degree.items()) |
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80 yield (v, len(vs), ntriangles, gen_degree) |
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81 |
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82 |
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83 @not_implemented_for("multigraph") |
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84 def _weighted_triangles_and_degree_iter(G, nodes=None, weight="weight"): |
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85 """ Return an iterator of (node, degree, weighted_triangles). |
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86 |
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87 Used for weighted clustering. |
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88 |
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89 """ |
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90 if weight is None or G.number_of_edges() == 0: |
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91 max_weight = 1 |
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92 else: |
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93 max_weight = max(d.get(weight, 1) for u, v, d in G.edges(data=True)) |
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94 if nodes is None: |
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95 nodes_nbrs = G.adj.items() |
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96 else: |
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97 nodes_nbrs = ((n, G[n]) for n in G.nbunch_iter(nodes)) |
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98 |
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99 def wt(u, v): |
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100 return G[u][v].get(weight, 1) / max_weight |
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101 |
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102 for i, nbrs in nodes_nbrs: |
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103 inbrs = set(nbrs) - {i} |
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104 weighted_triangles = 0 |
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105 seen = set() |
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106 for j in inbrs: |
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107 seen.add(j) |
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108 # This prevents double counting. |
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109 jnbrs = set(G[j]) - seen |
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110 # Only compute the edge weight once, before the inner inner |
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111 # loop. |
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112 wij = wt(i, j) |
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113 weighted_triangles += sum( |
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114 (wij * wt(j, k) * wt(k, i)) ** (1 / 3) for k in inbrs & jnbrs |
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115 ) |
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116 yield (i, len(inbrs), 2 * weighted_triangles) |
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117 |
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118 |
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119 @not_implemented_for("multigraph") |
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120 def _directed_triangles_and_degree_iter(G, nodes=None): |
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121 """ Return an iterator of |
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122 (node, total_degree, reciprocal_degree, directed_triangles). |
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123 |
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124 Used for directed clustering. |
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125 |
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126 """ |
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127 nodes_nbrs = ((n, G._pred[n], G._succ[n]) for n in G.nbunch_iter(nodes)) |
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128 |
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129 for i, preds, succs in nodes_nbrs: |
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130 ipreds = set(preds) - {i} |
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131 isuccs = set(succs) - {i} |
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132 |
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133 directed_triangles = 0 |
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134 for j in chain(ipreds, isuccs): |
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135 jpreds = set(G._pred[j]) - {j} |
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136 jsuccs = set(G._succ[j]) - {j} |
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137 directed_triangles += sum( |
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138 1 |
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139 for k in chain( |
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140 (ipreds & jpreds), |
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141 (ipreds & jsuccs), |
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142 (isuccs & jpreds), |
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143 (isuccs & jsuccs), |
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144 ) |
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145 ) |
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146 dtotal = len(ipreds) + len(isuccs) |
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147 dbidirectional = len(ipreds & isuccs) |
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148 yield (i, dtotal, dbidirectional, directed_triangles) |
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149 |
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150 |
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151 @not_implemented_for("multigraph") |
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152 def _directed_weighted_triangles_and_degree_iter(G, nodes=None, weight="weight"): |
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153 """ Return an iterator of |
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154 (node, total_degree, reciprocal_degree, directed_weighted_triangles). |
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155 |
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156 Used for directed weighted clustering. |
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157 |
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158 """ |
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159 if weight is None or G.number_of_edges() == 0: |
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160 max_weight = 1 |
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161 else: |
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162 max_weight = max(d.get(weight, 1) for u, v, d in G.edges(data=True)) |
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163 |
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164 nodes_nbrs = ((n, G._pred[n], G._succ[n]) for n in G.nbunch_iter(nodes)) |
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165 |
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166 def wt(u, v): |
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167 return G[u][v].get(weight, 1) / max_weight |
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168 |
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169 for i, preds, succs in nodes_nbrs: |
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170 ipreds = set(preds) - {i} |
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171 isuccs = set(succs) - {i} |
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172 |
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173 directed_triangles = 0 |
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174 for j in ipreds: |
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175 jpreds = set(G._pred[j]) - {j} |
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176 jsuccs = set(G._succ[j]) - {j} |
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177 directed_triangles += sum( |
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178 (wt(j, i) * wt(k, i) * wt(k, j)) ** (1 / 3) for k in ipreds & jpreds |
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179 ) |
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180 directed_triangles += sum( |
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181 (wt(j, i) * wt(k, i) * wt(j, k)) ** (1 / 3) for k in ipreds & jsuccs |
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182 ) |
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183 directed_triangles += sum( |
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184 (wt(j, i) * wt(i, k) * wt(k, j)) ** (1 / 3) for k in isuccs & jpreds |
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185 ) |
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186 directed_triangles += sum( |
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187 (wt(j, i) * wt(i, k) * wt(j, k)) ** (1 / 3) for k in isuccs & jsuccs |
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188 ) |
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189 |
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190 for j in isuccs: |
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191 jpreds = set(G._pred[j]) - {j} |
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192 jsuccs = set(G._succ[j]) - {j} |
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193 directed_triangles += sum( |
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194 (wt(i, j) * wt(k, i) * wt(k, j)) ** (1 / 3) for k in ipreds & jpreds |
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195 ) |
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196 directed_triangles += sum( |
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197 (wt(i, j) * wt(k, i) * wt(j, k)) ** (1 / 3) for k in ipreds & jsuccs |
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198 ) |
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199 directed_triangles += sum( |
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200 (wt(i, j) * wt(i, k) * wt(k, j)) ** (1 / 3) for k in isuccs & jpreds |
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201 ) |
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202 directed_triangles += sum( |
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203 (wt(i, j) * wt(i, k) * wt(j, k)) ** (1 / 3) for k in isuccs & jsuccs |
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204 ) |
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205 |
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206 dtotal = len(ipreds) + len(isuccs) |
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207 dbidirectional = len(ipreds & isuccs) |
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208 yield (i, dtotal, dbidirectional, directed_triangles) |
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209 |
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210 |
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211 def average_clustering(G, nodes=None, weight=None, count_zeros=True): |
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212 r"""Compute the average clustering coefficient for the graph G. |
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213 |
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214 The clustering coefficient for the graph is the average, |
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215 |
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216 .. math:: |
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217 |
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218 C = \frac{1}{n}\sum_{v \in G} c_v, |
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219 |
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220 where :math:`n` is the number of nodes in `G`. |
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221 |
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222 Parameters |
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223 ---------- |
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224 G : graph |
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225 |
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226 nodes : container of nodes, optional (default=all nodes in G) |
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227 Compute average clustering for nodes in this container. |
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228 |
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229 weight : string or None, optional (default=None) |
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230 The edge attribute that holds the numerical value used as a weight. |
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231 If None, then each edge has weight 1. |
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232 |
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233 count_zeros : bool |
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234 If False include only the nodes with nonzero clustering in the average. |
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235 |
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236 Returns |
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237 ------- |
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238 avg : float |
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239 Average clustering |
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240 |
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241 Examples |
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242 -------- |
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243 >>> G = nx.complete_graph(5) |
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244 >>> print(nx.average_clustering(G)) |
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245 1.0 |
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246 |
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247 Notes |
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248 ----- |
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249 This is a space saving routine; it might be faster |
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250 to use the clustering function to get a list and then take the average. |
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251 |
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252 Self loops are ignored. |
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253 |
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254 References |
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255 ---------- |
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256 .. [1] Generalizations of the clustering coefficient to weighted |
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257 complex networks by J. Saramäki, M. Kivelä, J.-P. Onnela, |
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258 K. Kaski, and J. Kertész, Physical Review E, 75 027105 (2007). |
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259 http://jponnela.com/web_documents/a9.pdf |
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260 .. [2] Marcus Kaiser, Mean clustering coefficients: the role of isolated |
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261 nodes and leafs on clustering measures for small-world networks. |
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262 https://arxiv.org/abs/0802.2512 |
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263 """ |
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264 c = clustering(G, nodes, weight=weight).values() |
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265 if not count_zeros: |
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266 c = [v for v in c if v > 0] |
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267 return sum(c) / len(c) |
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268 |
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269 |
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270 def clustering(G, nodes=None, weight=None): |
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271 r"""Compute the clustering coefficient for nodes. |
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272 |
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273 For unweighted graphs, the clustering of a node :math:`u` |
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274 is the fraction of possible triangles through that node that exist, |
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275 |
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276 .. math:: |
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277 |
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278 c_u = \frac{2 T(u)}{deg(u)(deg(u)-1)}, |
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279 |
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280 where :math:`T(u)` is the number of triangles through node :math:`u` and |
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281 :math:`deg(u)` is the degree of :math:`u`. |
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282 |
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283 For weighted graphs, there are several ways to define clustering [1]_. |
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284 the one used here is defined |
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285 as the geometric average of the subgraph edge weights [2]_, |
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286 |
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287 .. math:: |
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288 |
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289 c_u = \frac{1}{deg(u)(deg(u)-1))} |
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290 \sum_{vw} (\hat{w}_{uv} \hat{w}_{uw} \hat{w}_{vw})^{1/3}. |
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291 |
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292 The edge weights :math:`\hat{w}_{uv}` are normalized by the maximum weight |
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293 in the network :math:`\hat{w}_{uv} = w_{uv}/\max(w)`. |
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294 |
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295 The value of :math:`c_u` is assigned to 0 if :math:`deg(u) < 2`. |
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296 |
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297 For directed graphs, the clustering is similarly defined as the fraction |
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298 of all possible directed triangles or geometric average of the subgraph |
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299 edge weights for unweighted and weighted directed graph respectively [3]_. |
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300 |
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301 .. math:: |
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302 |
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303 c_u = \frac{1}{deg^{tot}(u)(deg^{tot}(u)-1) - 2deg^{\leftrightarrow}(u)} |
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304 T(u), |
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305 |
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306 where :math:`T(u)` is the number of directed triangles through node |
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307 :math:`u`, :math:`deg^{tot}(u)` is the sum of in degree and out degree of |
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308 :math:`u` and :math:`deg^{\leftrightarrow}(u)` is the reciprocal degree of |
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309 :math:`u`. |
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310 |
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311 Parameters |
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312 ---------- |
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313 G : graph |
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314 |
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315 nodes : container of nodes, optional (default=all nodes in G) |
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316 Compute clustering for nodes in this container. |
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317 |
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318 weight : string or None, optional (default=None) |
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319 The edge attribute that holds the numerical value used as a weight. |
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320 If None, then each edge has weight 1. |
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321 |
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322 Returns |
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323 ------- |
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324 out : float, or dictionary |
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325 Clustering coefficient at specified nodes |
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326 |
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327 Examples |
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328 -------- |
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329 >>> G = nx.complete_graph(5) |
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330 >>> print(nx.clustering(G, 0)) |
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331 1.0 |
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332 >>> print(nx.clustering(G)) |
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333 {0: 1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0} |
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334 |
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335 Notes |
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336 ----- |
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337 Self loops are ignored. |
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338 |
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339 References |
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340 ---------- |
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341 .. [1] Generalizations of the clustering coefficient to weighted |
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342 complex networks by J. Saramäki, M. Kivelä, J.-P. Onnela, |
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343 K. Kaski, and J. Kertész, Physical Review E, 75 027105 (2007). |
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344 http://jponnela.com/web_documents/a9.pdf |
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345 .. [2] Intensity and coherence of motifs in weighted complex |
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346 networks by J. P. Onnela, J. Saramäki, J. Kertész, and K. Kaski, |
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347 Physical Review E, 71(6), 065103 (2005). |
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348 .. [3] Clustering in complex directed networks by G. Fagiolo, |
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349 Physical Review E, 76(2), 026107 (2007). |
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350 """ |
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351 if G.is_directed(): |
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352 if weight is not None: |
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353 td_iter = _directed_weighted_triangles_and_degree_iter(G, nodes, weight) |
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354 clusterc = { |
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355 v: 0 if t == 0 else t / ((dt * (dt - 1) - 2 * db) * 2) |
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356 for v, dt, db, t in td_iter |
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357 } |
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358 else: |
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359 td_iter = _directed_triangles_and_degree_iter(G, nodes) |
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360 clusterc = { |
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361 v: 0 if t == 0 else t / ((dt * (dt - 1) - 2 * db) * 2) |
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362 for v, dt, db, t in td_iter |
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363 } |
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364 else: |
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365 if weight is not None: |
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366 td_iter = _weighted_triangles_and_degree_iter(G, nodes, weight) |
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367 clusterc = {v: 0 if t == 0 else t / (d * (d - 1)) for v, d, t in td_iter} |
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368 else: |
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369 td_iter = _triangles_and_degree_iter(G, nodes) |
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370 clusterc = {v: 0 if t == 0 else t / (d * (d - 1)) for v, d, t, _ in td_iter} |
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371 if nodes in G: |
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372 # Return the value of the sole entry in the dictionary. |
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373 return clusterc[nodes] |
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374 return clusterc |
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375 |
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376 |
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377 def transitivity(G): |
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378 r"""Compute graph transitivity, the fraction of all possible triangles |
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379 present in G. |
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380 |
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381 Possible triangles are identified by the number of "triads" |
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382 (two edges with a shared vertex). |
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383 |
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384 The transitivity is |
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385 |
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386 .. math:: |
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387 |
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388 T = 3\frac{\#triangles}{\#triads}. |
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389 |
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390 Parameters |
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391 ---------- |
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392 G : graph |
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393 |
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394 Returns |
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395 ------- |
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396 out : float |
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397 Transitivity |
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398 |
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399 Examples |
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400 -------- |
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401 >>> G = nx.complete_graph(5) |
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402 >>> print(nx.transitivity(G)) |
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403 1.0 |
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404 """ |
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405 triangles = sum(t for v, d, t, _ in _triangles_and_degree_iter(G)) |
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406 contri = sum(d * (d - 1) for v, d, t, _ in _triangles_and_degree_iter(G)) |
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407 return 0 if triangles == 0 else triangles / contri |
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408 |
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409 |
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410 def square_clustering(G, nodes=None): |
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411 r""" Compute the squares clustering coefficient for nodes. |
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412 |
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413 For each node return the fraction of possible squares that exist at |
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414 the node [1]_ |
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415 |
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416 .. math:: |
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417 C_4(v) = \frac{ \sum_{u=1}^{k_v} |
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418 \sum_{w=u+1}^{k_v} q_v(u,w) }{ \sum_{u=1}^{k_v} |
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419 \sum_{w=u+1}^{k_v} [a_v(u,w) + q_v(u,w)]}, |
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420 |
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421 where :math:`q_v(u,w)` are the number of common neighbors of :math:`u` and |
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422 :math:`w` other than :math:`v` (ie squares), and :math:`a_v(u,w) = (k_u - |
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423 (1+q_v(u,w)+\theta_{uv}))(k_w - (1+q_v(u,w)+\theta_{uw}))`, where |
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424 :math:`\theta_{uw} = 1` if :math:`u` and :math:`w` are connected and 0 |
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425 otherwise. |
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426 |
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427 Parameters |
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428 ---------- |
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429 G : graph |
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430 |
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431 nodes : container of nodes, optional (default=all nodes in G) |
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432 Compute clustering for nodes in this container. |
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433 |
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434 Returns |
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435 ------- |
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436 c4 : dictionary |
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437 A dictionary keyed by node with the square clustering coefficient value. |
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438 |
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439 Examples |
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440 -------- |
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441 >>> G = nx.complete_graph(5) |
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442 >>> print(nx.square_clustering(G, 0)) |
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443 1.0 |
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444 >>> print(nx.square_clustering(G)) |
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445 {0: 1.0, 1: 1.0, 2: 1.0, 3: 1.0, 4: 1.0} |
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446 |
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447 Notes |
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448 ----- |
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449 While :math:`C_3(v)` (triangle clustering) gives the probability that |
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450 two neighbors of node v are connected with each other, :math:`C_4(v)` is |
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451 the probability that two neighbors of node v share a common |
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452 neighbor different from v. This algorithm can be applied to both |
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453 bipartite and unipartite networks. |
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454 |
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455 References |
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456 ---------- |
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457 .. [1] Pedro G. Lind, Marta C. González, and Hans J. Herrmann. 2005 |
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458 Cycles and clustering in bipartite networks. |
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459 Physical Review E (72) 056127. |
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460 """ |
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461 if nodes is None: |
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462 node_iter = G |
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463 else: |
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464 node_iter = G.nbunch_iter(nodes) |
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465 clustering = {} |
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466 for v in node_iter: |
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467 clustering[v] = 0 |
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468 potential = 0 |
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469 for u, w in combinations(G[v], 2): |
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470 squares = len((set(G[u]) & set(G[w])) - {v}) |
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471 clustering[v] += squares |
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472 degm = squares + 1 |
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473 if w in G[u]: |
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474 degm += 1 |
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475 potential += (len(G[u]) - degm) * (len(G[w]) - degm) + squares |
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476 if potential > 0: |
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477 clustering[v] /= potential |
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478 if nodes in G: |
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479 # Return the value of the sole entry in the dictionary. |
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480 return clustering[nodes] |
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481 return clustering |
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482 |
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483 |
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484 @not_implemented_for("directed") |
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485 def generalized_degree(G, nodes=None): |
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486 r""" Compute the generalized degree for nodes. |
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487 |
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488 For each node, the generalized degree shows how many edges of given |
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489 triangle multiplicity the node is connected to. The triangle multiplicity |
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490 of an edge is the number of triangles an edge participates in. The |
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491 generalized degree of node :math:`i` can be written as a vector |
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492 :math:`\mathbf{k}_i=(k_i^{(0)}, \dotsc, k_i^{(N-2)})` where |
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493 :math:`k_i^{(j)}` is the number of edges attached to node :math:`i` that |
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494 participate in :math:`j` triangles. |
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495 |
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496 Parameters |
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497 ---------- |
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498 G : graph |
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499 |
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500 nodes : container of nodes, optional (default=all nodes in G) |
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501 Compute the generalized degree for nodes in this container. |
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502 |
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503 Returns |
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504 ------- |
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505 out : Counter, or dictionary of Counters |
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506 Generalized degree of specified nodes. The Counter is keyed by edge |
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507 triangle multiplicity. |
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508 |
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509 Examples |
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510 -------- |
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511 >>> G = nx.complete_graph(5) |
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512 >>> print(nx.generalized_degree(G, 0)) |
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513 Counter({3: 4}) |
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514 >>> print(nx.generalized_degree(G)) |
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515 {0: Counter({3: 4}), 1: Counter({3: 4}), 2: Counter({3: 4}), 3: Counter({3: 4}), 4: Counter({3: 4})} |
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516 |
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517 To recover the number of triangles attached to a node: |
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518 |
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519 >>> k1 = nx.generalized_degree(G, 0) |
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520 >>> sum([k * v for k, v in k1.items()]) / 2 == nx.triangles(G, 0) |
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521 True |
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522 |
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523 Notes |
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524 ----- |
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525 In a network of N nodes, the highest triangle multiplicty an edge can have |
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526 is N-2. |
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527 |
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528 The return value does not include a `zero` entry if no edges of a |
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529 particular triangle multiplicity are present. |
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530 |
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531 The number of triangles node :math:`i` is attached to can be recovered from |
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532 the generalized degree :math:`\mathbf{k}_i=(k_i^{(0)}, \dotsc, |
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533 k_i^{(N-2)})` by :math:`(k_i^{(1)}+2k_i^{(2)}+\dotsc +(N-2)k_i^{(N-2)})/2`. |
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534 |
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535 References |
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536 ---------- |
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537 .. [1] Networks with arbitrary edge multiplicities by V. Zlatić, |
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538 D. Garlaschelli and G. Caldarelli, EPL (Europhysics Letters), |
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539 Volume 97, Number 2 (2012). |
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540 https://iopscience.iop.org/article/10.1209/0295-5075/97/28005 |
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541 """ |
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542 if nodes in G: |
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543 return next(_triangles_and_degree_iter(G, nodes))[3] |
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544 return {v: gd for v, d, t, gd in _triangles_and_degree_iter(G, nodes)} |