Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/generators/random_graphs.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 """ |
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2 Generators for random graphs. |
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3 |
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4 """ |
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5 |
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6 import itertools |
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7 import math |
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8 |
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9 import networkx as nx |
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10 from networkx.utils import py_random_state |
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11 from .classic import empty_graph, path_graph, complete_graph |
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12 from .degree_seq import degree_sequence_tree |
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13 from collections import defaultdict |
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14 |
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15 __all__ = [ |
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16 "fast_gnp_random_graph", |
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17 "gnp_random_graph", |
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18 "dense_gnm_random_graph", |
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19 "gnm_random_graph", |
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20 "erdos_renyi_graph", |
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21 "binomial_graph", |
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22 "newman_watts_strogatz_graph", |
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23 "watts_strogatz_graph", |
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24 "connected_watts_strogatz_graph", |
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25 "random_regular_graph", |
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26 "barabasi_albert_graph", |
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27 "dual_barabasi_albert_graph", |
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28 "extended_barabasi_albert_graph", |
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29 "powerlaw_cluster_graph", |
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30 "random_lobster", |
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31 "random_shell_graph", |
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32 "random_powerlaw_tree", |
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33 "random_powerlaw_tree_sequence", |
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34 "random_kernel_graph", |
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35 ] |
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36 |
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37 |
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38 @py_random_state(2) |
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39 def fast_gnp_random_graph(n, p, seed=None, directed=False): |
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40 """Returns a $G_{n,p}$ random graph, also known as an Erdős-Rényi graph or |
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41 a binomial graph. |
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42 |
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43 Parameters |
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44 ---------- |
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45 n : int |
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46 The number of nodes. |
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47 p : float |
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48 Probability for edge creation. |
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49 seed : integer, random_state, or None (default) |
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50 Indicator of random number generation state. |
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51 See :ref:`Randomness<randomness>`. |
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52 directed : bool, optional (default=False) |
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53 If True, this function returns a directed graph. |
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54 |
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55 Notes |
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56 ----- |
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57 The $G_{n,p}$ graph algorithm chooses each of the $[n (n - 1)] / 2$ |
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58 (undirected) or $n (n - 1)$ (directed) possible edges with probability $p$. |
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59 |
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60 This algorithm [1]_ runs in $O(n + m)$ time, where `m` is the expected number of |
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61 edges, which equals $p n (n - 1) / 2$. This should be faster than |
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62 :func:`gnp_random_graph` when $p$ is small and the expected number of edges |
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63 is small (that is, the graph is sparse). |
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64 |
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65 See Also |
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66 -------- |
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67 gnp_random_graph |
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68 |
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69 References |
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70 ---------- |
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71 .. [1] Vladimir Batagelj and Ulrik Brandes, |
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72 "Efficient generation of large random networks", |
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73 Phys. Rev. E, 71, 036113, 2005. |
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74 """ |
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75 G = empty_graph(n) |
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76 |
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77 if p <= 0 or p >= 1: |
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78 return nx.gnp_random_graph(n, p, seed=seed, directed=directed) |
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79 |
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80 w = -1 |
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81 lp = math.log(1.0 - p) |
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82 |
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83 if directed: |
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84 G = nx.DiGraph(G) |
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85 # Nodes in graph are from 0,n-1 (start with v as the first node index). |
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86 v = 0 |
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87 while v < n: |
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88 lr = math.log(1.0 - seed.random()) |
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89 w = w + 1 + int(lr / lp) |
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90 if v == w: # avoid self loops |
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91 w = w + 1 |
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92 while v < n <= w: |
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93 w = w - n |
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94 v = v + 1 |
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95 if v == w: # avoid self loops |
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96 w = w + 1 |
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97 if v < n: |
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98 G.add_edge(v, w) |
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99 else: |
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100 # Nodes in graph are from 0,n-1 (start with v as the second node index). |
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101 v = 1 |
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102 while v < n: |
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103 lr = math.log(1.0 - seed.random()) |
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104 w = w + 1 + int(lr / lp) |
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105 while w >= v and v < n: |
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106 w = w - v |
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107 v = v + 1 |
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108 if v < n: |
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109 G.add_edge(v, w) |
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110 return G |
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111 |
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112 |
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113 @py_random_state(2) |
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114 def gnp_random_graph(n, p, seed=None, directed=False): |
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115 """Returns a $G_{n,p}$ random graph, also known as an Erdős-Rényi graph |
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116 or a binomial graph. |
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117 |
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118 The $G_{n,p}$ model chooses each of the possible edges with probability $p$. |
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119 |
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120 Parameters |
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121 ---------- |
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122 n : int |
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123 The number of nodes. |
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124 p : float |
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125 Probability for edge creation. |
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126 seed : integer, random_state, or None (default) |
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127 Indicator of random number generation state. |
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128 See :ref:`Randomness<randomness>`. |
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129 directed : bool, optional (default=False) |
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130 If True, this function returns a directed graph. |
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131 |
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132 See Also |
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133 -------- |
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134 fast_gnp_random_graph |
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135 |
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136 Notes |
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137 ----- |
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138 This algorithm [2]_ runs in $O(n^2)$ time. For sparse graphs (that is, for |
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139 small values of $p$), :func:`fast_gnp_random_graph` is a faster algorithm. |
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140 |
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141 :func:`binomial_graph` and :func:`erdos_renyi_graph` are |
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142 aliases for :func:`gnp_random_graph`. |
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143 |
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144 >>> nx.binomial_graph is nx.gnp_random_graph |
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145 True |
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146 >>> nx.erdos_renyi_graph is nx.gnp_random_graph |
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147 True |
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148 |
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149 References |
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150 ---------- |
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151 .. [1] P. Erdős and A. Rényi, On Random Graphs, Publ. Math. 6, 290 (1959). |
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152 .. [2] E. N. Gilbert, Random Graphs, Ann. Math. Stat., 30, 1141 (1959). |
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153 """ |
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154 if directed: |
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155 edges = itertools.permutations(range(n), 2) |
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156 G = nx.DiGraph() |
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157 else: |
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158 edges = itertools.combinations(range(n), 2) |
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159 G = nx.Graph() |
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160 G.add_nodes_from(range(n)) |
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161 if p <= 0: |
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162 return G |
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163 if p >= 1: |
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164 return complete_graph(n, create_using=G) |
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165 |
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166 for e in edges: |
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167 if seed.random() < p: |
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168 G.add_edge(*e) |
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169 return G |
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170 |
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171 |
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172 # add some aliases to common names |
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173 binomial_graph = gnp_random_graph |
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174 erdos_renyi_graph = gnp_random_graph |
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175 |
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176 |
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177 @py_random_state(2) |
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178 def dense_gnm_random_graph(n, m, seed=None): |
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179 """Returns a $G_{n,m}$ random graph. |
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180 |
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181 In the $G_{n,m}$ model, a graph is chosen uniformly at random from the set |
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182 of all graphs with $n$ nodes and $m$ edges. |
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183 |
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184 This algorithm should be faster than :func:`gnm_random_graph` for dense |
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185 graphs. |
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186 |
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187 Parameters |
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188 ---------- |
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189 n : int |
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190 The number of nodes. |
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191 m : int |
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192 The number of edges. |
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193 seed : integer, random_state, or None (default) |
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194 Indicator of random number generation state. |
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195 See :ref:`Randomness<randomness>`. |
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196 |
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197 See Also |
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198 -------- |
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199 gnm_random_graph() |
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200 |
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201 Notes |
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202 ----- |
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203 Algorithm by Keith M. Briggs Mar 31, 2006. |
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204 Inspired by Knuth's Algorithm S (Selection sampling technique), |
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205 in section 3.4.2 of [1]_. |
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206 |
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207 References |
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208 ---------- |
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209 .. [1] Donald E. Knuth, The Art of Computer Programming, |
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210 Volume 2/Seminumerical algorithms, Third Edition, Addison-Wesley, 1997. |
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211 """ |
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212 mmax = n * (n - 1) / 2 |
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213 if m >= mmax: |
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214 G = complete_graph(n) |
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215 else: |
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216 G = empty_graph(n) |
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217 |
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218 if n == 1 or m >= mmax: |
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219 return G |
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220 |
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221 u = 0 |
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222 v = 1 |
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223 t = 0 |
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224 k = 0 |
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225 while True: |
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226 if seed.randrange(mmax - t) < m - k: |
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227 G.add_edge(u, v) |
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228 k += 1 |
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229 if k == m: |
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230 return G |
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231 t += 1 |
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232 v += 1 |
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233 if v == n: # go to next row of adjacency matrix |
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234 u += 1 |
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235 v = u + 1 |
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236 |
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237 |
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238 @py_random_state(2) |
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239 def gnm_random_graph(n, m, seed=None, directed=False): |
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240 """Returns a $G_{n,m}$ random graph. |
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241 |
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242 In the $G_{n,m}$ model, a graph is chosen uniformly at random from the set |
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243 of all graphs with $n$ nodes and $m$ edges. |
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244 |
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245 This algorithm should be faster than :func:`dense_gnm_random_graph` for |
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246 sparse graphs. |
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247 |
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248 Parameters |
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249 ---------- |
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250 n : int |
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251 The number of nodes. |
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252 m : int |
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253 The number of edges. |
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254 seed : integer, random_state, or None (default) |
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255 Indicator of random number generation state. |
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256 See :ref:`Randomness<randomness>`. |
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257 directed : bool, optional (default=False) |
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258 If True return a directed graph |
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259 |
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260 See also |
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261 -------- |
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262 dense_gnm_random_graph |
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263 |
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264 """ |
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265 if directed: |
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266 G = nx.DiGraph() |
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267 else: |
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268 G = nx.Graph() |
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269 G.add_nodes_from(range(n)) |
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270 |
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271 if n == 1: |
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272 return G |
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273 max_edges = n * (n - 1) |
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274 if not directed: |
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275 max_edges /= 2.0 |
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276 if m >= max_edges: |
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277 return complete_graph(n, create_using=G) |
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278 |
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279 nlist = list(G) |
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280 edge_count = 0 |
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281 while edge_count < m: |
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282 # generate random edge,u,v |
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283 u = seed.choice(nlist) |
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284 v = seed.choice(nlist) |
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285 if u == v or G.has_edge(u, v): |
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286 continue |
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287 else: |
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288 G.add_edge(u, v) |
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289 edge_count = edge_count + 1 |
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290 return G |
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291 |
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292 |
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293 @py_random_state(3) |
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294 def newman_watts_strogatz_graph(n, k, p, seed=None): |
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295 """Returns a Newman–Watts–Strogatz small-world graph. |
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296 |
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297 Parameters |
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298 ---------- |
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299 n : int |
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300 The number of nodes. |
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301 k : int |
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302 Each node is joined with its `k` nearest neighbors in a ring |
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303 topology. |
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304 p : float |
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305 The probability of adding a new edge for each edge. |
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306 seed : integer, random_state, or None (default) |
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307 Indicator of random number generation state. |
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308 See :ref:`Randomness<randomness>`. |
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309 |
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310 Notes |
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311 ----- |
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312 First create a ring over $n$ nodes [1]_. Then each node in the ring is |
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313 connected with its $k$ nearest neighbors (or $k - 1$ neighbors if $k$ |
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314 is odd). Then shortcuts are created by adding new edges as follows: for |
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315 each edge $(u, v)$ in the underlying "$n$-ring with $k$ nearest |
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316 neighbors" with probability $p$ add a new edge $(u, w)$ with |
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317 randomly-chosen existing node $w$. In contrast with |
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318 :func:`watts_strogatz_graph`, no edges are removed. |
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319 |
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320 See Also |
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321 -------- |
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322 watts_strogatz_graph() |
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323 |
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324 References |
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325 ---------- |
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326 .. [1] M. E. J. Newman and D. J. Watts, |
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327 Renormalization group analysis of the small-world network model, |
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328 Physics Letters A, 263, 341, 1999. |
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329 https://doi.org/10.1016/S0375-9601(99)00757-4 |
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330 """ |
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331 if k > n: |
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332 raise nx.NetworkXError("k>=n, choose smaller k or larger n") |
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333 |
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334 # If k == n the graph return is a complete graph |
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335 if k == n: |
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336 return nx.complete_graph(n) |
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337 |
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338 G = empty_graph(n) |
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339 nlist = list(G.nodes()) |
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340 fromv = nlist |
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341 # connect the k/2 neighbors |
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342 for j in range(1, k // 2 + 1): |
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343 tov = fromv[j:] + fromv[0:j] # the first j are now last |
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344 for i in range(len(fromv)): |
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345 G.add_edge(fromv[i], tov[i]) |
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346 # for each edge u-v, with probability p, randomly select existing |
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347 # node w and add new edge u-w |
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348 e = list(G.edges()) |
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349 for (u, v) in e: |
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350 if seed.random() < p: |
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351 w = seed.choice(nlist) |
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352 # no self-loops and reject if edge u-w exists |
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353 # is that the correct NWS model? |
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354 while w == u or G.has_edge(u, w): |
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355 w = seed.choice(nlist) |
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356 if G.degree(u) >= n - 1: |
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357 break # skip this rewiring |
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358 else: |
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359 G.add_edge(u, w) |
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360 return G |
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361 |
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362 |
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363 @py_random_state(3) |
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364 def watts_strogatz_graph(n, k, p, seed=None): |
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365 """Returns a Watts–Strogatz small-world graph. |
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366 |
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367 Parameters |
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368 ---------- |
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369 n : int |
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370 The number of nodes |
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371 k : int |
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372 Each node is joined with its `k` nearest neighbors in a ring |
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373 topology. |
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374 p : float |
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375 The probability of rewiring each edge |
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376 seed : integer, random_state, or None (default) |
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377 Indicator of random number generation state. |
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378 See :ref:`Randomness<randomness>`. |
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379 |
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380 See Also |
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381 -------- |
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382 newman_watts_strogatz_graph() |
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383 connected_watts_strogatz_graph() |
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384 |
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385 Notes |
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386 ----- |
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387 First create a ring over $n$ nodes [1]_. Then each node in the ring is joined |
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388 to its $k$ nearest neighbors (or $k - 1$ neighbors if $k$ is odd). |
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389 Then shortcuts are created by replacing some edges as follows: for each |
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390 edge $(u, v)$ in the underlying "$n$-ring with $k$ nearest neighbors" |
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391 with probability $p$ replace it with a new edge $(u, w)$ with uniformly |
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392 random choice of existing node $w$. |
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393 |
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394 In contrast with :func:`newman_watts_strogatz_graph`, the random rewiring |
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395 does not increase the number of edges. The rewired graph is not guaranteed |
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396 to be connected as in :func:`connected_watts_strogatz_graph`. |
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397 |
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398 References |
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399 ---------- |
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400 .. [1] Duncan J. Watts and Steven H. Strogatz, |
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401 Collective dynamics of small-world networks, |
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402 Nature, 393, pp. 440--442, 1998. |
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403 """ |
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404 if k > n: |
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405 raise nx.NetworkXError("k>n, choose smaller k or larger n") |
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406 |
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407 # If k == n, the graph is complete not Watts-Strogatz |
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408 if k == n: |
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409 return nx.complete_graph(n) |
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410 |
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411 G = nx.Graph() |
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412 nodes = list(range(n)) # nodes are labeled 0 to n-1 |
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413 # connect each node to k/2 neighbors |
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414 for j in range(1, k // 2 + 1): |
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415 targets = nodes[j:] + nodes[0:j] # first j nodes are now last in list |
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416 G.add_edges_from(zip(nodes, targets)) |
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417 # rewire edges from each node |
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418 # loop over all nodes in order (label) and neighbors in order (distance) |
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419 # no self loops or multiple edges allowed |
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420 for j in range(1, k // 2 + 1): # outer loop is neighbors |
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421 targets = nodes[j:] + nodes[0:j] # first j nodes are now last in list |
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422 # inner loop in node order |
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423 for u, v in zip(nodes, targets): |
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424 if seed.random() < p: |
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425 w = seed.choice(nodes) |
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426 # Enforce no self-loops or multiple edges |
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427 while w == u or G.has_edge(u, w): |
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428 w = seed.choice(nodes) |
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429 if G.degree(u) >= n - 1: |
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430 break # skip this rewiring |
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431 else: |
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432 G.remove_edge(u, v) |
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433 G.add_edge(u, w) |
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434 return G |
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435 |
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436 |
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437 @py_random_state(4) |
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438 def connected_watts_strogatz_graph(n, k, p, tries=100, seed=None): |
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439 """Returns a connected Watts–Strogatz small-world graph. |
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440 |
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441 Attempts to generate a connected graph by repeated generation of |
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442 Watts–Strogatz small-world graphs. An exception is raised if the maximum |
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443 number of tries is exceeded. |
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444 |
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445 Parameters |
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446 ---------- |
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447 n : int |
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448 The number of nodes |
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449 k : int |
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450 Each node is joined with its `k` nearest neighbors in a ring |
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451 topology. |
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452 p : float |
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453 The probability of rewiring each edge |
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454 tries : int |
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455 Number of attempts to generate a connected graph. |
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456 seed : integer, random_state, or None (default) |
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457 Indicator of random number generation state. |
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458 See :ref:`Randomness<randomness>`. |
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459 |
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460 Notes |
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461 ----- |
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462 First create a ring over $n$ nodes [1]_. Then each node in the ring is joined |
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463 to its $k$ nearest neighbors (or $k - 1$ neighbors if $k$ is odd). |
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464 Then shortcuts are created by replacing some edges as follows: for each |
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465 edge $(u, v)$ in the underlying "$n$-ring with $k$ nearest neighbors" |
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466 with probability $p$ replace it with a new edge $(u, w)$ with uniformly |
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467 random choice of existing node $w$. |
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468 The entire process is repeated until a connected graph results. |
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469 |
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470 See Also |
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471 -------- |
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472 newman_watts_strogatz_graph() |
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473 watts_strogatz_graph() |
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474 |
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475 References |
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476 ---------- |
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477 .. [1] Duncan J. Watts and Steven H. Strogatz, |
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478 Collective dynamics of small-world networks, |
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479 Nature, 393, pp. 440--442, 1998. |
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480 """ |
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481 for i in range(tries): |
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482 # seed is an RNG so should change sequence each call |
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483 G = watts_strogatz_graph(n, k, p, seed) |
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484 if nx.is_connected(G): |
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485 return G |
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486 raise nx.NetworkXError("Maximum number of tries exceeded") |
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487 |
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488 |
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489 @py_random_state(2) |
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490 def random_regular_graph(d, n, seed=None): |
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491 r"""Returns a random $d$-regular graph on $n$ nodes. |
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492 |
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493 The resulting graph has no self-loops or parallel edges. |
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494 |
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495 Parameters |
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496 ---------- |
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497 d : int |
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498 The degree of each node. |
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499 n : integer |
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500 The number of nodes. The value of $n \times d$ must be even. |
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501 seed : integer, random_state, or None (default) |
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502 Indicator of random number generation state. |
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503 See :ref:`Randomness<randomness>`. |
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504 |
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505 Notes |
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506 ----- |
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507 The nodes are numbered from $0$ to $n - 1$. |
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508 |
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509 Kim and Vu's paper [2]_ shows that this algorithm samples in an |
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510 asymptotically uniform way from the space of random graphs when |
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511 $d = O(n^{1 / 3 - \epsilon})$. |
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512 |
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513 Raises |
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514 ------ |
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515 |
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516 NetworkXError |
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517 If $n \times d$ is odd or $d$ is greater than or equal to $n$. |
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518 |
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519 References |
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520 ---------- |
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521 .. [1] A. Steger and N. Wormald, |
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522 Generating random regular graphs quickly, |
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523 Probability and Computing 8 (1999), 377-396, 1999. |
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524 http://citeseer.ist.psu.edu/steger99generating.html |
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525 |
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526 .. [2] Jeong Han Kim and Van H. Vu, |
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527 Generating random regular graphs, |
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528 Proceedings of the thirty-fifth ACM symposium on Theory of computing, |
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529 San Diego, CA, USA, pp 213--222, 2003. |
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530 http://portal.acm.org/citation.cfm?id=780542.780576 |
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531 """ |
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532 if (n * d) % 2 != 0: |
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533 raise nx.NetworkXError("n * d must be even") |
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534 |
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535 if not 0 <= d < n: |
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536 raise nx.NetworkXError("the 0 <= d < n inequality must be satisfied") |
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537 |
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538 if d == 0: |
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539 return empty_graph(n) |
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540 |
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541 def _suitable(edges, potential_edges): |
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542 # Helper subroutine to check if there are suitable edges remaining |
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543 # If False, the generation of the graph has failed |
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544 if not potential_edges: |
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545 return True |
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546 for s1 in potential_edges: |
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547 for s2 in potential_edges: |
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548 # Two iterators on the same dictionary are guaranteed |
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549 # to visit it in the same order if there are no |
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550 # intervening modifications. |
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551 if s1 == s2: |
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552 # Only need to consider s1-s2 pair one time |
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553 break |
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554 if s1 > s2: |
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555 s1, s2 = s2, s1 |
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556 if (s1, s2) not in edges: |
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557 return True |
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558 return False |
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559 |
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560 def _try_creation(): |
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561 # Attempt to create an edge set |
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562 |
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563 edges = set() |
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564 stubs = list(range(n)) * d |
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565 |
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566 while stubs: |
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567 potential_edges = defaultdict(lambda: 0) |
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568 seed.shuffle(stubs) |
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569 stubiter = iter(stubs) |
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570 for s1, s2 in zip(stubiter, stubiter): |
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571 if s1 > s2: |
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572 s1, s2 = s2, s1 |
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573 if s1 != s2 and ((s1, s2) not in edges): |
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574 edges.add((s1, s2)) |
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575 else: |
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576 potential_edges[s1] += 1 |
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577 potential_edges[s2] += 1 |
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578 |
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579 if not _suitable(edges, potential_edges): |
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580 return None # failed to find suitable edge set |
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581 |
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582 stubs = [ |
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583 node |
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584 for node, potential in potential_edges.items() |
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585 for _ in range(potential) |
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586 ] |
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587 return edges |
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588 |
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589 # Even though a suitable edge set exists, |
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590 # the generation of such a set is not guaranteed. |
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591 # Try repeatedly to find one. |
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592 edges = _try_creation() |
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593 while edges is None: |
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594 edges = _try_creation() |
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595 |
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596 G = nx.Graph() |
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597 G.add_edges_from(edges) |
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598 |
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599 return G |
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600 |
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601 |
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602 def _random_subset(seq, m, rng): |
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603 """ Return m unique elements from seq. |
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604 |
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605 This differs from random.sample which can return repeated |
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606 elements if seq holds repeated elements. |
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607 |
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608 Note: rng is a random.Random or numpy.random.RandomState instance. |
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609 """ |
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610 targets = set() |
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611 while len(targets) < m: |
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612 x = rng.choice(seq) |
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613 targets.add(x) |
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614 return targets |
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615 |
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616 |
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617 @py_random_state(2) |
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618 def barabasi_albert_graph(n, m, seed=None): |
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619 """Returns a random graph according to the Barabási–Albert preferential |
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620 attachment model. |
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621 |
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622 A graph of $n$ nodes is grown by attaching new nodes each with $m$ |
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623 edges that are preferentially attached to existing nodes with high degree. |
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624 |
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625 Parameters |
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626 ---------- |
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627 n : int |
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628 Number of nodes |
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629 m : int |
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630 Number of edges to attach from a new node to existing nodes |
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631 seed : integer, random_state, or None (default) |
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632 Indicator of random number generation state. |
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633 See :ref:`Randomness<randomness>`. |
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634 |
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635 Returns |
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636 ------- |
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637 G : Graph |
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638 |
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639 Raises |
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640 ------ |
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641 NetworkXError |
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642 If `m` does not satisfy ``1 <= m < n``. |
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643 |
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644 References |
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645 ---------- |
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646 .. [1] A. L. Barabási and R. Albert "Emergence of scaling in |
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647 random networks", Science 286, pp 509-512, 1999. |
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648 """ |
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649 |
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650 if m < 1 or m >= n: |
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651 raise nx.NetworkXError( |
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652 f"Barabási–Albert network must have m >= 1 and m < n, m = {m}, n = {n}" |
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653 ) |
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654 |
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655 # Add m initial nodes (m0 in barabasi-speak) |
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656 G = empty_graph(m) |
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657 # Target nodes for new edges |
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658 targets = list(range(m)) |
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659 # List of existing nodes, with nodes repeated once for each adjacent edge |
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660 repeated_nodes = [] |
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661 # Start adding the other n-m nodes. The first node is m. |
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662 source = m |
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663 while source < n: |
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664 # Add edges to m nodes from the source. |
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665 G.add_edges_from(zip([source] * m, targets)) |
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666 # Add one node to the list for each new edge just created. |
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667 repeated_nodes.extend(targets) |
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668 # And the new node "source" has m edges to add to the list. |
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669 repeated_nodes.extend([source] * m) |
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670 # Now choose m unique nodes from the existing nodes |
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671 # Pick uniformly from repeated_nodes (preferential attachment) |
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672 targets = _random_subset(repeated_nodes, m, seed) |
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673 source += 1 |
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674 return G |
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675 |
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676 |
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677 @py_random_state(4) |
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678 def dual_barabasi_albert_graph(n, m1, m2, p, seed=None): |
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679 """Returns a random graph according to the dual Barabási–Albert preferential |
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680 attachment model. |
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681 |
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682 A graph of $n$ nodes is grown by attaching new nodes each with either $m_1$ |
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683 edges (with probability $p$) or $m_2$ edges (with probability $1-p$) that |
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684 are preferentially attached to existing nodes with high degree. |
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685 |
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686 Parameters |
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687 ---------- |
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688 n : int |
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689 Number of nodes |
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690 m1 : int |
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691 Number of edges to attach from a new node to existing nodes with probability $p$ |
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692 m2 : int |
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693 Number of edges to attach from a new node to existing nodes with probability $1-p$ |
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694 p : float |
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695 The probability of attaching $m_1$ edges (as opposed to $m_2$ edges) |
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696 seed : integer, random_state, or None (default) |
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697 Indicator of random number generation state. |
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698 See :ref:`Randomness<randomness>`. |
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699 |
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700 Returns |
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701 ------- |
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702 G : Graph |
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703 |
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704 Raises |
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705 ------ |
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706 NetworkXError |
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707 If `m1` and `m2` do not satisfy ``1 <= m1,m2 < n`` or `p` does not satisfy ``0 <= p <= 1``. |
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708 |
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709 References |
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710 ---------- |
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711 .. [1] N. Moshiri "The dual-Barabasi-Albert model", arXiv:1810.10538. |
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712 """ |
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713 |
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714 if m1 < 1 or m1 >= n: |
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715 raise nx.NetworkXError( |
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716 f"Dual Barabási–Albert network must have m1 >= 1 and m1 < n, m1 = {m1}, n = {n}" |
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717 ) |
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718 if m2 < 1 or m2 >= n: |
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719 raise nx.NetworkXError( |
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720 f"Dual Barabási–Albert network must have m2 >= 1 and m2 < n, m2 = {m2}, n = {n}" |
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721 ) |
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722 if p < 0 or p > 1: |
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723 raise nx.NetworkXError( |
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724 f"Dual Barabási–Albert network must have 0 <= p <= 1, p = {p}" |
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725 ) |
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726 |
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727 # For simplicity, if p == 0 or 1, just return BA |
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728 if p == 1: |
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729 return barabasi_albert_graph(n, m1, seed) |
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730 elif p == 0: |
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731 return barabasi_albert_graph(n, m2, seed) |
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732 |
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733 # Add max(m1,m2) initial nodes (m0 in barabasi-speak) |
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734 G = empty_graph(max(m1, m2)) |
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735 # Target nodes for new edges |
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736 targets = list(range(max(m1, m2))) |
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737 # List of existing nodes, with nodes repeated once for each adjacent edge |
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738 repeated_nodes = [] |
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739 # Start adding the remaining nodes. |
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740 source = max(m1, m2) |
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741 # Pick which m to use first time (m1 or m2) |
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742 if seed.random() < p: |
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743 m = m1 |
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744 else: |
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745 m = m2 |
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746 while source < n: |
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747 # Add edges to m nodes from the source. |
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748 G.add_edges_from(zip([source] * m, targets)) |
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749 # Add one node to the list for each new edge just created. |
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750 repeated_nodes.extend(targets) |
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751 # And the new node "source" has m edges to add to the list. |
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752 repeated_nodes.extend([source] * m) |
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753 # Pick which m to use next time (m1 or m2) |
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754 if seed.random() < p: |
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755 m = m1 |
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756 else: |
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757 m = m2 |
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758 # Now choose m unique nodes from the existing nodes |
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759 # Pick uniformly from repeated_nodes (preferential attachment) |
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760 targets = _random_subset(repeated_nodes, m, seed) |
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761 source += 1 |
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762 return G |
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763 |
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764 |
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765 @py_random_state(4) |
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766 def extended_barabasi_albert_graph(n, m, p, q, seed=None): |
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767 """Returns an extended Barabási–Albert model graph. |
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768 |
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769 An extended Barabási–Albert model graph is a random graph constructed |
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770 using preferential attachment. The extended model allows new edges, |
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771 rewired edges or new nodes. Based on the probabilities $p$ and $q$ |
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772 with $p + q < 1$, the growing behavior of the graph is determined as: |
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773 |
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774 1) With $p$ probability, $m$ new edges are added to the graph, |
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775 starting from randomly chosen existing nodes and attached preferentially at the other end. |
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776 |
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777 2) With $q$ probability, $m$ existing edges are rewired |
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778 by randomly choosing an edge and rewiring one end to a preferentially chosen node. |
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779 |
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780 3) With $(1 - p - q)$ probability, $m$ new nodes are added to the graph |
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781 with edges attached preferentially. |
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782 |
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783 When $p = q = 0$, the model behaves just like the Barabási–Alber model. |
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784 |
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785 Parameters |
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786 ---------- |
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787 n : int |
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788 Number of nodes |
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789 m : int |
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790 Number of edges with which a new node attaches to existing nodes |
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791 p : float |
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792 Probability value for adding an edge between existing nodes. p + q < 1 |
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793 q : float |
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794 Probability value of rewiring of existing edges. p + q < 1 |
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795 seed : integer, random_state, or None (default) |
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796 Indicator of random number generation state. |
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797 See :ref:`Randomness<randomness>`. |
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798 |
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799 Returns |
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800 ------- |
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801 G : Graph |
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802 |
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803 Raises |
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804 ------ |
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805 NetworkXError |
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806 If `m` does not satisfy ``1 <= m < n`` or ``1 >= p + q`` |
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807 |
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808 References |
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809 ---------- |
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810 .. [1] Albert, R., & Barabási, A. L. (2000) |
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811 Topology of evolving networks: local events and universality |
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812 Physical review letters, 85(24), 5234. |
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813 """ |
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814 if m < 1 or m >= n: |
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815 msg = f"Extended Barabasi-Albert network needs m>=1 and m<n, m={m}, n={n}" |
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816 raise nx.NetworkXError(msg) |
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817 if p + q >= 1: |
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818 msg = f"Extended Barabasi-Albert network needs p + q <= 1, p={p}, q={q}" |
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819 raise nx.NetworkXError(msg) |
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820 |
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821 # Add m initial nodes (m0 in barabasi-speak) |
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822 G = empty_graph(m) |
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823 |
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824 # List of nodes to represent the preferential attachment random selection. |
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825 # At the creation of the graph, all nodes are added to the list |
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826 # so that even nodes that are not connected have a chance to get selected, |
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827 # for rewiring and adding of edges. |
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828 # With each new edge, nodes at the ends of the edge are added to the list. |
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829 attachment_preference = [] |
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830 attachment_preference.extend(range(m)) |
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831 |
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832 # Start adding the other n-m nodes. The first node is m. |
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833 new_node = m |
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834 while new_node < n: |
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835 a_probability = seed.random() |
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836 |
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837 # Total number of edges of a Clique of all the nodes |
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838 clique_degree = len(G) - 1 |
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839 clique_size = (len(G) * clique_degree) / 2 |
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840 |
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841 # Adding m new edges, if there is room to add them |
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842 if a_probability < p and G.size() <= clique_size - m: |
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843 # Select the nodes where an edge can be added |
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844 elligible_nodes = [nd for nd, deg in G.degree() if deg < clique_degree] |
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845 for i in range(m): |
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846 # Choosing a random source node from elligible_nodes |
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847 src_node = seed.choice(elligible_nodes) |
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848 |
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849 # Picking a possible node that is not 'src_node' or |
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850 # neighbor with 'src_node', with preferential attachment |
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851 prohibited_nodes = list(G[src_node]) |
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852 prohibited_nodes.append(src_node) |
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853 # This will raise an exception if the sequence is empty |
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854 dest_node = seed.choice( |
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855 [nd for nd in attachment_preference if nd not in prohibited_nodes] |
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856 ) |
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857 # Adding the new edge |
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858 G.add_edge(src_node, dest_node) |
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859 |
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860 # Appending both nodes to add to their preferential attachment |
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861 attachment_preference.append(src_node) |
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862 attachment_preference.append(dest_node) |
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863 |
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864 # Adjusting the elligible nodes. Degree may be saturated. |
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865 if G.degree(src_node) == clique_degree: |
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866 elligible_nodes.remove(src_node) |
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867 if ( |
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868 G.degree(dest_node) == clique_degree |
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869 and dest_node in elligible_nodes |
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870 ): |
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871 elligible_nodes.remove(dest_node) |
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872 |
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873 # Rewiring m edges, if there are enough edges |
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874 elif p <= a_probability < (p + q) and m <= G.size() < clique_size: |
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875 # Selecting nodes that have at least 1 edge but that are not |
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876 # fully connected to ALL other nodes (center of star). |
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877 # These nodes are the pivot nodes of the edges to rewire |
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878 elligible_nodes = [nd for nd, deg in G.degree() if 0 < deg < clique_degree] |
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879 for i in range(m): |
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880 # Choosing a random source node |
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881 node = seed.choice(elligible_nodes) |
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882 |
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883 # The available nodes do have a neighbor at least. |
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884 neighbor_nodes = list(G[node]) |
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885 |
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886 # Choosing the other end that will get dettached |
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887 src_node = seed.choice(neighbor_nodes) |
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888 |
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889 # Picking a target node that is not 'node' or |
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890 # neighbor with 'node', with preferential attachment |
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891 neighbor_nodes.append(node) |
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892 dest_node = seed.choice( |
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893 [nd for nd in attachment_preference if nd not in neighbor_nodes] |
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894 ) |
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895 # Rewire |
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896 G.remove_edge(node, src_node) |
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897 G.add_edge(node, dest_node) |
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898 |
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899 # Adjusting the preferential attachment list |
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900 attachment_preference.remove(src_node) |
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901 attachment_preference.append(dest_node) |
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902 |
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903 # Adjusting the elligible nodes. |
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904 # nodes may be saturated or isolated. |
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905 if G.degree(src_node) == 0 and src_node in elligible_nodes: |
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906 elligible_nodes.remove(src_node) |
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907 if dest_node in elligible_nodes: |
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908 if G.degree(dest_node) == clique_degree: |
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909 elligible_nodes.remove(dest_node) |
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910 else: |
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911 if G.degree(dest_node) == 1: |
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912 elligible_nodes.append(dest_node) |
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913 |
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914 # Adding new node with m edges |
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915 else: |
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916 # Select the edges' nodes by preferential attachment |
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917 targets = _random_subset(attachment_preference, m, seed) |
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918 G.add_edges_from(zip([new_node] * m, targets)) |
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919 |
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920 # Add one node to the list for each new edge just created. |
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921 attachment_preference.extend(targets) |
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922 # The new node has m edges to it, plus itself: m + 1 |
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923 attachment_preference.extend([new_node] * (m + 1)) |
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924 new_node += 1 |
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925 return G |
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926 |
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927 |
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928 @py_random_state(3) |
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929 def powerlaw_cluster_graph(n, m, p, seed=None): |
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930 """Holme and Kim algorithm for growing graphs with powerlaw |
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931 degree distribution and approximate average clustering. |
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932 |
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933 Parameters |
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934 ---------- |
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935 n : int |
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936 the number of nodes |
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937 m : int |
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938 the number of random edges to add for each new node |
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939 p : float, |
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940 Probability of adding a triangle after adding a random edge |
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941 seed : integer, random_state, or None (default) |
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942 Indicator of random number generation state. |
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943 See :ref:`Randomness<randomness>`. |
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944 |
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945 Notes |
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946 ----- |
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947 The average clustering has a hard time getting above a certain |
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948 cutoff that depends on `m`. This cutoff is often quite low. The |
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949 transitivity (fraction of triangles to possible triangles) seems to |
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950 decrease with network size. |
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951 |
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952 It is essentially the Barabási–Albert (BA) growth model with an |
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953 extra step that each random edge is followed by a chance of |
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954 making an edge to one of its neighbors too (and thus a triangle). |
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955 |
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956 This algorithm improves on BA in the sense that it enables a |
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957 higher average clustering to be attained if desired. |
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958 |
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959 It seems possible to have a disconnected graph with this algorithm |
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960 since the initial `m` nodes may not be all linked to a new node |
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961 on the first iteration like the BA model. |
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962 |
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963 Raises |
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964 ------ |
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965 NetworkXError |
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966 If `m` does not satisfy ``1 <= m <= n`` or `p` does not |
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967 satisfy ``0 <= p <= 1``. |
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968 |
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969 References |
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970 ---------- |
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971 .. [1] P. Holme and B. J. Kim, |
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972 "Growing scale-free networks with tunable clustering", |
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973 Phys. Rev. E, 65, 026107, 2002. |
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974 """ |
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975 |
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976 if m < 1 or n < m: |
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977 raise nx.NetworkXError(f"NetworkXError must have m>1 and m<n, m={m},n={n}") |
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978 |
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979 if p > 1 or p < 0: |
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980 raise nx.NetworkXError(f"NetworkXError p must be in [0,1], p={p}") |
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981 |
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982 G = empty_graph(m) # add m initial nodes (m0 in barabasi-speak) |
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983 repeated_nodes = list(G.nodes()) # list of existing nodes to sample from |
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984 # with nodes repeated once for each adjacent edge |
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985 source = m # next node is m |
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986 while source < n: # Now add the other n-1 nodes |
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987 possible_targets = _random_subset(repeated_nodes, m, seed) |
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988 # do one preferential attachment for new node |
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989 target = possible_targets.pop() |
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990 G.add_edge(source, target) |
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991 repeated_nodes.append(target) # add one node to list for each new link |
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992 count = 1 |
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993 while count < m: # add m-1 more new links |
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994 if seed.random() < p: # clustering step: add triangle |
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995 neighborhood = [ |
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996 nbr |
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997 for nbr in G.neighbors(target) |
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998 if not G.has_edge(source, nbr) and not nbr == source |
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999 ] |
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1000 if neighborhood: # if there is a neighbor without a link |
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1001 nbr = seed.choice(neighborhood) |
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1002 G.add_edge(source, nbr) # add triangle |
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1003 repeated_nodes.append(nbr) |
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1004 count = count + 1 |
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1005 continue # go to top of while loop |
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1006 # else do preferential attachment step if above fails |
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1007 target = possible_targets.pop() |
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1008 G.add_edge(source, target) |
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1009 repeated_nodes.append(target) |
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1010 count = count + 1 |
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1011 |
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1012 repeated_nodes.extend([source] * m) # add source node to list m times |
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1013 source += 1 |
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1014 return G |
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1015 |
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1016 |
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1017 @py_random_state(3) |
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1018 def random_lobster(n, p1, p2, seed=None): |
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1019 """Returns a random lobster graph. |
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1020 |
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1021 A lobster is a tree that reduces to a caterpillar when pruning all |
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1022 leaf nodes. A caterpillar is a tree that reduces to a path graph |
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1023 when pruning all leaf nodes; setting `p2` to zero produces a caterpillar. |
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1024 |
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1025 This implementation iterates on the probabilities `p1` and `p2` to add |
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1026 edges at levels 1 and 2, respectively. Graphs are therefore constructed |
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1027 iteratively with uniform randomness at each level rather than being selected |
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1028 uniformly at random from the set of all possible lobsters. |
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1029 |
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1030 Parameters |
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1031 ---------- |
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1032 n : int |
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1033 The expected number of nodes in the backbone |
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1034 p1 : float |
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1035 Probability of adding an edge to the backbone |
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1036 p2 : float |
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1037 Probability of adding an edge one level beyond backbone |
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1038 seed : integer, random_state, or None (default) |
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1039 Indicator of random number generation state. |
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1040 See :ref:`Randomness<randomness>`. |
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1041 |
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1042 Raises |
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1043 ------ |
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1044 NetworkXError |
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1045 If `p1` or `p2` parameters are >= 1 because the while loops would never finish. |
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1046 """ |
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1047 p1, p2 = abs(p1), abs(p2) |
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1048 if any([p >= 1 for p in [p1, p2]]): |
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1049 raise nx.NetworkXError("Probability values for `p1` and `p2` must both be < 1.") |
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1050 |
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1051 # a necessary ingredient in any self-respecting graph library |
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1052 llen = int(2 * seed.random() * n + 0.5) |
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1053 L = path_graph(llen) |
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1054 # build caterpillar: add edges to path graph with probability p1 |
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1055 current_node = llen - 1 |
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1056 for n in range(llen): |
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1057 while seed.random() < p1: # add fuzzy caterpillar parts |
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1058 current_node += 1 |
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1059 L.add_edge(n, current_node) |
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1060 cat_node = current_node |
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1061 while seed.random() < p2: # add crunchy lobster bits |
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1062 current_node += 1 |
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1063 L.add_edge(cat_node, current_node) |
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1064 return L # voila, un lobster! |
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1065 |
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1066 |
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1067 @py_random_state(1) |
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1068 def random_shell_graph(constructor, seed=None): |
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1069 """Returns a random shell graph for the constructor given. |
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1070 |
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1071 Parameters |
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1072 ---------- |
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1073 constructor : list of three-tuples |
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1074 Represents the parameters for a shell, starting at the center |
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1075 shell. Each element of the list must be of the form `(n, m, |
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1076 d)`, where `n` is the number of nodes in the shell, `m` is |
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1077 the number of edges in the shell, and `d` is the ratio of |
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1078 inter-shell (next) edges to intra-shell edges. If `d` is zero, |
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1079 there will be no intra-shell edges, and if `d` is one there |
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1080 will be all possible intra-shell edges. |
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1081 seed : integer, random_state, or None (default) |
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1082 Indicator of random number generation state. |
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1083 See :ref:`Randomness<randomness>`. |
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1084 |
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1085 Examples |
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1086 -------- |
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1087 >>> constructor = [(10, 20, 0.8), (20, 40, 0.8)] |
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1088 >>> G = nx.random_shell_graph(constructor) |
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1089 |
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1090 """ |
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1091 G = empty_graph(0) |
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1092 |
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1093 glist = [] |
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1094 intra_edges = [] |
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1095 nnodes = 0 |
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1096 # create gnm graphs for each shell |
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1097 for (n, m, d) in constructor: |
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1098 inter_edges = int(m * d) |
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1099 intra_edges.append(m - inter_edges) |
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1100 g = nx.convert_node_labels_to_integers( |
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1101 gnm_random_graph(n, inter_edges, seed=seed), first_label=nnodes |
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1102 ) |
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1103 glist.append(g) |
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1104 nnodes += n |
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1105 G = nx.operators.union(G, g) |
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1106 |
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1107 # connect the shells randomly |
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1108 for gi in range(len(glist) - 1): |
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1109 nlist1 = list(glist[gi]) |
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1110 nlist2 = list(glist[gi + 1]) |
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1111 total_edges = intra_edges[gi] |
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1112 edge_count = 0 |
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1113 while edge_count < total_edges: |
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1114 u = seed.choice(nlist1) |
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1115 v = seed.choice(nlist2) |
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1116 if u == v or G.has_edge(u, v): |
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1117 continue |
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1118 else: |
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1119 G.add_edge(u, v) |
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1120 edge_count = edge_count + 1 |
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1121 return G |
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1122 |
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1123 |
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1124 @py_random_state(2) |
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1125 def random_powerlaw_tree(n, gamma=3, seed=None, tries=100): |
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1126 """Returns a tree with a power law degree distribution. |
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1127 |
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1128 Parameters |
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1129 ---------- |
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1130 n : int |
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1131 The number of nodes. |
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1132 gamma : float |
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1133 Exponent of the power law. |
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1134 seed : integer, random_state, or None (default) |
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1135 Indicator of random number generation state. |
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1136 See :ref:`Randomness<randomness>`. |
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1137 tries : int |
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1138 Number of attempts to adjust the sequence to make it a tree. |
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1139 |
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1140 Raises |
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1141 ------ |
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1142 NetworkXError |
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1143 If no valid sequence is found within the maximum number of |
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1144 attempts. |
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1145 |
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1146 Notes |
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1147 ----- |
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1148 A trial power law degree sequence is chosen and then elements are |
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1149 swapped with new elements from a powerlaw distribution until the |
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1150 sequence makes a tree (by checking, for example, that the number of |
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1151 edges is one smaller than the number of nodes). |
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1152 |
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1153 """ |
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1154 # This call may raise a NetworkXError if the number of tries is succeeded. |
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1155 seq = random_powerlaw_tree_sequence(n, gamma=gamma, seed=seed, tries=tries) |
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1156 G = degree_sequence_tree(seq) |
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1157 return G |
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1158 |
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1159 |
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1160 @py_random_state(2) |
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1161 def random_powerlaw_tree_sequence(n, gamma=3, seed=None, tries=100): |
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1162 """Returns a degree sequence for a tree with a power law distribution. |
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1163 |
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1164 Parameters |
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1165 ---------- |
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1166 n : int, |
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1167 The number of nodes. |
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1168 gamma : float |
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1169 Exponent of the power law. |
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1170 seed : integer, random_state, or None (default) |
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1171 Indicator of random number generation state. |
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1172 See :ref:`Randomness<randomness>`. |
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1173 tries : int |
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1174 Number of attempts to adjust the sequence to make it a tree. |
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1175 |
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1176 Raises |
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1177 ------ |
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1178 NetworkXError |
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1179 If no valid sequence is found within the maximum number of |
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1180 attempts. |
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1181 |
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1182 Notes |
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1183 ----- |
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1184 A trial power law degree sequence is chosen and then elements are |
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1185 swapped with new elements from a power law distribution until |
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1186 the sequence makes a tree (by checking, for example, that the number of |
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1187 edges is one smaller than the number of nodes). |
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1188 |
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1189 """ |
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1190 # get trial sequence |
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1191 z = nx.utils.powerlaw_sequence(n, exponent=gamma, seed=seed) |
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1192 # round to integer values in the range [0,n] |
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1193 zseq = [min(n, max(int(round(s)), 0)) for s in z] |
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1194 |
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1195 # another sequence to swap values from |
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1196 z = nx.utils.powerlaw_sequence(tries, exponent=gamma, seed=seed) |
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1197 # round to integer values in the range [0,n] |
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1198 swap = [min(n, max(int(round(s)), 0)) for s in z] |
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1199 |
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1200 for deg in swap: |
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1201 # If this degree sequence can be the degree sequence of a tree, return |
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1202 # it. It can be a tree if the number of edges is one fewer than the |
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1203 # number of nodes, or in other words, `n - sum(zseq) / 2 == 1`. We |
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1204 # use an equivalent condition below that avoids floating point |
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1205 # operations. |
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1206 if 2 * n - sum(zseq) == 2: |
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1207 return zseq |
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1208 index = seed.randint(0, n - 1) |
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1209 zseq[index] = swap.pop() |
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1210 |
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1211 raise nx.NetworkXError( |
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1212 f"Exceeded max ({tries}) attempts for a valid tree sequence." |
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1213 ) |
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1214 |
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1215 |
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1216 @py_random_state(3) |
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1217 def random_kernel_graph(n, kernel_integral, kernel_root=None, seed=None): |
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1218 r"""Returns an random graph based on the specified kernel. |
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1219 |
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1220 The algorithm chooses each of the $[n(n-1)]/2$ possible edges with |
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1221 probability specified by a kernel $\kappa(x,y)$ [1]_. The kernel |
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1222 $\kappa(x,y)$ must be a symmetric (in $x,y$), non-negative, |
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1223 bounded function. |
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1224 |
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1225 Parameters |
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1226 ---------- |
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1227 n : int |
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1228 The number of nodes |
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1229 kernal_integral : function |
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1230 Function that returns the definite integral of the kernel $\kappa(x,y)$, |
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1231 $F(y,a,b) := \int_a^b \kappa(x,y)dx$ |
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1232 kernel_root: function (optional) |
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1233 Function that returns the root $b$ of the equation $F(y,a,b) = r$. |
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1234 If None, the root is found using :func:`scipy.optimize.brentq` |
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1235 (this requires SciPy). |
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1236 seed : integer, random_state, or None (default) |
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1237 Indicator of random number generation state. |
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1238 See :ref:`Randomness<randomness>`. |
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1239 |
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1240 Notes |
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1241 ----- |
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1242 The kernel is specified through its definite integral which must be |
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1243 provided as one of the arguments. If the integral and root of the |
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1244 kernel integral can be found in $O(1)$ time then this algorithm runs in |
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1245 time $O(n+m)$ where m is the expected number of edges [2]_. |
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1246 |
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1247 The nodes are set to integers from $0$ to $n-1$. |
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1248 |
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1249 Examples |
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1250 -------- |
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1251 Generate an Erdős–Rényi random graph $G(n,c/n)$, with kernel |
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1252 $\kappa(x,y)=c$ where $c$ is the mean expected degree. |
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1253 |
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1254 >>> def integral(u, w, z): |
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1255 ... return c * (z - w) |
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1256 >>> def root(u, w, r): |
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1257 ... return r / c + w |
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1258 >>> c = 1 |
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1259 >>> graph = nx.random_kernel_graph(1000, integral, root) |
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1260 |
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1261 See Also |
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1262 -------- |
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1263 gnp_random_graph |
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1264 expected_degree_graph |
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1265 |
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1266 References |
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1267 ---------- |
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1268 .. [1] Bollobás, Béla, Janson, S. and Riordan, O. |
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1269 "The phase transition in inhomogeneous random graphs", |
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1270 *Random Structures Algorithms*, 31, 3--122, 2007. |
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1271 |
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1272 .. [2] Hagberg A, Lemons N (2015), |
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1273 "Fast Generation of Sparse Random Kernel Graphs". |
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1274 PLoS ONE 10(9): e0135177, 2015. doi:10.1371/journal.pone.0135177 |
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1275 """ |
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1276 if kernel_root is None: |
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1277 import scipy.optimize as optimize |
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1278 |
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1279 def kernel_root(y, a, r): |
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1280 def my_function(b): |
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1281 return kernel_integral(y, a, b) - r |
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1282 |
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1283 return optimize.brentq(my_function, a, 1) |
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1284 |
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1285 graph = nx.Graph() |
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1286 graph.add_nodes_from(range(n)) |
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1287 (i, j) = (1, 1) |
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1288 while i < n: |
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1289 r = -math.log(1 - seed.random()) # (1-seed.random()) in (0, 1] |
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1290 if kernel_integral(i / n, j / n, 1) <= r: |
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1291 i, j = i + 1, i + 1 |
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1292 else: |
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1293 j = int(math.ceil(n * kernel_root(i / n, j / n, r))) |
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1294 graph.add_edge(i - 1, j - 1) |
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1295 return graph |