Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/algorithms/smallworld.py @ 0:4f3585e2f14b draft default tip
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author | shellac |
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date | Mon, 22 Mar 2021 18:12:50 +0000 |
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1 """Functions for estimating the small-world-ness of graphs. |
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2 |
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3 A small world network is characterized by a small average shortest path length, |
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4 and a large clustering coefficient. |
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5 |
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6 Small-worldness is commonly measured with the coefficient sigma or omega. |
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7 |
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8 Both coefficients compare the average clustering coefficient and shortest path |
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9 length of a given graph against the same quantities for an equivalent random |
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10 or lattice graph. |
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11 |
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12 For more information, see the Wikipedia article on small-world network [1]_. |
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13 |
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14 .. [1] Small-world network:: https://en.wikipedia.org/wiki/Small-world_network |
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15 |
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16 """ |
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17 import networkx as nx |
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18 from networkx.utils import not_implemented_for |
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19 from networkx.utils import py_random_state |
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20 |
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21 __all__ = ["random_reference", "lattice_reference", "sigma", "omega"] |
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22 |
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23 |
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24 @py_random_state(3) |
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25 @not_implemented_for("directed") |
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26 @not_implemented_for("multigraph") |
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27 def random_reference(G, niter=1, connectivity=True, seed=None): |
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28 """Compute a random graph by swapping edges of a given graph. |
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29 |
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30 Parameters |
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31 ---------- |
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32 G : graph |
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33 An undirected graph with 4 or more nodes. |
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34 |
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35 niter : integer (optional, default=1) |
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36 An edge is rewired approximately `niter` times. |
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37 |
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38 connectivity : boolean (optional, default=True) |
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39 When True, ensure connectivity for the randomized graph. |
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40 |
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41 seed : integer, random_state, or None (default) |
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42 Indicator of random number generation state. |
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43 See :ref:`Randomness<randomness>`. |
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44 |
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45 Returns |
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46 ------- |
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47 G : graph |
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48 The randomized graph. |
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49 |
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50 Notes |
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51 ----- |
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52 The implementation is adapted from the algorithm by Maslov and Sneppen |
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53 (2002) [1]_. |
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54 |
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55 References |
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56 ---------- |
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57 .. [1] Maslov, Sergei, and Kim Sneppen. |
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58 "Specificity and stability in topology of protein networks." |
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59 Science 296.5569 (2002): 910-913. |
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60 """ |
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61 if G.is_directed(): |
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62 msg = "random_reference() not defined for directed graphs." |
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63 raise nx.NetworkXError(msg) |
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64 if len(G) < 4: |
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65 raise nx.NetworkXError("Graph has less than four nodes.") |
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66 |
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67 from networkx.utils import cumulative_distribution, discrete_sequence |
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68 |
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69 local_conn = nx.connectivity.local_edge_connectivity |
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70 |
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71 G = G.copy() |
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72 keys, degrees = zip(*G.degree()) # keys, degree |
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73 cdf = cumulative_distribution(degrees) # cdf of degree |
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74 nnodes = len(G) |
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75 nedges = nx.number_of_edges(G) |
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76 niter = niter * nedges |
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77 ntries = int(nnodes * nedges / (nnodes * (nnodes - 1) / 2)) |
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78 swapcount = 0 |
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79 |
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80 for i in range(niter): |
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81 n = 0 |
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82 while n < ntries: |
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83 # pick two random edges without creating edge list |
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84 # choose source node indices from discrete distribution |
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85 (ai, ci) = discrete_sequence(2, cdistribution=cdf, seed=seed) |
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86 if ai == ci: |
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87 continue # same source, skip |
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88 a = keys[ai] # convert index to label |
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89 c = keys[ci] |
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90 # choose target uniformly from neighbors |
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91 b = seed.choice(list(G.neighbors(a))) |
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92 d = seed.choice(list(G.neighbors(c))) |
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93 bi = keys.index(b) |
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94 di = keys.index(d) |
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95 if b in [a, c, d] or d in [a, b, c]: |
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96 continue # all vertices should be different |
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97 |
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98 # don't create parallel edges |
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99 if (d not in G[a]) and (b not in G[c]): |
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100 G.add_edge(a, d) |
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101 G.add_edge(c, b) |
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102 G.remove_edge(a, b) |
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103 G.remove_edge(c, d) |
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104 |
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105 # Check if the graph is still connected |
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106 if connectivity and local_conn(G, a, b) == 0: |
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107 # Not connected, revert the swap |
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108 G.remove_edge(a, d) |
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109 G.remove_edge(c, b) |
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110 G.add_edge(a, b) |
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111 G.add_edge(c, d) |
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112 else: |
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113 swapcount += 1 |
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114 break |
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115 n += 1 |
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116 return G |
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117 |
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118 |
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119 @py_random_state(4) |
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120 @not_implemented_for("directed") |
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121 @not_implemented_for("multigraph") |
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122 def lattice_reference(G, niter=1, D=None, connectivity=True, seed=None): |
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123 """Latticize the given graph by swapping edges. |
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124 |
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125 Parameters |
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126 ---------- |
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127 G : graph |
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128 An undirected graph with 4 or more nodes. |
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129 |
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130 niter : integer (optional, default=1) |
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131 An edge is rewired approximatively niter times. |
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132 |
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133 D : numpy.array (optional, default=None) |
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134 Distance to the diagonal matrix. |
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135 |
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136 connectivity : boolean (optional, default=True) |
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137 Ensure connectivity for the latticized graph when set to True. |
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138 |
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139 seed : integer, random_state, or None (default) |
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140 Indicator of random number generation state. |
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141 See :ref:`Randomness<randomness>`. |
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142 |
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143 Returns |
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144 ------- |
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145 G : graph |
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146 The latticized graph. |
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147 |
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148 Notes |
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149 ----- |
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150 The implementation is adapted from the algorithm by Sporns et al. [1]_. |
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151 which is inspired from the original work by Maslov and Sneppen(2002) [2]_. |
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152 |
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153 References |
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154 ---------- |
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155 .. [1] Sporns, Olaf, and Jonathan D. Zwi. |
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156 "The small world of the cerebral cortex." |
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157 Neuroinformatics 2.2 (2004): 145-162. |
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158 .. [2] Maslov, Sergei, and Kim Sneppen. |
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159 "Specificity and stability in topology of protein networks." |
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160 Science 296.5569 (2002): 910-913. |
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161 """ |
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162 import numpy as np |
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163 from networkx.utils import cumulative_distribution, discrete_sequence |
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164 |
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165 local_conn = nx.connectivity.local_edge_connectivity |
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166 |
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167 if G.is_directed(): |
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168 msg = "lattice_reference() not defined for directed graphs." |
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169 raise nx.NetworkXError(msg) |
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170 if len(G) < 4: |
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171 raise nx.NetworkXError("Graph has less than four nodes.") |
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172 # Instead of choosing uniformly at random from a generated edge list, |
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173 # this algorithm chooses nonuniformly from the set of nodes with |
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174 # probability weighted by degree. |
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175 G = G.copy() |
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176 keys, degrees = zip(*G.degree()) # keys, degree |
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177 cdf = cumulative_distribution(degrees) # cdf of degree |
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178 |
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179 nnodes = len(G) |
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180 nedges = nx.number_of_edges(G) |
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181 if D is None: |
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182 D = np.zeros((nnodes, nnodes)) |
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183 un = np.arange(1, nnodes) |
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184 um = np.arange(nnodes - 1, 0, -1) |
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185 u = np.append((0,), np.where(un < um, un, um)) |
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186 |
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187 for v in range(int(np.ceil(nnodes / 2))): |
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188 D[nnodes - v - 1, :] = np.append(u[v + 1 :], u[: v + 1]) |
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189 D[v, :] = D[nnodes - v - 1, :][::-1] |
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190 |
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191 niter = niter * nedges |
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192 ntries = int(nnodes * nedges / (nnodes * (nnodes - 1) / 2)) |
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193 swapcount = 0 |
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194 |
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195 for i in range(niter): |
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196 n = 0 |
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197 while n < ntries: |
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198 # pick two random edges without creating edge list |
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199 # choose source node indices from discrete distribution |
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200 (ai, ci) = discrete_sequence(2, cdistribution=cdf, seed=seed) |
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201 if ai == ci: |
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202 continue # same source, skip |
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203 a = keys[ai] # convert index to label |
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204 c = keys[ci] |
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205 # choose target uniformly from neighbors |
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206 b = seed.choice(list(G.neighbors(a))) |
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207 d = seed.choice(list(G.neighbors(c))) |
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208 bi = keys.index(b) |
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209 di = keys.index(d) |
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210 |
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211 if b in [a, c, d] or d in [a, b, c]: |
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212 continue # all vertices should be different |
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213 |
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214 # don't create parallel edges |
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215 if (d not in G[a]) and (b not in G[c]): |
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216 if D[ai, bi] + D[ci, di] >= D[ai, ci] + D[bi, di]: |
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217 # only swap if we get closer to the diagonal |
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218 G.add_edge(a, d) |
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219 G.add_edge(c, b) |
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220 G.remove_edge(a, b) |
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221 G.remove_edge(c, d) |
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222 |
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223 # Check if the graph is still connected |
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224 if connectivity and local_conn(G, a, b) == 0: |
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225 # Not connected, revert the swap |
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226 G.remove_edge(a, d) |
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227 G.remove_edge(c, b) |
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228 G.add_edge(a, b) |
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229 G.add_edge(c, d) |
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230 else: |
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231 swapcount += 1 |
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232 break |
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233 n += 1 |
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234 |
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235 return G |
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236 |
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237 |
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238 @py_random_state(3) |
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239 @not_implemented_for("directed") |
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240 @not_implemented_for("multigraph") |
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241 def sigma(G, niter=100, nrand=10, seed=None): |
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242 """Returns the small-world coefficient (sigma) of the given graph. |
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243 |
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244 The small-world coefficient is defined as: |
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245 sigma = C/Cr / L/Lr |
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246 where C and L are respectively the average clustering coefficient and |
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247 average shortest path length of G. Cr and Lr are respectively the average |
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248 clustering coefficient and average shortest path length of an equivalent |
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249 random graph. |
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250 |
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251 A graph is commonly classified as small-world if sigma>1. |
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252 |
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253 Parameters |
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254 ---------- |
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255 G : NetworkX graph |
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256 An undirected graph. |
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257 niter : integer (optional, default=100) |
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258 Approximate number of rewiring per edge to compute the equivalent |
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259 random graph. |
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260 nrand : integer (optional, default=10) |
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261 Number of random graphs generated to compute the average clustering |
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262 coefficient (Cr) and average shortest path length (Lr). |
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263 seed : integer, random_state, or None (default) |
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264 Indicator of random number generation state. |
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265 See :ref:`Randomness<randomness>`. |
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266 |
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267 Returns |
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268 ------- |
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269 sigma : float |
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270 The small-world coefficient of G. |
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271 |
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272 Notes |
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273 ----- |
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274 The implementation is adapted from Humphries et al. [1]_ [2]_. |
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275 |
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276 References |
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277 ---------- |
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278 .. [1] The brainstem reticular formation is a small-world, not scale-free, |
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279 network M. D. Humphries, K. Gurney and T. J. Prescott, |
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280 Proc. Roy. Soc. B 2006 273, 503-511, doi:10.1098/rspb.2005.3354. |
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281 .. [2] Humphries and Gurney (2008). |
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282 "Network 'Small-World-Ness': A Quantitative Method for Determining |
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283 Canonical Network Equivalence". |
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284 PLoS One. 3 (4). PMID 18446219. doi:10.1371/journal.pone.0002051. |
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285 """ |
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286 import numpy as np |
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287 |
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288 # Compute the mean clustering coefficient and average shortest path length |
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289 # for an equivalent random graph |
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290 randMetrics = {"C": [], "L": []} |
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291 for i in range(nrand): |
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292 Gr = random_reference(G, niter=niter, seed=seed) |
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293 randMetrics["C"].append(nx.transitivity(Gr)) |
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294 randMetrics["L"].append(nx.average_shortest_path_length(Gr)) |
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295 |
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296 C = nx.transitivity(G) |
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297 L = nx.average_shortest_path_length(G) |
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298 Cr = np.mean(randMetrics["C"]) |
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299 Lr = np.mean(randMetrics["L"]) |
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300 |
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301 sigma = (C / Cr) / (L / Lr) |
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302 |
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303 return sigma |
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304 |
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305 |
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306 @py_random_state(3) |
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307 @not_implemented_for("directed") |
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308 @not_implemented_for("multigraph") |
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309 def omega(G, niter=100, nrand=10, seed=None): |
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310 """Returns the small-world coefficient (omega) of a graph |
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311 |
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312 The small-world coefficient of a graph G is: |
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313 |
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314 omega = Lr/L - C/Cl |
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315 |
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316 where C and L are respectively the average clustering coefficient and |
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317 average shortest path length of G. Lr is the average shortest path length |
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318 of an equivalent random graph and Cl is the average clustering coefficient |
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319 of an equivalent lattice graph. |
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320 |
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321 The small-world coefficient (omega) ranges between -1 and 1. Values close |
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322 to 0 means the G features small-world characteristics. Values close to -1 |
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323 means G has a lattice shape whereas values close to 1 means G is a random |
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324 graph. |
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325 |
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326 Parameters |
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327 ---------- |
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328 G : NetworkX graph |
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329 An undirected graph. |
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330 |
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331 niter: integer (optional, default=100) |
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332 Approximate number of rewiring per edge to compute the equivalent |
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333 random graph. |
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334 |
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335 nrand: integer (optional, default=10) |
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336 Number of random graphs generated to compute the average clustering |
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337 coefficient (Cr) and average shortest path length (Lr). |
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338 |
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339 seed : integer, random_state, or None (default) |
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340 Indicator of random number generation state. |
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341 See :ref:`Randomness<randomness>`. |
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342 |
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343 |
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344 Returns |
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345 ------- |
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346 omega : float |
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347 The small-work coefficient (omega) |
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348 |
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349 Notes |
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350 ----- |
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351 The implementation is adapted from the algorithm by Telesford et al. [1]_. |
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352 |
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353 References |
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354 ---------- |
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355 .. [1] Telesford, Joyce, Hayasaka, Burdette, and Laurienti (2011). |
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356 "The Ubiquity of Small-World Networks". |
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357 Brain Connectivity. 1 (0038): 367-75. PMC 3604768. PMID 22432451. |
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358 doi:10.1089/brain.2011.0038. |
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359 """ |
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360 import numpy as np |
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361 |
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362 # Compute the mean clustering coefficient and average shortest path length |
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363 # for an equivalent random graph |
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364 randMetrics = {"C": [], "L": []} |
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365 for i in range(nrand): |
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366 Gr = random_reference(G, niter=niter, seed=seed) |
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367 Gl = lattice_reference(G, niter=niter, seed=seed) |
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368 randMetrics["C"].append(nx.transitivity(Gl)) |
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369 randMetrics["L"].append(nx.average_shortest_path_length(Gr)) |
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370 |
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371 C = nx.transitivity(G) |
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372 L = nx.average_shortest_path_length(G) |
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373 Cl = np.mean(randMetrics["C"]) |
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374 Lr = np.mean(randMetrics["L"]) |
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375 |
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376 omega = (Lr / L) - (C / Cl) |
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377 |
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378 return omega |