Mercurial > repos > shellac > sam_consensus_v3
annotate env/lib/python3.9/site-packages/networkx/generators/geometric.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 """Generators for geometric graphs. |
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2 """ |
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3 |
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4 from bisect import bisect_left |
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5 from itertools import accumulate, combinations, product |
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6 from math import sqrt |
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7 import math |
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8 |
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9 try: |
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10 from scipy.spatial import cKDTree as KDTree |
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11 except ImportError: |
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12 _is_scipy_available = False |
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13 else: |
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14 _is_scipy_available = True |
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15 |
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16 import networkx as nx |
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17 from networkx.utils import nodes_or_number, py_random_state |
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18 |
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19 __all__ = [ |
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20 "geographical_threshold_graph", |
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21 "waxman_graph", |
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22 "navigable_small_world_graph", |
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23 "random_geometric_graph", |
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24 "soft_random_geometric_graph", |
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25 "thresholded_random_geometric_graph", |
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26 ] |
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27 |
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28 |
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29 def euclidean(x, y): |
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30 """Returns the Euclidean distance between the vectors ``x`` and ``y``. |
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31 |
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32 Each of ``x`` and ``y`` can be any iterable of numbers. The |
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33 iterables must be of the same length. |
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34 |
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35 """ |
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36 return sqrt(sum((a - b) ** 2 for a, b in zip(x, y))) |
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37 |
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38 |
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39 def _fast_edges(G, radius, p): |
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40 """Returns edge list of node pairs within `radius` of each other |
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41 using scipy KDTree and Minkowski distance metric `p` |
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42 |
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43 Requires scipy to be installed. |
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44 """ |
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45 pos = nx.get_node_attributes(G, "pos") |
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46 nodes, coords = list(zip(*pos.items())) |
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47 kdtree = KDTree(coords) # Cannot provide generator. |
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48 edge_indexes = kdtree.query_pairs(radius, p) |
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49 edges = ((nodes[u], nodes[v]) for u, v in edge_indexes) |
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50 return edges |
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51 |
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52 |
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53 def _slow_edges(G, radius, p): |
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54 """Returns edge list of node pairs within `radius` of each other |
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55 using Minkowski distance metric `p` |
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56 |
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57 Works without scipy, but in `O(n^2)` time. |
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58 """ |
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59 # TODO This can be parallelized. |
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60 edges = [] |
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61 for (u, pu), (v, pv) in combinations(G.nodes(data="pos"), 2): |
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62 if sum(abs(a - b) ** p for a, b in zip(pu, pv)) <= radius ** p: |
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63 edges.append((u, v)) |
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64 return edges |
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65 |
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66 |
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67 @py_random_state(5) |
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68 @nodes_or_number(0) |
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69 def random_geometric_graph(n, radius, dim=2, pos=None, p=2, seed=None): |
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70 """Returns a random geometric graph in the unit cube of dimensions `dim`. |
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71 |
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72 The random geometric graph model places `n` nodes uniformly at |
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73 random in the unit cube. Two nodes are joined by an edge if the |
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74 distance between the nodes is at most `radius`. |
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75 |
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76 Edges are determined using a KDTree when SciPy is available. |
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77 This reduces the time complexity from $O(n^2)$ to $O(n)$. |
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78 |
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79 Parameters |
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80 ---------- |
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81 n : int or iterable |
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82 Number of nodes or iterable of nodes |
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83 radius: float |
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84 Distance threshold value |
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85 dim : int, optional |
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86 Dimension of graph |
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87 pos : dict, optional |
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88 A dictionary keyed by node with node positions as values. |
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89 p : float, optional |
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90 Which Minkowski distance metric to use. `p` has to meet the condition |
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91 ``1 <= p <= infinity``. |
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92 |
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93 If this argument is not specified, the :math:`L^2` metric |
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94 (the Euclidean distance metric), p = 2 is used. |
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95 This should not be confused with the `p` of an Erdős-Rényi random |
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96 graph, which represents probability. |
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97 seed : integer, random_state, or None (default) |
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98 Indicator of random number generation state. |
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99 See :ref:`Randomness<randomness>`. |
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100 |
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101 Returns |
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102 ------- |
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103 Graph |
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104 A random geometric graph, undirected and without self-loops. |
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105 Each node has a node attribute ``'pos'`` that stores the |
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106 position of that node in Euclidean space as provided by the |
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107 ``pos`` keyword argument or, if ``pos`` was not provided, as |
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108 generated by this function. |
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109 |
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110 Examples |
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111 -------- |
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112 Create a random geometric graph on twenty nodes where nodes are joined by |
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113 an edge if their distance is at most 0.1:: |
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114 |
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115 >>> G = nx.random_geometric_graph(20, 0.1) |
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116 |
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117 Notes |
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118 ----- |
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119 This uses a *k*-d tree to build the graph. |
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120 |
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121 The `pos` keyword argument can be used to specify node positions so you |
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122 can create an arbitrary distribution and domain for positions. |
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123 |
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124 For example, to use a 2D Gaussian distribution of node positions with mean |
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125 (0, 0) and standard deviation 2:: |
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126 |
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127 >>> import random |
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128 >>> n = 20 |
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129 >>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)} |
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130 >>> G = nx.random_geometric_graph(n, 0.2, pos=pos) |
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131 |
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132 References |
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133 ---------- |
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134 .. [1] Penrose, Mathew, *Random Geometric Graphs*, |
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135 Oxford Studies in Probability, 5, 2003. |
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136 |
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137 """ |
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138 # TODO Is this function just a special case of the geographical |
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139 # threshold graph? |
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140 # |
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141 # n_name, nodes = n |
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142 # half_radius = {v: radius / 2 for v in nodes} |
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143 # return geographical_threshold_graph(nodes, theta=1, alpha=1, |
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144 # weight=half_radius) |
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145 # |
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146 n_name, nodes = n |
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147 G = nx.Graph() |
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148 G.add_nodes_from(nodes) |
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149 # If no positions are provided, choose uniformly random vectors in |
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150 # Euclidean space of the specified dimension. |
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151 if pos is None: |
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152 pos = {v: [seed.random() for i in range(dim)] for v in nodes} |
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153 nx.set_node_attributes(G, pos, "pos") |
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154 |
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155 if _is_scipy_available: |
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156 edges = _fast_edges(G, radius, p) |
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157 else: |
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158 edges = _slow_edges(G, radius, p) |
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159 G.add_edges_from(edges) |
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160 |
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161 return G |
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162 |
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163 |
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164 @py_random_state(6) |
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165 @nodes_or_number(0) |
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166 def soft_random_geometric_graph( |
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167 n, radius, dim=2, pos=None, p=2, p_dist=None, seed=None |
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168 ): |
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169 r"""Returns a soft random geometric graph in the unit cube. |
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170 |
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171 The soft random geometric graph [1] model places `n` nodes uniformly at |
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172 random in the unit cube in dimension `dim`. Two nodes of distance, `dist`, |
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173 computed by the `p`-Minkowski distance metric are joined by an edge with |
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174 probability `p_dist` if the computed distance metric value of the nodes |
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175 is at most `radius`, otherwise they are not joined. |
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176 |
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177 Edges within `radius` of each other are determined using a KDTree when |
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178 SciPy is available. This reduces the time complexity from :math:`O(n^2)` |
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179 to :math:`O(n)`. |
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180 |
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181 Parameters |
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182 ---------- |
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183 n : int or iterable |
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184 Number of nodes or iterable of nodes |
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185 radius: float |
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186 Distance threshold value |
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187 dim : int, optional |
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188 Dimension of graph |
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189 pos : dict, optional |
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190 A dictionary keyed by node with node positions as values. |
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191 p : float, optional |
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192 Which Minkowski distance metric to use. |
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193 `p` has to meet the condition ``1 <= p <= infinity``. |
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194 |
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195 If this argument is not specified, the :math:`L^2` metric |
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196 (the Euclidean distance metric), p = 2 is used. |
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197 |
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198 This should not be confused with the `p` of an Erdős-Rényi random |
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199 graph, which represents probability. |
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200 p_dist : function, optional |
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201 A probability density function computing the probability of |
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202 connecting two nodes that are of distance, dist, computed by the |
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203 Minkowski distance metric. The probability density function, `p_dist`, |
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204 must be any function that takes the metric value as input |
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205 and outputs a single probability value between 0-1. The scipy.stats |
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206 package has many probability distribution functions implemented and |
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207 tools for custom probability distribution definitions [2], and passing |
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208 the .pdf method of scipy.stats distributions can be used here. If the |
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209 probability function, `p_dist`, is not supplied, the default function |
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210 is an exponential distribution with rate parameter :math:`\lambda=1`. |
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211 seed : integer, random_state, or None (default) |
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212 Indicator of random number generation state. |
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213 See :ref:`Randomness<randomness>`. |
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214 |
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215 Returns |
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216 ------- |
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217 Graph |
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218 A soft random geometric graph, undirected and without self-loops. |
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219 Each node has a node attribute ``'pos'`` that stores the |
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220 position of that node in Euclidean space as provided by the |
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221 ``pos`` keyword argument or, if ``pos`` was not provided, as |
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222 generated by this function. |
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223 |
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224 Examples |
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225 -------- |
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226 Default Graph: |
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227 |
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228 G = nx.soft_random_geometric_graph(50, 0.2) |
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229 |
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230 Custom Graph: |
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231 |
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232 Create a soft random geometric graph on 100 uniformly distributed nodes |
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233 where nodes are joined by an edge with probability computed from an |
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234 exponential distribution with rate parameter :math:`\lambda=1` if their |
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235 Euclidean distance is at most 0.2. |
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236 |
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237 Notes |
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238 ----- |
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239 This uses a *k*-d tree to build the graph. |
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240 |
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241 The `pos` keyword argument can be used to specify node positions so you |
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242 can create an arbitrary distribution and domain for positions. |
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243 |
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244 For example, to use a 2D Gaussian distribution of node positions with mean |
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245 (0, 0) and standard deviation 2 |
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246 |
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247 The scipy.stats package can be used to define the probability distribution |
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248 with the .pdf method used as `p_dist`. |
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249 |
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250 :: |
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251 |
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252 >>> import random |
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253 >>> import math |
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254 >>> n = 100 |
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255 >>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)} |
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256 >>> p_dist = lambda dist: math.exp(-dist) |
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257 >>> G = nx.soft_random_geometric_graph(n, 0.2, pos=pos, p_dist=p_dist) |
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258 |
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259 References |
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260 ---------- |
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261 .. [1] Penrose, Mathew D. "Connectivity of soft random geometric graphs." |
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262 The Annals of Applied Probability 26.2 (2016): 986-1028. |
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263 [2] scipy.stats - |
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264 https://docs.scipy.org/doc/scipy/reference/tutorial/stats.html |
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265 |
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266 """ |
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267 n_name, nodes = n |
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268 G = nx.Graph() |
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269 G.name = f"soft_random_geometric_graph({n}, {radius}, {dim})" |
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270 G.add_nodes_from(nodes) |
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271 # If no positions are provided, choose uniformly random vectors in |
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272 # Euclidean space of the specified dimension. |
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273 if pos is None: |
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274 pos = {v: [seed.random() for i in range(dim)] for v in nodes} |
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275 nx.set_node_attributes(G, pos, "pos") |
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276 |
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277 # if p_dist function not supplied the default function is an exponential |
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278 # distribution with rate parameter :math:`\lambda=1`. |
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279 if p_dist is None: |
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280 |
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281 def p_dist(dist): |
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282 return math.exp(-dist) |
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283 |
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284 def should_join(pair): |
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285 u, v = pair |
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286 u_pos, v_pos = pos[u], pos[v] |
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287 dist = (sum(abs(a - b) ** p for a, b in zip(u_pos, v_pos))) ** (1 / p) |
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288 # Check if dist <= radius parameter. This check is redundant if scipy |
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289 # is available and _fast_edges routine is used, but provides the |
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290 # check in case scipy is not available and all edge combinations |
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291 # need to be checked |
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292 if dist <= radius: |
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293 return seed.random() < p_dist(dist) |
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294 else: |
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295 return False |
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296 |
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297 if _is_scipy_available: |
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298 edges = _fast_edges(G, radius, p) |
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299 G.add_edges_from(filter(should_join, edges)) |
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300 else: |
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301 G.add_edges_from(filter(should_join, combinations(G, 2))) |
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302 |
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303 return G |
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304 |
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305 |
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306 @py_random_state(7) |
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307 @nodes_or_number(0) |
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308 def geographical_threshold_graph( |
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309 n, theta, dim=2, pos=None, weight=None, metric=None, p_dist=None, seed=None |
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310 ): |
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311 r"""Returns a geographical threshold graph. |
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312 |
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313 The geographical threshold graph model places $n$ nodes uniformly at |
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314 random in a rectangular domain. Each node $u$ is assigned a weight |
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315 $w_u$. Two nodes $u$ and $v$ are joined by an edge if |
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316 |
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317 .. math:: |
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318 |
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319 (w_u + w_v)h(r) \ge \theta |
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320 |
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321 where `r` is the distance between `u` and `v`, h(r) is a probability of |
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322 connection as a function of `r`, and :math:`\theta` as the threshold |
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323 parameter. h(r) corresponds to the p_dist parameter. |
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324 |
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325 Parameters |
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326 ---------- |
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327 n : int or iterable |
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328 Number of nodes or iterable of nodes |
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329 theta: float |
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330 Threshold value |
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331 dim : int, optional |
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332 Dimension of graph |
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333 pos : dict |
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334 Node positions as a dictionary of tuples keyed by node. |
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335 weight : dict |
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336 Node weights as a dictionary of numbers keyed by node. |
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337 metric : function |
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338 A metric on vectors of numbers (represented as lists or |
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339 tuples). This must be a function that accepts two lists (or |
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340 tuples) as input and yields a number as output. The function |
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341 must also satisfy the four requirements of a `metric`_. |
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342 Specifically, if $d$ is the function and $x$, $y$, |
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343 and $z$ are vectors in the graph, then $d$ must satisfy |
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344 |
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345 1. $d(x, y) \ge 0$, |
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346 2. $d(x, y) = 0$ if and only if $x = y$, |
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347 3. $d(x, y) = d(y, x)$, |
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348 4. $d(x, z) \le d(x, y) + d(y, z)$. |
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349 |
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350 If this argument is not specified, the Euclidean distance metric is |
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351 used. |
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352 |
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353 .. _metric: https://en.wikipedia.org/wiki/Metric_%28mathematics%29 |
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354 p_dist : function, optional |
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355 A probability density function computing the probability of |
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356 connecting two nodes that are of distance, r, computed by metric. |
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357 The probability density function, `p_dist`, must |
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358 be any function that takes the metric value as input |
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359 and outputs a single probability value between 0-1. |
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360 The scipy.stats package has many probability distribution functions |
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361 implemented and tools for custom probability distribution |
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362 definitions [2], and passing the .pdf method of scipy.stats |
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363 distributions can be used here. If the probability |
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364 function, `p_dist`, is not supplied, the default exponential function |
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365 :math: `r^{-2}` is used. |
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366 seed : integer, random_state, or None (default) |
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367 Indicator of random number generation state. |
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368 See :ref:`Randomness<randomness>`. |
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369 |
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370 Returns |
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371 ------- |
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372 Graph |
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373 A random geographic threshold graph, undirected and without |
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374 self-loops. |
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375 |
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376 Each node has a node attribute ``pos`` that stores the |
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377 position of that node in Euclidean space as provided by the |
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378 ``pos`` keyword argument or, if ``pos`` was not provided, as |
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379 generated by this function. Similarly, each node has a node |
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380 attribute ``weight`` that stores the weight of that node as |
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381 provided or as generated. |
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382 |
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383 Examples |
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384 -------- |
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385 Specify an alternate distance metric using the ``metric`` keyword |
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386 argument. For example, to use the `taxicab metric`_ instead of the |
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387 default `Euclidean metric`_:: |
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388 |
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389 >>> dist = lambda x, y: sum(abs(a - b) for a, b in zip(x, y)) |
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390 >>> G = nx.geographical_threshold_graph(10, 0.1, metric=dist) |
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391 |
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392 .. _taxicab metric: https://en.wikipedia.org/wiki/Taxicab_geometry |
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393 .. _Euclidean metric: https://en.wikipedia.org/wiki/Euclidean_distance |
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394 |
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395 Notes |
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396 ----- |
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397 If weights are not specified they are assigned to nodes by drawing randomly |
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398 from the exponential distribution with rate parameter $\lambda=1$. |
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399 To specify weights from a different distribution, use the `weight` keyword |
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400 argument:: |
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401 |
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402 >>> import random |
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403 >>> n = 20 |
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404 >>> w = {i: random.expovariate(5.0) for i in range(n)} |
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405 >>> G = nx.geographical_threshold_graph(20, 50, weight=w) |
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406 |
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407 If node positions are not specified they are randomly assigned from the |
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408 uniform distribution. |
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409 |
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410 References |
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411 ---------- |
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412 .. [1] Masuda, N., Miwa, H., Konno, N.: |
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413 Geographical threshold graphs with small-world and scale-free |
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414 properties. |
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415 Physical Review E 71, 036108 (2005) |
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416 .. [2] Milan Bradonjić, Aric Hagberg and Allon G. Percus, |
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417 Giant component and connectivity in geographical threshold graphs, |
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418 in Algorithms and Models for the Web-Graph (WAW 2007), |
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419 Antony Bonato and Fan Chung (Eds), pp. 209--216, 2007 |
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420 """ |
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421 n_name, nodes = n |
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422 G = nx.Graph() |
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423 G.add_nodes_from(nodes) |
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424 # If no weights are provided, choose them from an exponential |
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425 # distribution. |
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426 if weight is None: |
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427 weight = {v: seed.expovariate(1) for v in G} |
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428 # If no positions are provided, choose uniformly random vectors in |
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429 # Euclidean space of the specified dimension. |
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430 if pos is None: |
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431 pos = {v: [seed.random() for i in range(dim)] for v in nodes} |
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432 # If no distance metric is provided, use Euclidean distance. |
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433 if metric is None: |
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434 metric = euclidean |
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435 nx.set_node_attributes(G, weight, "weight") |
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436 nx.set_node_attributes(G, pos, "pos") |
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437 |
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438 # if p_dist is not supplied, use default r^-2 |
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439 if p_dist is None: |
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440 |
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441 def p_dist(r): |
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442 return r ** -2 |
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443 |
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444 # Returns ``True`` if and only if the nodes whose attributes are |
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445 # ``du`` and ``dv`` should be joined, according to the threshold |
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446 # condition. |
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447 def should_join(pair): |
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448 u, v = pair |
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449 u_pos, v_pos = pos[u], pos[v] |
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450 u_weight, v_weight = weight[u], weight[v] |
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451 return (u_weight + v_weight) * p_dist(metric(u_pos, v_pos)) >= theta |
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452 |
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453 G.add_edges_from(filter(should_join, combinations(G, 2))) |
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454 return G |
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455 |
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456 |
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457 @py_random_state(6) |
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458 @nodes_or_number(0) |
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459 def waxman_graph( |
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460 n, beta=0.4, alpha=0.1, L=None, domain=(0, 0, 1, 1), metric=None, seed=None |
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461 ): |
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462 r"""Returns a Waxman random graph. |
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463 |
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464 The Waxman random graph model places `n` nodes uniformly at random |
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465 in a rectangular domain. Each pair of nodes at distance `d` is |
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466 joined by an edge with probability |
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467 |
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468 .. math:: |
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469 p = \beta \exp(-d / \alpha L). |
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470 |
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471 This function implements both Waxman models, using the `L` keyword |
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472 argument. |
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473 |
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474 * Waxman-1: if `L` is not specified, it is set to be the maximum distance |
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475 between any pair of nodes. |
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476 * Waxman-2: if `L` is specified, the distance between a pair of nodes is |
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477 chosen uniformly at random from the interval `[0, L]`. |
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478 |
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479 Parameters |
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480 ---------- |
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481 n : int or iterable |
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482 Number of nodes or iterable of nodes |
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483 beta: float |
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484 Model parameter |
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485 alpha: float |
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486 Model parameter |
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487 L : float, optional |
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488 Maximum distance between nodes. If not specified, the actual distance |
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489 is calculated. |
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490 domain : four-tuple of numbers, optional |
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491 Domain size, given as a tuple of the form `(x_min, y_min, x_max, |
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492 y_max)`. |
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493 metric : function |
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494 A metric on vectors of numbers (represented as lists or |
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495 tuples). This must be a function that accepts two lists (or |
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496 tuples) as input and yields a number as output. The function |
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497 must also satisfy the four requirements of a `metric`_. |
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498 Specifically, if $d$ is the function and $x$, $y$, |
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499 and $z$ are vectors in the graph, then $d$ must satisfy |
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500 |
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501 1. $d(x, y) \ge 0$, |
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502 2. $d(x, y) = 0$ if and only if $x = y$, |
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503 3. $d(x, y) = d(y, x)$, |
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504 4. $d(x, z) \le d(x, y) + d(y, z)$. |
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505 |
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506 If this argument is not specified, the Euclidean distance metric is |
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507 used. |
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508 |
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509 .. _metric: https://en.wikipedia.org/wiki/Metric_%28mathematics%29 |
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510 |
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511 seed : integer, random_state, or None (default) |
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512 Indicator of random number generation state. |
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513 See :ref:`Randomness<randomness>`. |
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514 |
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515 Returns |
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516 ------- |
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517 Graph |
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518 A random Waxman graph, undirected and without self-loops. Each |
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519 node has a node attribute ``'pos'`` that stores the position of |
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520 that node in Euclidean space as generated by this function. |
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521 |
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522 Examples |
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523 -------- |
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524 Specify an alternate distance metric using the ``metric`` keyword |
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525 argument. For example, to use the "`taxicab metric`_" instead of the |
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526 default `Euclidean metric`_:: |
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527 |
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528 >>> dist = lambda x, y: sum(abs(a - b) for a, b in zip(x, y)) |
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529 >>> G = nx.waxman_graph(10, 0.5, 0.1, metric=dist) |
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530 |
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531 .. _taxicab metric: https://en.wikipedia.org/wiki/Taxicab_geometry |
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532 .. _Euclidean metric: https://en.wikipedia.org/wiki/Euclidean_distance |
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533 |
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534 Notes |
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535 ----- |
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536 Starting in NetworkX 2.0 the parameters alpha and beta align with their |
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537 usual roles in the probability distribution. In earlier versions their |
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538 positions in the expression were reversed. Their position in the calling |
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539 sequence reversed as well to minimize backward incompatibility. |
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540 |
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541 References |
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542 ---------- |
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543 .. [1] B. M. Waxman, *Routing of multipoint connections*. |
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544 IEEE J. Select. Areas Commun. 6(9),(1988) 1617--1622. |
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545 """ |
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546 n_name, nodes = n |
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547 G = nx.Graph() |
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548 G.add_nodes_from(nodes) |
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549 (xmin, ymin, xmax, ymax) = domain |
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550 # Each node gets a uniformly random position in the given rectangle. |
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551 pos = {v: (seed.uniform(xmin, xmax), seed.uniform(ymin, ymax)) for v in G} |
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552 nx.set_node_attributes(G, pos, "pos") |
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553 # If no distance metric is provided, use Euclidean distance. |
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554 if metric is None: |
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555 metric = euclidean |
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556 # If the maximum distance L is not specified (that is, we are in the |
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557 # Waxman-1 model), then find the maximum distance between any pair |
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558 # of nodes. |
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559 # |
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560 # In the Waxman-1 model, join nodes randomly based on distance. In |
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561 # the Waxman-2 model, join randomly based on random l. |
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562 if L is None: |
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563 L = max(metric(x, y) for x, y in combinations(pos.values(), 2)) |
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564 |
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565 def dist(u, v): |
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566 return metric(pos[u], pos[v]) |
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567 |
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568 else: |
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569 |
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570 def dist(u, v): |
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571 return seed.random() * L |
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572 |
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573 # `pair` is the pair of nodes to decide whether to join. |
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574 def should_join(pair): |
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575 return seed.random() < beta * math.exp(-dist(*pair) / (alpha * L)) |
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576 |
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577 G.add_edges_from(filter(should_join, combinations(G, 2))) |
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578 return G |
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579 |
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580 |
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581 @py_random_state(5) |
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582 def navigable_small_world_graph(n, p=1, q=1, r=2, dim=2, seed=None): |
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583 r"""Returns a navigable small-world graph. |
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584 |
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585 A navigable small-world graph is a directed grid with additional long-range |
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586 connections that are chosen randomly. |
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587 |
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588 [...] we begin with a set of nodes [...] that are identified with the set |
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589 of lattice points in an $n \times n$ square, |
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590 $\{(i, j): i \in \{1, 2, \ldots, n\}, j \in \{1, 2, \ldots, n\}\}$, |
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591 and we define the *lattice distance* between two nodes $(i, j)$ and |
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592 $(k, l)$ to be the number of "lattice steps" separating them: |
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593 $d((i, j), (k, l)) = |k - i| + |l - j|$. |
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594 |
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595 For a universal constant $p >= 1$, the node $u$ has a directed edge to |
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596 every other node within lattice distance $p$---these are its *local |
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597 contacts*. For universal constants $q >= 0$ and $r >= 0$ we also |
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598 construct directed edges from $u$ to $q$ other nodes (the *long-range |
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599 contacts*) using independent random trials; the $i$th directed edge from |
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600 $u$ has endpoint $v$ with probability proportional to $[d(u,v)]^{-r}$. |
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601 |
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602 -- [1]_ |
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603 |
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604 Parameters |
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605 ---------- |
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606 n : int |
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607 The length of one side of the lattice; the number of nodes in |
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608 the graph is therefore $n^2$. |
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609 p : int |
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610 The diameter of short range connections. Each node is joined with every |
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611 other node within this lattice distance. |
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612 q : int |
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613 The number of long-range connections for each node. |
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614 r : float |
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615 Exponent for decaying probability of connections. The probability of |
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616 connecting to a node at lattice distance $d$ is $1/d^r$. |
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617 dim : int |
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618 Dimension of grid |
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619 seed : integer, random_state, or None (default) |
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620 Indicator of random number generation state. |
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621 See :ref:`Randomness<randomness>`. |
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622 |
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623 References |
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624 ---------- |
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625 .. [1] J. Kleinberg. The small-world phenomenon: An algorithmic |
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626 perspective. Proc. 32nd ACM Symposium on Theory of Computing, 2000. |
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627 """ |
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628 if p < 1: |
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629 raise nx.NetworkXException("p must be >= 1") |
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630 if q < 0: |
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631 raise nx.NetworkXException("q must be >= 0") |
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632 if r < 0: |
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633 raise nx.NetworkXException("r must be >= 1") |
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634 |
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635 G = nx.DiGraph() |
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636 nodes = list(product(range(n), repeat=dim)) |
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637 for p1 in nodes: |
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638 probs = [0] |
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639 for p2 in nodes: |
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640 if p1 == p2: |
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641 continue |
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642 d = sum((abs(b - a) for a, b in zip(p1, p2))) |
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643 if d <= p: |
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644 G.add_edge(p1, p2) |
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645 probs.append(d ** -r) |
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646 cdf = list(accumulate(probs)) |
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647 for _ in range(q): |
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648 target = nodes[bisect_left(cdf, seed.uniform(0, cdf[-1]))] |
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649 G.add_edge(p1, target) |
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650 return G |
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651 |
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652 |
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653 @py_random_state(7) |
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654 @nodes_or_number(0) |
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655 def thresholded_random_geometric_graph( |
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656 n, radius, theta, dim=2, pos=None, weight=None, p=2, seed=None |
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657 ): |
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658 r"""Returns a thresholded random geometric graph in the unit cube. |
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659 |
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660 The thresholded random geometric graph [1] model places `n` nodes |
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661 uniformly at random in the unit cube of dimensions `dim`. Each node |
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662 `u` is assigned a weight :math:`w_u`. Two nodes `u` and `v` are |
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663 joined by an edge if they are within the maximum connection distance, |
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664 `radius` computed by the `p`-Minkowski distance and the summation of |
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665 weights :math:`w_u` + :math:`w_v` is greater than or equal |
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666 to the threshold parameter `theta`. |
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667 |
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668 Edges within `radius` of each other are determined using a KDTree when |
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669 SciPy is available. This reduces the time complexity from :math:`O(n^2)` |
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670 to :math:`O(n)`. |
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671 |
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672 Parameters |
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673 ---------- |
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674 n : int or iterable |
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675 Number of nodes or iterable of nodes |
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676 radius: float |
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677 Distance threshold value |
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678 theta: float |
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679 Threshold value |
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680 dim : int, optional |
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681 Dimension of graph |
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682 pos : dict, optional |
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683 A dictionary keyed by node with node positions as values. |
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684 weight : dict, optional |
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685 Node weights as a dictionary of numbers keyed by node. |
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686 p : float, optional |
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687 Which Minkowski distance metric to use. `p` has to meet the condition |
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688 ``1 <= p <= infinity``. |
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689 |
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690 If this argument is not specified, the :math:`L^2` metric |
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691 (the Euclidean distance metric), p = 2 is used. |
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692 |
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693 This should not be confused with the `p` of an Erdős-Rényi random |
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694 graph, which represents probability. |
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695 seed : integer, random_state, or None (default) |
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696 Indicator of random number generation state. |
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697 See :ref:`Randomness<randomness>`. |
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698 |
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699 Returns |
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700 ------- |
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701 Graph |
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702 A thresholded random geographic graph, undirected and without |
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703 self-loops. |
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704 |
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705 Each node has a node attribute ``'pos'`` that stores the |
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706 position of that node in Euclidean space as provided by the |
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707 ``pos`` keyword argument or, if ``pos`` was not provided, as |
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708 generated by this function. Similarly, each node has a nodethre |
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709 attribute ``'weight'`` that stores the weight of that node as |
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710 provided or as generated. |
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711 |
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712 Examples |
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713 -------- |
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714 Default Graph: |
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715 |
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716 G = nx.thresholded_random_geometric_graph(50, 0.2, 0.1) |
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717 |
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718 Custom Graph: |
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719 |
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720 Create a thresholded random geometric graph on 50 uniformly distributed |
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721 nodes where nodes are joined by an edge if their sum weights drawn from |
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722 a exponential distribution with rate = 5 are >= theta = 0.1 and their |
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723 Euclidean distance is at most 0.2. |
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724 |
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725 Notes |
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726 ----- |
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727 This uses a *k*-d tree to build the graph. |
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728 |
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729 The `pos` keyword argument can be used to specify node positions so you |
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730 can create an arbitrary distribution and domain for positions. |
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731 |
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732 For example, to use a 2D Gaussian distribution of node positions with mean |
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733 (0, 0) and standard deviation 2 |
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734 |
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735 If weights are not specified they are assigned to nodes by drawing randomly |
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736 from the exponential distribution with rate parameter :math:`\lambda=1`. |
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737 To specify weights from a different distribution, use the `weight` keyword |
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738 argument:: |
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739 |
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740 :: |
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741 |
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742 >>> import random |
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743 >>> import math |
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744 >>> n = 50 |
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745 >>> pos = {i: (random.gauss(0, 2), random.gauss(0, 2)) for i in range(n)} |
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746 >>> w = {i: random.expovariate(5.0) for i in range(n)} |
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747 >>> G = nx.thresholded_random_geometric_graph(n, 0.2, 0.1, 2, pos, w) |
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748 |
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749 References |
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750 ---------- |
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751 .. [1] http://cole-maclean.github.io/blog/files/thesis.pdf |
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752 |
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753 """ |
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754 |
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755 n_name, nodes = n |
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756 G = nx.Graph() |
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757 G.name = f"thresholded_random_geometric_graph({n}, {radius}, {theta}, {dim})" |
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758 G.add_nodes_from(nodes) |
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759 # If no weights are provided, choose them from an exponential |
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760 # distribution. |
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761 if weight is None: |
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762 weight = {v: seed.expovariate(1) for v in G} |
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763 # If no positions are provided, choose uniformly random vectors in |
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764 # Euclidean space of the specified dimension. |
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765 if pos is None: |
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766 pos = {v: [seed.random() for i in range(dim)] for v in nodes} |
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767 # If no distance metric is provided, use Euclidean distance. |
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768 |
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769 nx.set_node_attributes(G, weight, "weight") |
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770 nx.set_node_attributes(G, pos, "pos") |
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771 |
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772 # Returns ``True`` if and only if the nodes whose attributes are |
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773 # ``du`` and ``dv`` should be joined, according to the threshold |
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774 # condition and node pairs are within the maximum connection |
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775 # distance, ``radius``. |
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776 def should_join(pair): |
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777 u, v = pair |
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778 u_weight, v_weight = weight[u], weight[v] |
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779 u_pos, v_pos = pos[u], pos[v] |
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780 dist = (sum(abs(a - b) ** p for a, b in zip(u_pos, v_pos))) ** (1 / p) |
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781 # Check if dist is <= radius parameter. This check is redundant if |
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782 # scipy is available and _fast_edges routine is used, but provides |
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783 # the check in case scipy is not available and all edge combinations |
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784 # need to be checked |
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785 if dist <= radius: |
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786 return theta <= u_weight + v_weight |
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787 else: |
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788 return False |
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789 |
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790 if _is_scipy_available: |
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791 edges = _fast_edges(G, radius, p) |
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792 G.add_edges_from(filter(should_join, edges)) |
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793 else: |
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794 G.add_edges_from(filter(should_join, combinations(G, 2))) |
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795 |
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796 return G |