annotate env/lib/python3.9/site-packages/networkx/generators/geometric.py @ 0:4f3585e2f14b draft default tip

"planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
author shellac
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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diff changeset
630 if q < 0:
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631 raise nx.NetworkXException("q must be >= 0")
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diff changeset
632 if r < 0:
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diff changeset
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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diff changeset
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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diff changeset
639 for p2 in nodes:
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diff changeset
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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parents:
diff changeset
646 cdf = list(accumulate(probs))
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parents:
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647 for _ in range(q):
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diff changeset
648 target = nodes[bisect_left(cdf, seed.uniform(0, cdf[-1]))]
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diff changeset
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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diff changeset
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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diff changeset
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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diff changeset
694 graph, which represents probability.
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695 seed : integer, random_state, or None (default)
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diff changeset
696 Indicator of random number generation state.
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diff changeset
697 See :ref:`Randomness<randomness>`.
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diff changeset
698
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699 Returns
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700 -------
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diff changeset
701 Graph
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702 A thresholded random geographic graph, undirected and without
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parents:
diff changeset
703 self-loops.
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parents:
diff changeset
704
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diff changeset
705 Each node has a node attribute ``'pos'`` that stores the
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diff changeset
706 position of that node in Euclidean space as provided by the
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diff changeset
707 ``pos`` keyword argument or, if ``pos`` was not provided, as
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diff changeset
708 generated by this function. Similarly, each node has a nodethre
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parents:
diff changeset
709 attribute ``'weight'`` that stores the weight of that node as
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parents:
diff changeset
710 provided or as generated.
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diff changeset
711
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diff changeset
712 Examples
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diff changeset
713 --------
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parents:
diff changeset
714 Default Graph:
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parents:
diff changeset
715
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diff changeset
716 G = nx.thresholded_random_geometric_graph(50, 0.2, 0.1)
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parents:
diff changeset
717
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parents:
diff changeset
718 Custom Graph:
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diff changeset
719
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diff changeset
720 Create a thresholded random geometric graph on 50 uniformly distributed
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parents:
diff changeset
721 nodes where nodes are joined by an edge if their sum weights drawn from
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parents:
diff changeset
722 a exponential distribution with rate = 5 are >= theta = 0.1 and their
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parents:
diff changeset
723 Euclidean distance is at most 0.2.
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diff changeset
724
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diff changeset
725 Notes
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diff changeset
726 -----
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parents:
diff changeset
727 This uses a *k*-d tree to build the graph.
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parents:
diff changeset
728
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diff changeset
729 The `pos` keyword argument can be used to specify node positions so you
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parents:
diff changeset
730 can create an arbitrary distribution and domain for positions.
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parents:
diff changeset
731
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diff changeset
732 For example, to use a 2D Gaussian distribution of node positions with mean
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parents:
diff changeset
733 (0, 0) and standard deviation 2
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diff changeset
734
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diff changeset
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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parents:
diff changeset
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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parents:
diff changeset
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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parents:
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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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diff changeset
755 n_name, nodes = n
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diff changeset
756 G = nx.Graph()
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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757 G.name = f"thresholded_random_geometric_graph({n}, {radius}, {theta}, {dim})"
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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758 G.add_nodes_from(nodes)
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
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759 # If no weights are provided, choose them from an exponential
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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760 # distribution.
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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761 if weight is None:
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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762 weight = {v: seed.expovariate(1) for v in G}
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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763 # If no positions are provided, choose uniformly random vectors in
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff changeset
764 # Euclidean space of the specified dimension.
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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765 if pos is None:
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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766 pos = {v: [seed.random() for i in range(dim)] for v in nodes}
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
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767 # If no distance metric is provided, use Euclidean distance.
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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768
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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769 nx.set_node_attributes(G, weight, "weight")
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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770 nx.set_node_attributes(G, pos, "pos")
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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parents:
diff changeset
771
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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parents:
diff changeset
772 # Returns ``True`` if and only if the nodes whose attributes are
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff changeset
773 # ``du`` and ``dv`` should be joined, according to the threshold
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff changeset
774 # condition and node pairs are within the maximum connection
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
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775 # distance, ``radius``.
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
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776 def should_join(pair):
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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777 u, v = pair
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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778 u_weight, v_weight = weight[u], weight[v]
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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779 u_pos, v_pos = pos[u], pos[v]
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff changeset
780 dist = (sum(abs(a - b) ** p for a, b in zip(u_pos, v_pos))) ** (1 / p)
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
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781 # Check if dist is <= radius parameter. This check is redundant if
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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782 # scipy is available and _fast_edges routine is used, but provides
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff changeset
783 # the check in case scipy is not available and all edge combinations
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff changeset
784 # need to be checked
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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785 if dist <= radius:
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff changeset
786 return theta <= u_weight + v_weight
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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787 else:
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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788 return False
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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789
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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parents:
diff changeset
790 if _is_scipy_available:
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
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791 edges = _fast_edges(G, radius, p)
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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792 G.add_edges_from(filter(should_join, edges))
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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793 else:
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
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diff changeset
794 G.add_edges_from(filter(should_join, combinations(G, 2)))
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff changeset
795
4f3585e2f14b "planemo upload commit 60cee0fc7c0cda8592644e1aad72851dec82c959"
shellac
parents:
diff changeset
796 return G