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 Tiago Peixoto committed Apr 10, 2008 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 #! /usr/bin/env python # graph_tool.py -- a general graph manipulation python module # # Copyright (C) 2007 Tiago de Paula Peixoto # # This program is free software: you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by # the Free Software Foundation, either version 3 of the License, or # (at your option) any later version. # # This program is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the # GNU General Public License for more details. # # You should have received a copy of the GNU General Public License # along with this program. If not, see .  Tiago Peixoto committed Jul 15, 2009 19 20 21 22 23 """ graph_tool.generation - Random Graph Generation --------------------------------------------------- """  Tiago Peixoto committed Oct 26, 2008 24 25 from .. dl_import import dl_import dl_import("import libgraph_tool_generation")  Tiago Peixoto committed Apr 10, 2008 26 27  from .. core import Graph  Tiago Peixoto committed Feb 06, 2009 28 import sys, numpy  Tiago Peixoto committed Apr 10, 2008 29   Tiago Peixoto committed Aug 04, 2009 30 __all__ = ["random_graph", "random_rewire"]  Tiago Peixoto committed Apr 10, 2008 31   Tiago Peixoto committed Jul 15, 2009 32 33 34 def _corr_wrap(i, j, corr): return corr(i[1], j[1])  Tiago Peixoto committed Apr 10, 2008 35 36 37 def random_graph(N, deg_sampler, deg_corr=None, directed=True, parallel=False, self_loops=False, seed=0, verbose=False):  Tiago Peixoto committed Aug 02, 2009 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84  r""" Generate a random graph, with a given degree distribution and correlation. Parameters ---------- N : int Number of vertices in the graph. deg_sampler : function A degree sampler function which is called without arguments, and returns a tuple of ints representing the in and out-degree of a given vertex (or a single int for undirected graphs, representing the out-degree). This function is called once per vertex, but may be called more times, if the degree sequence cannot be used to build a graph. deg_corr : function (optional, default: None) A function which give the degree correlation of the graph. It should be callable with two parameters: the in,out-degree pair of the source vertex an edge, and the in,out-degree pair of the target of the same edge (for undirected graphs, both parameters are single values). The function should return a number proportional to the probability of such an edge existing in the generated graph. directed : bool (optional, default: True) Whether the generated graph should be directed. parallel : bool (optional, default: False) If True, parallel edges are allowed. self_loops : bool (optional, default: False) If True, self-loops are allowed. seed : int (optional, default: 0) Seed for the random number generator. If seed=0, a random value is chosen. Returns ------- random_graph : Graph The generated graph. See Also -------- random_rewire: in place graph shuffling Notes ----- The algorithm maintains a list of all available source and target degree pairs, such that the deg_corr function is called only once with the same parameters. The uncorrelated case, the complexity is :math:O(V+E). For the correlated case the worst-case complexity is :math:O(V^2), but the typical case has  Tiago Peixoto committed Aug 04, 2009 85 86  complexity :math:O(V + E\log N_k + N_k^2), where :math:N_k < V is the number of different degrees sampled (or in,out-degree pairs).  Tiago Peixoto committed Aug 02, 2009 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116  Examples -------- >>> from numpy.random import randint, random, seed, poisson >>> from pylab import * >>> seed(42) This is a degree sampler which uses rejection sampling to sample from the distribution :math:P(k)\propto 1/k, up to a maximum. >>> def sample_k(max): ... accept = False ... while not accept: ... k = randint(1,max+1) ... accept = random() < 1.0/k ... return k ... The following generates a random undirected graph with degree distribution :math:P(k)\propto 1/k (with k_max=40) and an *assortative* degree correlation of the form: .. math:: P(i,k) \propto \frac{1}{1+|i-k|} >>> g = gt.random_graph(1000, lambda: sample_k(40), ... lambda i,k: 1.0/(1+abs(i-k)), directed=False) >>> gt.scalar_assortativity(g, "out")  Tiago Peixoto committed Aug 04, 2009 117  (0.59472179721535989, 0.011919463022240388)  Tiago Peixoto committed Aug 02, 2009 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201  The following samples an in,out-degree pair from the joint distribution: .. math:: p(j,k) = \frac{1}{2}\frac{e^{-m_1}m_1^j}{j!}\frac{e^{-m_1}m_1^k}{k!} + \frac{1}{2}\frac{e^{-m_2}m_2^j}{j!}\frac{e^{-m_2}m_2^k}{k!} with :math:m_1 = 4 and :math:m_2 = 20. >>> def deg_sample(): ... if random() > 0.5: ... return poisson(4), poisson(4) ... else: ... return poisson(20), poisson(20) ... The following generates a random directed graph with this distribution, and plots the combined degree correlation. >>> g = gt.random_graph(20000, deg_sample) >>> >>> hist = gt.combined_corr_hist(g, "in", "out") >>> imshow(hist[0], interpolation="nearest") <...> >>> colorbar() <...> >>> xlabel("in degree") <...> >>> ylabel("out degree") <...> >>> savefig("combined-deg-hist.png") .. figure:: combined-deg-hist.png :align: center Combined degree histogram. A correlated directed graph can be build as follows. Consider the following degree correlation: .. math:: P(j',k'|j,k)=\frac{e^{-k}k^{j'}}{j'!} \frac{e^{-(20-j)}(20-j)^{k'}}{k'!} i.e., the in->out correlation is "disassortative", the out->in correlation is "assortative", and everything else is uncorrelated. We will use a flat degree distribution in the range [1,20). >>> p = scipy.stats.poisson >>> g = gt.random_graph(20000, lambda: (sample_k(19), sample_k(19)), ... lambda a,b: (p.pmf(a[0],b[1])* ... p.pmf(a[1],20-b[0]))) Lets plot the average degree correlations to check. >>> clf() >>> corr = gt.avg_neighbour_corr(g, "in", "in") >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", label=" vs in") (...) >>> corr = gt.avg_neighbour_corr(g, "in", "out") >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", label=" vs in") (...) >>> corr = gt.avg_neighbour_corr(g, "out", "in") >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", label=" vs out") (...) >>> corr = gt.avg_neighbour_corr(g, "out", "out") >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", label=" vs out") (...) >>> legend(loc="best") <...> >>> xlabel("source degree") <...> >>> ylabel("average target degree") <...> >>> savefig("deg-corr-dir.png") .. figure:: deg-corr-dir.png :align: center Average nearest neighbour correlations. """  Tiago Peixoto committed Feb 06, 2009 202 203  if seed == 0: seed = numpy.random.randint(0, sys.maxint)  Tiago Peixoto committed Apr 10, 2008 204 205 206 207 208  g = Graph() if deg_corr == None: uncorrelated = True else: uncorrelated = False  Tiago Peixoto committed Jul 15, 2009 209 210 211 212  if not directed and deg_corr != None: corr = lambda i,j: _corr_wrap(i, j, deg_corr) else: corr = deg_corr  Tiago Peixoto committed Jul 21, 2008 213  libgraph_tool_generation.gen_random_graph(g._Graph__graph, N,  Tiago Peixoto committed Jul 15, 2009 214  deg_sampler, corr,  Tiago Peixoto committed Apr 10, 2008 215 216 217 218 219  uncorrelated, not parallel, not self_loops, not directed, seed, verbose) g.set_directed(directed) return g  Tiago Peixoto committed Aug 04, 2009 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234  def random_rewire(g, strat="uncorrelated", self_loops = False, parallel_edges = False, seed = 0): if seed != 0: seed = random.randint(0, sys.maxint) if g.is_reversed(): was_reversed = True else: was_reversed = False g.set_reversed(False) libgraph_tool_generation.random_rewire(g._Graph__graph, strat, self_loops, parallel_edges, seed) if was_reversed: g.set_reversed(True)