__init__.py 13.7 KB
 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   Tiago Peixoto committed Aug 17, 2009 27 from .. core import Graph, _check_prop_scalar, _prop  Tiago Peixoto committed Feb 06, 2009 28 import sys, numpy  Tiago Peixoto committed Apr 10, 2008 29   Tiago Peixoto committed Aug 28, 2009 30 __all__ = ["random_graph", "random_rewire", "predecessor_tree", "line_graph"]  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  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 201 202  if seed == 0: seed = numpy.random.randint(0, sys.maxint)  Tiago Peixoto committed Apr 10, 2008 203 204 205 206 207  g = Graph() if deg_corr == None: uncorrelated = True else: uncorrelated = False  Tiago Peixoto committed Jul 15, 2009 208 209 210 211  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 212  libgraph_tool_generation.gen_random_graph(g._Graph__graph, N,  Tiago Peixoto committed Jul 15, 2009 213  deg_sampler, corr,  Tiago Peixoto committed Apr 10, 2008 214 215 216 217 218  uncorrelated, not parallel, not self_loops, not directed, seed, verbose) g.set_directed(directed) return g  Tiago Peixoto committed Aug 04, 2009 219   Tiago Peixoto committed Aug 07, 2009 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 def random_rewire(g, strat= "uncorrelated", parallel_edges = False, self_loops = False, seed = 0): r""" Shuffled the graph in-place. The degrees (either in or out) of each vertex are always the same, but otherwise the edges are randomly placed. If strat == "correlated", the degree correlations are also maintained: The new source and target of each edge both have the same in and out-degree. Parameters ---------- g : Graph Graph to be shuffled. The graph will be modified. strat : string (optional, default: "uncorrelated") If strat == "uncorrelated" only the degrees of the vertices will be maintained, nothing else. If strat == "correlated", additionally the new source and target of each edge both have the same in and out-degree. 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 ------- None See Also -------- random_graph: random graph generation Notes ----- Each edge gets swapped at least once, so the overall complexity is :math:O(E). Examples -------- Some small graphs for visualization. >>> from numpy.random import zipf, seed >>> from pylab import * >>> seed(42) >>> g = gt.random_graph(1000, lambda: sample_k(10), ... lambda i,j: exp(abs(i-j)), directed=False) >>> gt.graph_draw(g, output="rewire_orig.png") (...) >>> gt.random_rewire(g, "correlated") >>> gt.graph_draw(g, output="rewire_corr.png") (...) >>> gt.random_rewire(g) >>> gt.graph_draw(g, output="rewire_uncorr.png") (...) .. figure:: rewire_orig.png :align: center Original graph. (It is a ridiculogram _). .. figure:: rewire_corr.png :align: center Shuffled graph, with degree correlations. .. figure:: rewire_uncorr.png :align: center Shuffled graph, without degree correlations. We can try some larger graphs to get better statistics. >>> clf() >>> g = gt.random_graph(20000, lambda: sample_k(20), ... lambda i,j: exp(abs(i-j)), directed=False) >>> corr = gt.avg_neighbour_corr(g, "out", "out") >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o", label="original") (...) >>> gt.random_rewire(g, "correlated") >>> corr = gt.avg_neighbour_corr(g, "out", "out") >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o", label="correlated") (...) >>> gt.random_rewire(g) >>> corr = gt.avg_neighbour_corr(g, "out", "out") >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o", label="uncorrelated") (...) >>> xlabel("$k$") <...> >>> ylabel(r"$\left$") <...> >>> legend(loc="best") <...> >>> savefig("shuffled-stats.png") .. figure:: shuffled-stats.png :align: center Average degree correlations for the different shuffled and non-shuffled graphs. The shuffled graph with correlations displays exactly the same correlation as the original graph. Now let's do it for a directed graph. See :func:~graph_tool.generation.random_graph for more details. >>> 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]))) >>> clf() >>> 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") (...) >>> gt.random_rewire(g, "correlated") >>> corr = gt.avg_neighbour_corr(g, "in", "out") >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", ... label=" vs in, correlated") (...) >>> corr = gt.avg_neighbour_corr(g, "out", "in") >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", ... label=" vs out, correlated") (...) >>> gt.random_rewire(g, "uncorrelated") >>> corr = gt.avg_neighbour_corr(g, "in", "out") >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", ... label=" vs in, uncorrelated") (...) >>> corr = gt.avg_neighbour_corr(g, "out", "in") >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", ... label=" vs out, uncorrelated") (...) >>> legend(loc="best") <...> >>> xlabel("source degree") <...> >>> ylabel("average target degree") <...> >>> savefig("shuffled-deg-corr-dir.png") .. figure:: shuffled-deg-corr-dir.png :align: center Average degree correlations for the different shuffled and non-shuffled directed graphs. The shuffled graph with correlations displays exactly the same correlation as the original graph. """  Tiago Peixoto committed Aug 04, 2009 372 373  if seed != 0: seed = random.randint(0, sys.maxint)  Tiago Peixoto committed Aug 07, 2009 374 375  g.stash_filter(reversed=True)  Tiago Peixoto committed Aug 04, 2009 376 377  libgraph_tool_generation.random_rewire(g._Graph__graph, strat, self_loops, parallel_edges, seed)  Tiago Peixoto committed Aug 07, 2009 378  g.pop_filter(reversed=True)  Tiago Peixoto committed Aug 17, 2009 379 380 381 382 383 384 385 386 387 388 389  def predecessor_tree(g, pred_map): """Return a graph from a list of predecessors given by the 'pred_map' vertex property.""" _check_prop_scalar(pred_map, "pred_map") pg = Graph() libgraph_tool_generation.predecessor_graph(g._Graph__graph, pg._Graph__graph, _prop("v", g, pred_map)) return pg  Tiago Peixoto committed Aug 28, 2009 390 391 392 393 394 395 396 397 398 399  def line_graph(g): lg = Graph(directed=g.is_directed()) vertex_map = lg.new_vertex_property("int64_t") libgraph_tool_generation.line_graph(g._Graph__graph, lg._Graph__graph, _prop("v", lg, vertex_map)) return lg, vertex_map