__init__.py 34 KB
 Tiago Peixoto committed Apr 10, 2008 1 #! /usr/bin/env python  Tiago Peixoto committed Mar 07, 2010 2 # -*- coding: utf-8 -*-  Tiago Peixoto committed Apr 10, 2008 3 #  Tiago Peixoto committed Mar 07, 2010 4 5 # graph_tool -- a general graph manipulation python module #  Tiago Peixoto committed Feb 10, 2011 6 # Copyright (C) 2007-2011 Tiago de Paula Peixoto  Tiago Peixoto committed Apr 10, 2008 7 8 9 10 11 12 13 14 15 16 17 18 19 20 # # 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 21 """  Tiago Peixoto committed Oct 05, 2009 22 graph_tool.generation - Random graph generation  Tiago Peixoto committed Jul 15, 2009 23 ---------------------------------------------------  Tiago Peixoto committed Oct 05, 2009 24 25 26 27 28 29 30 31 32 33 34 35  Summary +++++++ .. autosummary:: :nosignatures: random_graph random_rewire predecessor_tree line_graph graph_union  Tiago Peixoto committed Dec 06, 2009 36  triangulation  Tiago Peixoto committed Oct 04, 2010 37 38  lattice geometric_graph  Tiago Peixoto committed Nov 13, 2010 39  price_network  Tiago Peixoto committed Oct 05, 2009 40 41 42  Contents ++++++++  Tiago Peixoto committed Jul 15, 2009 43 44 """  Tiago Peixoto committed Oct 26, 2008 45 46 from .. dl_import import dl_import dl_import("import libgraph_tool_generation")  Tiago Peixoto committed Apr 10, 2008 47   Tiago Peixoto committed Dec 06, 2009 48 from .. core import Graph, _check_prop_scalar, _prop, _limit_args  Tiago Peixoto committed Feb 20, 2010 49 from .. stats import label_parallel_edges, label_self_loops  Tiago Peixoto committed Oct 05, 2009 50 import sys, numpy, numpy.random  Tiago Peixoto committed Apr 10, 2008 51   Tiago Peixoto committed Sep 06, 2009 52 __all__ = ["random_graph", "random_rewire", "predecessor_tree", "line_graph",  Tiago Peixoto committed Nov 13, 2010 53 54  "graph_union", "triangulation", "lattice", "geometric_graph", "price_network"]  Tiago Peixoto committed Apr 10, 2008 55   Tiago Peixoto committed May 03, 2010 56   Tiago Peixoto committed Apr 10, 2008 57 def random_graph(N, deg_sampler, deg_corr=None, directed=True,  Tiago Peixoto committed Feb 20, 2010 58 59  parallel_edges=False, self_loops=False, random=True, verbose=False):  Tiago Peixoto committed Aug 02, 2009 60 61 62 63 64 65 66 67 68 69 70 71 72 73  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)  Tiago Peixoto committed Dec 21, 2009 74  A function which gives the degree correlation of the graph. It should be  Tiago Peixoto committed Aug 02, 2009 75 76 77 78 79 80 81  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.  Tiago Peixoto committed Feb 20, 2010 82  parallel_edges : bool (optional, default: False)  Tiago Peixoto committed Aug 02, 2009 83 84 85  If True, parallel edges are allowed. self_loops : bool (optional, default: False) If True, self-loops are allowed.  Tiago Peixoto committed Feb 20, 2010 86 87  random : bool (optional, default: True) If True, the returned graph is randomized.  Tiago Peixoto committed Feb 20, 2010 88 89  verbose : bool (optional, default: False) If True, verbose information is displayed.  Tiago Peixoto committed Aug 02, 2009 90 91 92  Returns -------  Tiago Peixoto committed Oct 05, 2009 93  random_graph : :class:~graph_tool.Graph  Tiago Peixoto committed Aug 02, 2009 94 95 96 97 98 99 100 101  The generated graph. See Also -------- random_rewire: in place graph shuffling Notes -----  Tiago Peixoto committed Feb 20, 2010 102 103 104 105 106 107  The algorithm makes sure the degree sequence is graphical (i.e. realizable) and keeps re-sampling the degrees if is not. With a valid degree sequence, the edges are placed deterministically, and later the graph is shuffled with the :func:~graph_tool.generation.random_rewire function. The complexity is :math:O(V+E) if parallel edges are allowed, and  Tiago Peixoto committed Mar 07, 2010 108 109  :math:O(V+E\log N_k) if parallel edges are not allowed, where :math:N_k < V is the number of different degrees sampled (or in,out-degree pairs).  Tiago Peixoto committed Aug 02, 2009 110   Tiago Peixoto committed Feb 20, 2010 111 112 113  References ---------- [deg-sequence] http://en.wikipedia.org/wiki/Degree_%28graph_theory%29#Degree_sequence  Tiago Peixoto committed Aug 02, 2009 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143  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 May 03, 2010 144  (0.62986894481988553, 0.011101504846821255)  Tiago Peixoto committed Aug 02, 2009 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  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() <...>  Tiago Peixoto committed Oct 05, 2009 172  >>> xlabel("in-degree")  Tiago Peixoto committed Aug 02, 2009 173  <...>  Tiago Peixoto committed Oct 05, 2009 174  >>> ylabel("out-degree")  Tiago Peixoto committed Aug 02, 2009 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  <...> >>> 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.  Tiago Peixoto committed Dec 06, 2009 202 203 204 205  >>> figure(figsize=(6,3)) <...> >>> axes([0.1,0.15,0.63,0.8]) <...>  Tiago Peixoto committed Aug 02, 2009 206  >>> corr = gt.avg_neighbour_corr(g, "in", "in")  Tiago Peixoto committed Jul 13, 2010 207  >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",  Tiago Peixoto committed Oct 05, 2009 208  ... label=r"$\left<\text{in}\right>$ vs in")  Tiago Peixoto committed Aug 02, 2009 209 210  (...) >>> corr = gt.avg_neighbour_corr(g, "in", "out")  Tiago Peixoto committed Jul 13, 2010 211  >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",  Tiago Peixoto committed Oct 05, 2009 212  ... label=r"$\left<\text{out}\right>$ vs in")  Tiago Peixoto committed Aug 02, 2009 213 214  (...) >>> corr = gt.avg_neighbour_corr(g, "out", "in")  Tiago Peixoto committed Jul 13, 2010 215  >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",  Tiago Peixoto committed Oct 05, 2009 216  ... label=r"$\left<\text{in}\right>$ vs out")  Tiago Peixoto committed Aug 02, 2009 217 218  (...) >>> corr = gt.avg_neighbour_corr(g, "out", "out")  Tiago Peixoto committed Jul 13, 2010 219  >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",  Tiago Peixoto committed Oct 05, 2009 220  ... label=r"$\left<\text{out}\right>$ vs out")  Tiago Peixoto committed Aug 02, 2009 221  (...)  Tiago Peixoto committed Dec 06, 2009 222  >>> legend(loc=(1.05,0.5))  Tiago Peixoto committed Aug 02, 2009 223 224 225 226 227 228 229 230 231 232 233 234  <...> >>> 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 Oct 05, 2009 235  seed = numpy.random.randint(0, sys.maxint)  Tiago Peixoto committed Apr 10, 2008 236 237 238 239 240  g = Graph() if deg_corr == None: uncorrelated = True else: uncorrelated = False  Tiago Peixoto committed Feb 20, 2010 241 242  libgraph_tool_generation.gen_random_graph(g._Graph__graph, N, deg_sampler, uncorrelated, not parallel_edges,  Tiago Peixoto committed Apr 10, 2008 243  not self_loops, not directed,  Tiago Peixoto committed May 03, 2010 244  seed, verbose, True)  Tiago Peixoto committed Apr 10, 2008 245  g.set_directed(directed)  Tiago Peixoto committed Feb 20, 2010 246  if random:  Tiago Peixoto committed May 03, 2010 247 248  random_rewire(g, parallel_edges=parallel_edges, self_loops=self_loops, verbose=verbose)  Tiago Peixoto committed Feb 20, 2010 249  if deg_corr != None:  Tiago Peixoto committed May 03, 2010 250 251 252  random_rewire(g, strat="probabilistic", parallel_edges=parallel_edges, deg_corr=deg_corr, self_loops=self_loops, verbose=verbose)  Tiago Peixoto committed Apr 10, 2008 253  return g  Tiago Peixoto committed Aug 04, 2009 254   Tiago Peixoto committed May 03, 2010 255   Tiago Peixoto committed May 03, 2010 256 257 258 @_limit_args({"strat": ["erdos", "correlated", "uncorrelated", "probabilistic"]}) def random_rewire(g, strat="uncorrelated", parallel_edges=False, self_loops=False, deg_corr=None, verbose=False):  Tiago Peixoto committed Aug 07, 2009 259  r"""  Tiago Peixoto committed Dec 21, 2009 260 261  Shuffle the graph in-place. If strat != "erdos", the degrees (either in or out) of each vertex are always the same, but otherwise the edges are  Tiago Peixoto committed Feb 20, 2010 262  randomly placed. If strat = "correlated", the degree correlations are  Tiago Peixoto committed Dec 21, 2009 263  also maintained: The new source and target of each edge both have the same  Tiago Peixoto committed Feb 20, 2010 264 265  in and out-degree. If strat = "probabilistic", than edges are rewired according to the degree correlation given by the parameter deg_corr.  Tiago Peixoto committed Aug 07, 2009 266 267 268  Parameters ----------  Tiago Peixoto committed Oct 05, 2009 269  g : :class:~graph_tool.Graph  Tiago Peixoto committed Aug 07, 2009 270 271  Graph to be shuffled. The graph will be modified. strat : string (optional, default: "uncorrelated")  Tiago Peixoto committed Mar 07, 2010 272 273 274  If strat = "erdos", the resulting graph will be entirely random. If strat = "uncorrelated" only the degrees of the vertices will be maintained, nothing else. If strat = "correlated", additionally the  Tiago Peixoto committed Dec 21, 2009 275  new source and target of each edge both have the same in and out-degree.  Tiago Peixoto committed Mar 07, 2010 276  If strat = "probabilistic", than edges are rewired according to the  Tiago Peixoto committed Feb 20, 2010 277  degree correlation given by the parameter deg_corr.  Tiago Peixoto committed Aug 07, 2009 278 279 280 281  parallel : bool (optional, default: False) If True, parallel edges are allowed. self_loops : bool (optional, default: False) If True, self-loops are allowed.  Tiago Peixoto committed Feb 20, 2010 282 283 284 285 286 287 288 289 290 291  deg_corr : function (optional, default: None) A function which gives 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. This parameter is ignored, unless strat = "probabilistic". verbose : bool (optional, default: False) If True, verbose information is displayed.  Tiago Peixoto committed Aug 07, 2009 292 293 294 295 296 297 298  See Also -------- random_graph: random graph generation Notes -----  Tiago Peixoto committed Feb 20, 2010 299  This algorithm iterates through all the edges in the network and tries to  Tiago Peixoto committed May 21, 2010 300  swap its target or source with the target or source of another edge.  Tiago Peixoto committed Feb 20, 2010 301 302 303 304 305 306 307  .. note:: If parallel_edges = False, parallel edges are not placed during rewiring. In this case, for some special graphs it may be necessary to call the function more than once to obtain a graph which corresponds to a uniform sample from the ensemble. But typically, if the graph is sufficiently large, a single call should be enough.  Tiago Peixoto committed Aug 07, 2009 308 309  Each edge gets swapped at least once, so the overall complexity is  Tiago Peixoto committed Feb 20, 2010 310 311 312 313  :math:O(E). If strat = "probabilistic" the complexity is :math:O(E\log N_k), where :math:N_k < V is the number of different degrees (or in,out-degree pairs).  Tiago Peixoto committed Aug 07, 2009 314 315 316 317 318 319  Examples -------- Some small graphs for visualization.  Tiago Peixoto committed Dec 06, 2009 320  >>> from numpy.random import random, seed  Tiago Peixoto committed Aug 07, 2009 321 322  >>> from pylab import * >>> seed(42)  Tiago Peixoto committed Dec 06, 2009 323  >>> g, pos = gt.triangulation(random((1000,2)))  Tiago Peixoto committed Oct 05, 2009 324  >>> gt.graph_draw(g, layout="arf", output="rewire_orig.png", size=(6,6))  Tiago Peixoto committed Sep 03, 2009 325  <...>  Tiago Peixoto committed Aug 07, 2009 326  >>> gt.random_rewire(g, "correlated")  Tiago Peixoto committed Oct 05, 2009 327  >>> gt.graph_draw(g, layout="arf", output="rewire_corr.png", size=(6,6))  Tiago Peixoto committed Sep 03, 2009 328  <...>  Tiago Peixoto committed Aug 07, 2009 329  >>> gt.random_rewire(g)  Tiago Peixoto committed Oct 05, 2009 330  >>> gt.graph_draw(g, layout="arf", output="rewire_uncorr.png", size=(6,6))  Tiago Peixoto committed Sep 03, 2009 331  <...>  Tiago Peixoto committed Dec 21, 2009 332 333 334  >>> gt.random_rewire(g, "erdos") >>> gt.graph_draw(g, layout="arf", output="rewire_erdos.png", size=(6,6)) <...>  Tiago Peixoto committed Aug 07, 2009 335   Tiago Peixoto committed Oct 05, 2009 336  Some ridiculograms _ :  Tiago Peixoto committed Aug 07, 2009 337   Tiago Peixoto committed Oct 05, 2009 338 339 340  .. image:: rewire_orig.png .. image:: rewire_corr.png .. image:: rewire_uncorr.png  Tiago Peixoto committed Dec 21, 2009 341  .. image:: rewire_erdos.png  Tiago Peixoto committed Aug 07, 2009 342   Tiago Peixoto committed Dec 21, 2009 343 344 345  *From left to right:* Original graph; Shuffled graph, with degree correlations; Shuffled graph, without degree correlations; Shuffled graph, with random degrees.  Tiago Peixoto committed Aug 07, 2009 346 347 348  We can try some larger graphs to get better statistics.  Tiago Peixoto committed Dec 06, 2009 349 350  >>> figure() <...>  Tiago Peixoto committed Dec 21, 2009 351  >>> g = gt.random_graph(30000, lambda: sample_k(20),  Tiago Peixoto committed Aug 07, 2009 352 353  ... lambda i,j: exp(abs(i-j)), directed=False) >>> corr = gt.avg_neighbour_corr(g, "out", "out")  Tiago Peixoto committed Jul 13, 2010 354  >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-", label="original")  Tiago Peixoto committed Aug 07, 2009 355 356 357  (...) >>> gt.random_rewire(g, "correlated") >>> corr = gt.avg_neighbour_corr(g, "out", "out")  Tiago Peixoto committed Jul 13, 2010 358  >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="*", label="correlated")  Tiago Peixoto committed Aug 07, 2009 359 360 361  (...) >>> gt.random_rewire(g) >>> corr = gt.avg_neighbour_corr(g, "out", "out")  Tiago Peixoto committed Jul 13, 2010 362  >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-", label="uncorrelated")  Tiago Peixoto committed Aug 07, 2009 363  (...)  Tiago Peixoto committed Dec 21, 2009 364 365  >>> gt.random_rewire(g, "erdos") >>> corr = gt.avg_neighbour_corr(g, "out", "out")  Tiago Peixoto committed Jul 13, 2010 366  >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-", label="Erdos")  Tiago Peixoto committed Dec 21, 2009 367  (...)  Tiago Peixoto committed Aug 07, 2009 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387  >>> 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)),  Tiago Peixoto committed Dec 21, 2009 388  ... lambda a,b: (p.pmf(a[0],b[1])*p.pmf(a[1],20-b[0])))  Tiago Peixoto committed Dec 06, 2009 389 390 391 392  >>> figure(figsize=(6,3)) <...> >>> axes([0.1,0.15,0.6,0.8]) <...>  Tiago Peixoto committed Aug 07, 2009 393  >>> corr = gt.avg_neighbour_corr(g, "in", "out")  Tiago Peixoto committed Jul 13, 2010 394  >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",  Tiago Peixoto committed Oct 05, 2009 395  ... label=r"$\left<\text{o}\right>$ vs i")  Tiago Peixoto committed Aug 07, 2009 396 397  (...) >>> corr = gt.avg_neighbour_corr(g, "out", "in")  Tiago Peixoto committed Jul 13, 2010 398  >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",  Tiago Peixoto committed Oct 05, 2009 399  ... label=r"$\left<\text{i}\right>$ vs o")  Tiago Peixoto committed Aug 07, 2009 400 401 402  (...) >>> gt.random_rewire(g, "correlated") >>> corr = gt.avg_neighbour_corr(g, "in", "out")  Tiago Peixoto committed Jul 13, 2010 403  >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",  Tiago Peixoto committed Oct 05, 2009 404  ... label=r"$\left<\text{o}\right>$ vs i, corr.")  Tiago Peixoto committed Aug 07, 2009 405 406  (...) >>> corr = gt.avg_neighbour_corr(g, "out", "in")  Tiago Peixoto committed Jul 13, 2010 407  >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",  Tiago Peixoto committed Oct 05, 2009 408  ... label=r"$\left<\text{i}\right>$ vs o, corr.")  Tiago Peixoto committed Aug 07, 2009 409 410 411  (...) >>> gt.random_rewire(g, "uncorrelated") >>> corr = gt.avg_neighbour_corr(g, "in", "out")  Tiago Peixoto committed Jul 13, 2010 412  >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",  Tiago Peixoto committed Oct 05, 2009 413  ... label=r"$\left<\text{o}\right>$ vs i, uncorr.")  Tiago Peixoto committed Aug 07, 2009 414 415  (...) >>> corr = gt.avg_neighbour_corr(g, "out", "in")  Tiago Peixoto committed Jul 13, 2010 416  >>> errorbar(corr[2][:-1], corr[0], yerr=corr[1], fmt="o-",  Tiago Peixoto committed Oct 05, 2009 417  ... label=r"$\left<\text{i}\right>$ vs o, uncorr.")  Tiago Peixoto committed Aug 07, 2009 418  (...)  Tiago Peixoto committed Dec 06, 2009 419  >>> legend(loc=(1.05,0.45))  Tiago Peixoto committed Aug 07, 2009 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434  <...> >>> 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 Oct 05, 2009 435  seed = numpy.random.randint(0, sys.maxint)  Tiago Peixoto committed Aug 07, 2009 436   Tiago Peixoto committed Feb 20, 2010 437 438 439 440 441 442 443 444 445 446 447 448 449 450  if not parallel_edges: p = label_parallel_edges(g) if p.a.max() != 0: raise ValueError("Parallel edge detected. Can't rewire " + "graph without parallel edges if it " + "already contains parallel edges!") if not self_loops: l = label_self_loops(g) if l.a.max() != 0: raise ValueError("Self-loop detected. Can't rewire graph " + "without self-loops if it already contains" + " self-loops!") if deg_corr != None and not g.is_directed():  Tiago Peixoto committed May 03, 2010 451  corr = lambda i, j: deg_corr(i[1], j[1])  Tiago Peixoto committed Feb 20, 2010 452 453 454  else: corr = deg_corr  Tiago Peixoto committed Feb 20, 2010 455 456  if corr == None: g.stash_filter(reversed=True)  Tiago Peixoto committed Dec 21, 2009 457 458  try: libgraph_tool_generation.random_rewire(g._Graph__graph, strat,  Tiago Peixoto committed Feb 20, 2010 459 460  self_loops, parallel_edges, corr, seed, verbose)  Tiago Peixoto committed Dec 21, 2009 461  finally:  Tiago Peixoto committed Feb 20, 2010 462 463  if corr == None: g.pop_filter(reversed=True)  Tiago Peixoto committed Aug 17, 2009 464   Tiago Peixoto committed May 03, 2010 465   Tiago Peixoto committed Aug 17, 2009 466 def predecessor_tree(g, pred_map):  Tiago Peixoto committed Nov 13, 2010 467  """Return a graph from a list of predecessors given by the pred_map vertex property."""  Tiago Peixoto committed Aug 17, 2009 468 469 470 471 472 473 474  _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 475   Tiago Peixoto committed May 03, 2010 476   Tiago Peixoto committed Aug 28, 2009 477 def line_graph(g):  Tiago Peixoto committed Oct 05, 2009 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497  """Return the line graph of the given graph g. Notes ----- Given an undirected graph G, its line graph L(G) is a graph such that * each vertex of L(G) represents an edge of G; and * two vertices of L(G) are adjacent if and only if their corresponding edges share a common endpoint ("are adjacent") in G. For a directed graph, the second criterion becomes: * Two vertices representing directed edges from u to v and from w to x in G are connected by an edge from uv to wx in the line digraph when v = w. References ---------- .. [line-wiki] http://en.wikipedia.org/wiki/Line_graph """  Tiago Peixoto committed Aug 28, 2009 498 499 500 501 502 503 504 505  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  Tiago Peixoto committed Sep 06, 2009 506   Tiago Peixoto committed May 03, 2010 507 508  def graph_union(g1, g2, props=None, include=False):  Tiago Peixoto committed Oct 05, 2009 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533  """Return the union of graphs g1 and g2, composed of all edges and vertices of g1 and g2, without overlap. Parameters ---------- g1 : :class:~graph_tool.Graph First graph in the union. g2 : :class:~graph_tool.Graph Second graph in the union. props : list of tuples of :class:~graph_tool.PropertyMap (optional, default: []) Each element in this list must be a tuple of two PropertyMap objects. The first element must be a property of g1, and the second of g2. The values of the property maps are propagated into the union graph, and returned. include : bool (optional, default: False) If true, graph g2 is inserted into g1 which is modified. If false, a new graph is created, and both graphs remain unmodified. Returns ------- ug : :class:~graph_tool.Graph The union graph props : list of :class:~graph_tool.PropertyMap objects List of propagated properties. This is only returned if props is not empty.  Tiago Peixoto committed Dec 06, 2009 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553  Examples -------- >>> from numpy.random import random, seed >>> seed(42) >>> g = gt.triangulation(random((300,2)))[0] >>> ug = gt.graph_union(g, g) >>> uug = gt.graph_union(g, ug) >>> gt.graph_draw(g, layout="arf", size=(8,8), output="graph_original.png") <...> >>> gt.graph_draw(ug, layout="arf", size=(8,8), output="graph_union.png") <...> >>> gt.graph_draw(uug, layout="arf", size=(8,8), output="graph_union2.png") <...> .. image:: graph_original.png .. image:: graph_union.png .. image:: graph_union2.png  Tiago Peixoto committed Oct 05, 2009 554  """  Tiago Peixoto committed May 03, 2010 555 556  if props == None: props = []  Tiago Peixoto committed Sep 06, 2009 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594  if not include: g1 = Graph(g1) g1.stash_filter(directed=True) g1.set_directed(True) g2.stash_filter(directed=True) g2.set_directed(True) n_props = [] try: vmap, emap = libgraph_tool_generation.graph_union(g1._Graph__graph, g2._Graph__graph) for p in props: p1, p2 = p if not include: p1 = g1.copy_property(p1) if p2.value_type() != p1.value_type(): p2 = g2.copy_property(p2, value_type=p1.value_type()) if p1.key_type() == 'v': libgraph_tool_generation.\ vertex_property_union(g1._Graph__graph, g2._Graph__graph, vmap, emap, _prop(p1.key_type(), g1, p1), _prop(p2.key_type(), g2, p2)) else: libgraph_tool_generation.\ edge_property_union(g1._Graph__graph, g2._Graph__graph, vmap, emap, _prop(p1.key_type(), g1, p1), _prop(p2.key_type(), g2, p2)) n_props.append(p1) finally: g1.pop_filter(directed=True) g2.pop_filter(directed=True) if len(n_props) > 0: return g1, n_props else: return g1  Tiago Peixoto committed Dec 06, 2009 595   Tiago Peixoto committed May 03, 2010 596 597  @_limit_args({"type": ["simple", "delaunay"]})  Tiago Peixoto committed Jan 11, 2010 598 def triangulation(points, type="simple", periodic=False):  Tiago Peixoto committed Dec 06, 2009 599 600 601 602 603 604 605 606 607 608  r""" Generate a 2D or 3D triangulation graph from a given point set. Parameters ---------- points : :class:~numpy.ndarray Point set for the triangulation. It may be either a N x d array, where N is the number of points, and d is the space dimension (either 2 or 3). type : string (optional, default: 'simple') Type of triangulation. May be either 'simple' or 'delaunay'.  Tiago Peixoto committed Jan 11, 2010 609 610 611  periodic : bool (optional, default: False) If True, periodic boundary conditions will be used. This is parameter is valid only for type="delaunay", and is otherwise ignored.  Tiago Peixoto committed Dec 06, 2009 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626  Returns ------- triangulation_graph : :class:~graph_tool.Graph The generated graph. pos : :class:~graph_tool.PropertyMap Vertex property map with the Cartesian coordinates. See Also -------- random_graph: random graph generation Notes -----  Tiago Peixoto committed Dec 22, 2009 627  A triangulation [cgal-triang]_ is a division of the convex hull of a point  Tiago Peixoto committed Dec 22, 2009 628  set into triangles, using only that set as triangle vertices.  Tiago Peixoto committed Dec 06, 2009 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647  In simple triangulations (type="simple"), the insertion of a point is done by locating a face that contains the point, and splitting this face into three new faces (the order of insertion is therefore important). If the point falls outside the convex hull, the triangulation is restored by flips. Apart from the location, insertion takes a time O(1). This bound is only an amortized bound for points located outside the convex hull. Delaunay triangulations (type="delaunay") have the specific empty sphere property, that is, the circumscribing sphere of each cell of such a triangulation does not contain any other vertex of the triangulation in its interior. These triangulations are uniquely defined except in degenerate cases where five points are co-spherical. Note however that the CGAL implementation computes a unique triangulation even in these cases. Examples -------- >>> from numpy.random import seed, random >>> seed(42)  Tiago Peixoto committed Oct 04, 2010 648  >>> points = random((500, 2)) * 4  Tiago Peixoto committed Dec 06, 2009 649  >>> g, pos = gt.triangulation(points)  Tiago Peixoto committed Dec 22, 2009 650 651 652 653 654 655 656 657 658  >>> weight = g.new_edge_property("double") # Edge weights corresponding to ... # Euclidean distances >>> for e in g.edges(): ... weight[e] = sqrt(sum((array(pos[e.source()]) - ... array(pos[e.target()]))**2)) >>> b = gt.betweenness(g, weight=weight) >>> b[1].a *= 100 >>> gt.graph_draw(g, pos=pos, pin=True, size=(8,8), vsize=0.07, vcolor=b[0], ... eprops={"penwidth":b[1]}, output="triang.png")  Tiago Peixoto committed Dec 06, 2009 659 660  <...> >>> g, pos = gt.triangulation(points, type="delaunay")  Tiago Peixoto committed Dec 22, 2009 661 662 663 664 665 666 667 668  >>> weight = g.new_edge_property("double") >>> for e in g.edges(): ... weight[e] = sqrt(sum((array(pos[e.source()]) - ... array(pos[e.target()]))**2)) >>> b = gt.betweenness(g, weight=weight) >>> b[1].a *= 120 >>> gt.graph_draw(g, pos=pos, pin=True, size=(8,8), vsize=0.07, vcolor=b[0], ... eprops={"penwidth":b[1]}, output="triang-delaunay.png")  Tiago Peixoto committed Dec 06, 2009 669 670 671 672 673 674 675  <...> 2D triangulation of random points: .. image:: triang.png .. image:: triang-delaunay.png  Tiago Peixoto committed Dec 22, 2009 676 677 678  *Left:* Simple triangulation. *Right:* Delaunay triangulation. The vertex colors and the edge thickness correspond to the weighted betweenness centrality.  Tiago Peixoto committed Dec 06, 2009 679 680 681  References ----------  Tiago Peixoto committed Dec 22, 2009 682  .. [cgal-triang] http://www.cgal.org/Manual/last/doc_html/cgal_manual/Triangulation_3/Chapter_main.html  Tiago Peixoto committed Dec 06, 2009 683 684 685  """  Tiago Peixoto committed May 03, 2010 686  if points.shape[1] not in [2, 3]:  Tiago Peixoto committed Dec 06, 2009 687 688 689 690 691 692 693 694 695 696  raise ValueError("points array must have shape N x d, with d either 2 or 3.") # copy points to ensure continuity and correct data type points = numpy.array(points, dtype='float64') if points.shape[1] == 2: npoints = numpy.zeros((points.shape[0], 3)) npoints[:,:2] = points points = npoints g = Graph(directed=False) pos = g.new_vertex_property("vector") libgraph_tool_generation.triangulation(g._Graph__graph, points,  Tiago Peixoto committed Jan 11, 2010 697  _prop("v", g, pos), type, periodic)  Tiago Peixoto committed Dec 06, 2009 698  return g, pos  Tiago Peixoto committed Oct 04, 2010 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809  def lattice(shape, periodic=False): r""" Generate a N-dimensional square lattice. Parameters ---------- shape : list or :class:~numpy.ndarray List of sizes in each dimension. periodic : bool (optional, default: False) If True, periodic boundary conditions will be used. Returns ------- lattice_graph : :class:~graph_tool.Graph The generated graph. See Also -------- triangulation: 2D or 3D triangulation random_graph: random graph generation Examples -------- >>> g = gt.lattice([10,10]) >>> gt.graph_draw(g, size=(8,8), output="lattice.png") <...> >>> g = gt.lattice([10,20], periodic=True) >>> gt.graph_draw(g, size=(8,8), output="lattice_periodic.png") <...> >>> g = gt.lattice([10,10,10]) >>> gt.graph_draw(g, size=(8,8), output="lattice_3d.png") <...> .. image:: lattice.png .. image:: lattice_periodic.png .. image:: lattice_3d.png *Left:* 10x10 2D lattice. *Middle:* 10x20 2D periodic lattice (torus). *Right:* 10x10x10 3D lattice. References ---------- .. [lattice] http://en.wikipedia.org/wiki/Square_lattice """ g = Graph(directed=False) libgraph_tool_generation.lattice(g._Graph__graph, shape, periodic) return g def geometric_graph(points, radius, ranges=None): r""" Generate a geometric network form a set of N-dimensional points. Parameters ---------- points : list or :class:~numpy.ndarray List of points. This must be a two-dimensional array, where the rows are coordinates in a N-dimensional space. radius : float Pairs of points with an euclidean distance lower than this parameters will be connected. ranges : list or :class:~numpy.ndarray (optional, default: None) If provided, periodic boundary conditions will be assumed, and the values of this parameter it will be used as the ranges in all dimensions. It must be a two-dimensional array, where each row will cointain the lower and upper bound of each dimension. Returns ------- geometric_graph : :class:~graph_tool.Graph The generated graph. pos : :class:~graph_tool.PropertyMap A vertex property map with the position of each vertex. Notes ----- A geometric graph [geometric-graph]_ is generated by connecting points embedded in a N-dimensional euclidean space which are at a distance equal to or smaller than a given radius. See Also -------- triangulation: 2D or 3D triangulation random_graph: random graph generation lattice : N-dimensional square lattice Examples -------- >>> from numpy.random import seed, random >>> seed(42) >>> points = random((500, 2)) * 4 >>> g, pos = gt.geometric_graph(points, 0.3) >>> gt.graph_draw(g, pos=pos, pin=True, size=(8,8), output="geometric.png") <...> >>> g, pos = gt.geometric_graph(points, 0.3, [(0,4), (0,4)]) >>> gt.graph_draw(g, size=(8,8), output="geometric_periodic.png") <...> .. image:: geometric.png .. image:: geometric_periodic.png *Left:* Geometric network with random points. *Right:* Same network, but with periodic boundary conditions. References ---------- .. [geometric-graph] Jesper Dall and Michael Christensen, "Random geometric  Tiago Peixoto committed Dec 21, 2010 810  graphs", Phys. Rev. E 66, 016121 (2002), :doi:10.1103/PhysRevE.66.016121  Tiago Peixoto committed Oct 04, 2010 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833  """ g = Graph(directed=False) pos = g.new_vertex_property("vector") if type(points) != numpy.ndarray: points = numpy.array(points) if len(points.shape) < 2: raise ValueError("points list must be a two-dimensional array!") if ranges is not None: periodic = True if type(ranges) != numpy.ndarray: ranges = numpy.array(ranges, dtype="float") else: ranges = array(ranges, dtype="float") else: periodic = False ranges = () libgraph_tool_generation.geometric(g._Graph__graph, points, float(radius), ranges, periodic, _prop("v", g, pos)) return g, pos  Tiago Peixoto committed Nov 13, 2010 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925  def price_network(N, m=1, c=None, gamma=1, directed=True, seed_graph=None): r"""A generalized version of Price's -- or Barabási-Albert if undirected -- preferential attachment network model. Parameters ---------- N : int Size of the network. m : int (optional, default: 1) Out-degree of newly added vertices. c : float (optional, default: 1 if directed==True else 0) Constant factor added to the probability of a vertex receiving an edge (see notes below). gamma : float (optional, default: 1) Preferential attachment power (see notes below). directed : bool (optional, default: True) If True, a Price network is generated. If False, a Barabási-Albert network is generated. seed_graph : :class:~graph_tool.Graph (optional, default: None) If provided, this graph will be used as the starting point of the algorithm. Returns ------- price_graph : :class:~graph_tool.Graph The generated graph. Notes ----- The (generalized) [price]_ network is either a directed or undirected graph (the latter is called a Barabási-Albert network), generated dynamically by at each step adding a new vertex, and connecting it to :math:m other vertices, chosen with probability: .. math:: P \propto k^\gamma + c where :math:k is the in-degree of the vertex (or simply the degree in the undirected case). If :math:\gamma=1, the tail of resulting in-degree distribution of the directed case is given by .. math:: P_{k_\text{in}} \sim k_\text{in}^{-(2 + c/m)}, or for the undirected case .. math:: P_{k} \sim k^{-(3 + c/m)}. However, if :math:\gamma \ne 1, the in-degree distribution is not scale-free (see [dorogovtsev-evolution]_ for details). This algorithm runs in :math:O(N\log N) time. See Also -------- triangulation: 2D or 3D triangulation random_graph: random graph generation lattice : N-dimensional square lattice geometric_graph : N-dimensional geometric network Examples -------- >>> from numpy.random import seed, random >>> seed(42) >>> g = gt.price_network(100000) >>> gt.graph_draw(g, layout="sfdp", size=(12,12), vcolor=g.vertex_index, ... output="price-network.png") <...> >>> g = gt.price_network(100000, c=0.1) >>> gt.graph_draw(g, layout="sfdp", size=(12,12), vcolor=g.vertex_index, ... output="price-network-broader.png") <...> .. image:: price-network.png .. image:: price-network-broader.png Price networks with :math:N=10^5 nodes. **Left:** :math:c=1, **Right:** :math:c=0.1. The colors represent the order in which vertices were added. References ---------- .. [yule] Yule, G. U. "A Mathematical Theory of Evolution, based on the Conclusions of Dr. J. C. Willis, F.R.S.". Philosophical Transactions of the Royal Society of London, Ser. B 213: 21–87, 1925,  Tiago Peixoto committed Dec 21, 2010 926  :doi:10.1098/rstb.1925.0002  Tiago Peixoto committed Nov 13, 2010 927 928 929  .. [price] Derek De Solla Price, "A general theory of bibliometric and other cumulative advantage processes", Journal of the American Society for Information Science, Volume 27, Issue 5, pages 292–306, September 1976,  Tiago Peixoto committed Dec 21, 2010 930  :doi:10.1002/asi.4630270505  Tiago Peixoto committed Nov 13, 2010 931  .. [barabasi-albert] Barabási, A.-L., and Albert, R., "Emergence of  Tiago Peixoto committed Dec 21, 2010 932 933  scaling in random networks", Science, 286, 509, 1999, :doi:10.1126/science.286.5439.509  Tiago Peixoto committed Nov 13, 2010 934 935  .. [dorogovtsev-evolution] S. N. Dorogovtsev and J. F. F. Mendes, "Evolution of networks", Advances in Physics, 2002, Vol. 51, No. 4, 1079-1187,  Tiago Peixoto committed Dec 21, 2010 936  :doi:10.1080/00018730110112519  Tiago Peixoto committed Nov 13, 2010 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956  """ if c is None: c = 1 if directed else 0 if seed_graph is None: if directed: g = Graph() g.add_vertex(m) else: N_s = m + 1 if m % 2 != 0 else m + 2 g = random_graph(N_s, lambda: 1, directed=False) N -= g.num_vertices() else: g = seed_graph if g.num_vertices() < m: raise ValueError("seed_graph has number of vertices < m!") seed = numpy.random.randint(0, sys.maxint) libgraph_tool_generation.price(g._Graph__graph, N, gamma, c, m, seed) return g