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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 """  Tiago Peixoto committed Oct 05, 2009 20 graph_tool.generation - Random graph generation  Tiago Peixoto committed Jul 15, 2009 21 ---------------------------------------------------  Tiago Peixoto committed Oct 05, 2009 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36  Summary +++++++ .. autosummary:: :nosignatures: random_graph random_rewire predecessor_tree line_graph graph_union Contents ++++++++  Tiago Peixoto committed Jul 15, 2009 37 38 """  Tiago Peixoto committed Oct 26, 2008 39 40 from .. dl_import import dl_import dl_import("import libgraph_tool_generation")  Tiago Peixoto committed Apr 10, 2008 41   Tiago Peixoto committed Aug 17, 2009 42 from .. core import Graph, _check_prop_scalar, _prop  Tiago Peixoto committed Oct 05, 2009 43 import sys, numpy, numpy.random  Tiago Peixoto committed Apr 10, 2008 44   Tiago Peixoto committed Sep 06, 2009 45 46 __all__ = ["random_graph", "random_rewire", "predecessor_tree", "line_graph", "graph_union"]  Tiago Peixoto committed Apr 10, 2008 47   Tiago Peixoto committed Jul 15, 2009 48 49 50 def _corr_wrap(i, j, corr): return corr(i[1], j[1])  Tiago Peixoto committed Apr 10, 2008 51 def random_graph(N, deg_sampler, deg_corr=None, directed=True,  Tiago Peixoto committed Oct 05, 2009 52  parallel=False, self_loops=False, verbose=False):  Tiago Peixoto committed Aug 02, 2009 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  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. Returns -------  Tiago Peixoto committed Oct 05, 2009 82  random_graph : :class:~graph_tool.Graph  Tiago Peixoto committed Aug 02, 2009 83 84 85 86 87 88 89 90 91 92 93 94 95 96  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 97 98  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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128  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 129  (0.59472179721535989, 0.011919463022240388)  Tiago Peixoto committed Aug 02, 2009 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  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 157  >>> xlabel("in-degree")  Tiago Peixoto committed Aug 02, 2009 158  <...>  Tiago Peixoto committed Oct 05, 2009 159  >>> ylabel("out-degree")  Tiago Peixoto committed Aug 02, 2009 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  <...> >>> 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")  Tiago Peixoto committed Oct 05, 2009 189 190  >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", ... label=r"$\left<\text{in}\right>$ vs in")  Tiago Peixoto committed Aug 02, 2009 191 192  (...) >>> corr = gt.avg_neighbour_corr(g, "in", "out")  Tiago Peixoto committed Oct 05, 2009 193 194  >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", ... label=r"$\left<\text{out}\right>$ vs in")  Tiago Peixoto committed Aug 02, 2009 195 196  (...) >>> corr = gt.avg_neighbour_corr(g, "out", "in")  Tiago Peixoto committed Oct 05, 2009 197 198  >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", ... label=r"$\left<\text{in}\right>$ vs out")  Tiago Peixoto committed Aug 02, 2009 199 200  (...) >>> corr = gt.avg_neighbour_corr(g, "out", "out")  Tiago Peixoto committed Oct 05, 2009 201 202  >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", ... label=r"$\left<\text{out}\right>$ vs out")  Tiago Peixoto committed Aug 02, 2009 203 204 205 206 207 208 209 210 211 212 213 214 215 216  (...) >>> 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 Oct 05, 2009 217  seed = numpy.random.randint(0, sys.maxint)  Tiago Peixoto committed Apr 10, 2008 218 219 220 221 222  g = Graph() if deg_corr == None: uncorrelated = True else: uncorrelated = False  Tiago Peixoto committed Jul 15, 2009 223 224 225 226  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 227  libgraph_tool_generation.gen_random_graph(g._Graph__graph, N,  Tiago Peixoto committed Jul 15, 2009 228  deg_sampler, corr,  Tiago Peixoto committed Apr 10, 2008 229 230 231 232 233  uncorrelated, not parallel, not self_loops, not directed, seed, verbose) g.set_directed(directed) return g  Tiago Peixoto committed Aug 04, 2009 234   Tiago Peixoto committed Aug 07, 2009 235 def random_rewire(g, strat= "uncorrelated", parallel_edges = False,  Tiago Peixoto committed Oct 05, 2009 236  self_loops = False):  Tiago Peixoto committed Aug 07, 2009 237  r"""  Tiago Peixoto committed Oct 05, 2009 238  Shuffle the graph in-place. The degrees (either in or out) of each vertex  Tiago Peixoto committed Aug 07, 2009 239 240 241 242 243 244  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 ----------  Tiago Peixoto committed Oct 05, 2009 245  g : :class:~graph_tool.Graph  Tiago Peixoto committed Aug 07, 2009 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  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. 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)  Tiago Peixoto committed Oct 05, 2009 280  >>> gt.graph_draw(g, layout="arf", output="rewire_orig.png", size=(6,6))  Tiago Peixoto committed Sep 03, 2009 281  <...>  Tiago Peixoto committed Aug 07, 2009 282  >>> gt.random_rewire(g, "correlated")  Tiago Peixoto committed Oct 05, 2009 283  >>> gt.graph_draw(g, layout="arf", output="rewire_corr.png", size=(6,6))  Tiago Peixoto committed Sep 03, 2009 284  <...>  Tiago Peixoto committed Aug 07, 2009 285  >>> gt.random_rewire(g)  Tiago Peixoto committed Oct 05, 2009 286  >>> gt.graph_draw(g, layout="arf", output="rewire_uncorr.png", size=(6,6))  Tiago Peixoto committed Sep 03, 2009 287  <...>  Tiago Peixoto committed Aug 07, 2009 288   Tiago Peixoto committed Oct 05, 2009 289  Some ridiculograms _ :  Tiago Peixoto committed Aug 07, 2009 290   Tiago Peixoto committed Oct 05, 2009 291 292 293  .. image:: rewire_orig.png .. image:: rewire_corr.png .. image:: rewire_uncorr.png  Tiago Peixoto committed Aug 07, 2009 294   Tiago Peixoto committed Oct 05, 2009 295 296  *Left:* Original graph; *Middle:* Shuffled graph, with degree correlations; *Right:* Shuffled graph, without degree correlations.  Tiago Peixoto committed Aug 07, 2009 297 298 299 300 301 302 303  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")  Tiago Peixoto committed Oct 05, 2009 304  >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", label="original")  Tiago Peixoto committed Aug 07, 2009 305 306 307  (...) >>> gt.random_rewire(g, "correlated") >>> corr = gt.avg_neighbour_corr(g, "out", "out")  Tiago Peixoto committed Oct 05, 2009 308  >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", label="correlated")  Tiago Peixoto committed Aug 07, 2009 309 310 311  (...) >>> gt.random_rewire(g) >>> corr = gt.avg_neighbour_corr(g, "out", "out")  Tiago Peixoto committed Oct 05, 2009 312  >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", label="uncorrelated")  Tiago Peixoto committed Aug 07, 2009 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  (...) >>> 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")  Tiago Peixoto committed Oct 05, 2009 338 339  >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", ... label=r"$\left<\text{o}\right>$ vs i")  Tiago Peixoto committed Aug 07, 2009 340 341  (...) >>> corr = gt.avg_neighbour_corr(g, "out", "in")  Tiago Peixoto committed Oct 05, 2009 342 343  >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-", ... label=r"$\left<\text{i}\right>$ vs o")  Tiago Peixoto committed Aug 07, 2009 344 345 346 347  (...) >>> gt.random_rewire(g, "correlated") >>> corr = gt.avg_neighbour_corr(g, "in", "out") >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",  Tiago Peixoto committed Oct 05, 2009 348  ... label=r"$\left<\text{o}\right>$ vs i, corr.")  Tiago Peixoto committed Aug 07, 2009 349 350 351  (...) >>> corr = gt.avg_neighbour_corr(g, "out", "in") >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",  Tiago Peixoto committed Oct 05, 2009 352  ... label=r"$\left<\text{i}\right>$ vs o, corr.")  Tiago Peixoto committed Aug 07, 2009 353 354 355 356  (...) >>> gt.random_rewire(g, "uncorrelated") >>> corr = gt.avg_neighbour_corr(g, "in", "out") >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",  Tiago Peixoto committed Oct 05, 2009 357  ... label=r"$\left<\text{o}\right>$ vs i, uncorr.")  Tiago Peixoto committed Aug 07, 2009 358 359 360  (...) >>> corr = gt.avg_neighbour_corr(g, "out", "in") >>> errorbar(corr[2], corr[0], yerr=corr[1], fmt="o-",  Tiago Peixoto committed Oct 05, 2009 361  ... label=r"$\left<\text{i}\right>$ vs o, uncorr.")  Tiago Peixoto committed Aug 07, 2009 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378  (...) >>> 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 Oct 05, 2009 379  seed = numpy.random.randint(0, sys.maxint)  Tiago Peixoto committed Aug 07, 2009 380 381  g.stash_filter(reversed=True)  Tiago Peixoto committed Aug 04, 2009 382 383  libgraph_tool_generation.random_rewire(g._Graph__graph, strat, self_loops, parallel_edges, seed)  Tiago Peixoto committed Aug 07, 2009 384  g.pop_filter(reversed=True)  Tiago Peixoto committed Aug 17, 2009 385 386 387 388 389 390 391 392 393 394 395  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 396 397  def line_graph(g):  Tiago Peixoto committed Oct 05, 2009 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417  """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 418 419 420 421 422 423 424 425  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 426 427  def graph_union(g1, g2, props=[], include=False):  Tiago Peixoto committed Oct 05, 2009 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453  """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 Sep 06, 2009 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491  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