__init__.py 17.6 KB
 Tiago Peixoto committed Jul 15, 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 Jan 11, 2009 19 """  Tiago Peixoto committed Jan 18, 2009 20 21 graph_tool.clustering - Clustering coefficients ---------------------------------------------------  Tiago Peixoto committed Jan 11, 2009 22 23 24 25 26  Provides algorithms for calculation of clustering coefficients, aka. transitivity. """  Tiago Peixoto committed Oct 26, 2008 27 from .. dl_import import dl_import  Tiago Peixoto committed Jan 18, 2009 28 dl_import("import libgraph_tool_clustering as _gt")  Tiago Peixoto committed Jul 15, 2008 29   Tiago Peixoto committed Mar 09, 2009 30 31 from .. core import _degree, _prop, Graph from .. misc import isomorphism  Tiago Peixoto committed Jul 15, 2008 32 from numpy import *  Tiago Peixoto committed Mar 09, 2009 33 import sys  Tiago Peixoto committed Jul 15, 2008 34   Tiago Peixoto committed Mar 09, 2009 35 __all__ = ["local_clustering", "global_clustering", "extended_clustering",  Tiago Peixoto committed Apr 24, 2009 36  "motifs", "motif_significance"]  Tiago Peixoto committed Jul 15, 2008 37   Tiago Peixoto committed Jan 11, 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 def local_clustering(g, prop=None, undirected=False): r""" Return vertex property containing local clustering coefficients for all vertices. Parameters ---------- g : Graph Graph to be used. prop : PropertyMap or string, optional Vertex property map where results will be stored. If specified, this parameter will also be the return value. undirected : bool, optional Calculate the *undirected* clustering coefficient, if graph is directed (this option has no effect if the graph is undirected). Returns ------- prop : PropertyMap Vertex property containing the clustering coefficients. See Also -------- global_clustering: global clustering coefficient extended_clustering: extended (generalized) clustering coefficient  Tiago Peixoto committed Mar 09, 2009 63  motifs: motif counting  Tiago Peixoto committed Jan 11, 2009 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101  Notes ----- The local clustering coefficient [1]_ :math:c_i is defined as .. math:: c_i = \frac{|\{e_{jk}\}|}{k_i(k_i-1)} :\, v_j,v_k \in N_i,\, e_{jk} \in E where :math:k_i is the out-degree of vertex :math:i, and .. math:: N_i = \{v_j : e_{ij} \in E\} is the set of out-neighbours of vertex :math:i. For undirected graphs the value of :math:c_i is normalized as .. math:: c'_i = 2c_i. The implemented algorithm runs in :math:O(|V|\left< k\right>^3) time, where :math:\left< k\right> is the average out-degree. If enabled during compilation, this algorithm runs in parallel. Examples -------- >>> g = gt.random_graph(1000, lambda: (5,5), seed=42) >>> clust = gt.local_clustering(g) >>> print gt.vertex_average(g, clust) (0.0045633333333333333, 0.00041406305209606802) References ---------- .. [1] D. J. Watts and Steven Strogatz, "Collective dynamics of 'small-world' networks", Nature, vol. 393, pp 440-442, 1998. doi:10.1038/30918 """  Tiago Peixoto committed Jul 15, 2008 102 103  if prop == None: prop = g.new_vertex_property("double")  Tiago Peixoto committed Jan 11, 2009 104 105 106 107  was_directed = g.directed() if g.directed() and undirected: g.set_directed(False) try:  Tiago Peixoto committed Jan 18, 2009 108  _gt.extended_clustering(g._Graph__graph,  Tiago Peixoto committed Jan 11, 2009 109 110 111 112  [_prop("v", g, prop)]) finally: if was_directed and undirected: g.set_directed(True)  Tiago Peixoto committed Jul 15, 2008 113 114 115  return prop def global_clustering(g):  Tiago Peixoto committed Jan 11, 2009 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132  r""" Return global clustering coefficients for graphs. Parameters ---------- g : Graph Graph to be used. Returns ------- c : tuple of floats Global clustering coefficient and standard deviation (jacknife method) See Also -------- local_clustering: local clustering coefficient extended_clustering: extended (generalized) clustering coefficient  Tiago Peixoto committed Mar 09, 2009 133  motifs: motif counting  Tiago Peixoto committed Jan 11, 2009 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  Notes ----- The global clustering coefficient [1]_ :math:c is defined as .. math:: c = 3 \times \frac{\text{number of triangles}} {\text{number of connected triples}} The implemented algorithm runs in :math:O(|V|\left< k\right>^3) time, where :math:\left< k\right> is the average (total) degree. If enabled during compilation, this algorithm runs in parallel. Examples -------- >>> g = gt.random_graph(1000, lambda: (5,5), seed=42) >>> print gt.global_clustering(g) (0.0086380072318200073, 0.00044516087903790925) References ---------- .. [1] M. E. J. Newman, "The structure and function of complex networks", SIAM Review, vol. 45, pp. 167-256, 2003 """  Tiago Peixoto committed Jan 18, 2009 160  c =_gt.global_clustering(g._Graph__graph)  Tiago Peixoto committed Jul 15, 2008 161 162  return c  Tiago Peixoto committed Jan 11, 2009 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 def extended_clustering(g, props=None, max_depth=3, undirected=False): r""" Return a list of vertex properties containing the extended clustering coefficients for all vertices. Parameters ---------- g : Graph Graph to be used. props : list of PropertyMap objects, optional list of vertex property maps where results will be stored. If specified, this parameter will also be the return value. max_depth : int, optional Maximum clustering order (default: 3). undirected : bool, optional Calculate the *undirected* clustering coefficients, if graph is directed (this option has no effect if the graph is undirected). Returns ------- prop : list of PropertyMap objects List of vertex properties containing the clustering coefficients. See Also -------- local_clustering: local clustering coefficient global_clustering: global clustering coefficient  Tiago Peixoto committed Mar 09, 2009 190  motifs: motif counting  Tiago Peixoto committed Jan 11, 2009 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238  Notes ----- The definition of the extended clustering coefficient :math:c^d_i of order :math:d is defined as .. math:: c^d_i = \frac{\left|\right\{ \{u,v\}; u,v \in N_i | d_{G(V\diagdown \{i\})}(u,v) = d \left\}\right|}{\binom{\left|N_i\right|}{2}}, where :math:d_G(u,v) is the shortest distance from vertex :math:u to :math:v in graph :math:G, and .. math:: N_i = \{v_j : e_{ij} \in E\} is the set of out-neighbours of :math:i. According to the above definition, we have that the traditional local clustering coefficient is recovered for :math:d=1, i.e., :math:c^1_i = c_i. The implemented algorithm runs in :math:O(|V|\left< k\right>^{2+\text{max_depth}}) worst time, where :math:\left< k\right> is the average out-degree. If enabled during compilation, this algorithm runs in parallel. Examples -------- >>> g = gt.random_graph(1000, lambda: (5,5), seed=42) >>> clusts = gt.extended_clustering(g, max_depth=5) >>> for i in xrange(0, 5): ... print gt.vertex_average(g, clusts[i]) ... (0.0045633333333333333, 0.00041406305209606802) (0.027705, 0.0010493633929938454) (0.11730666666666667, 0.00201118990760307) (0.41394666666666663, 0.0030157036105470745) (0.41717499999999996, 0.0030272310298907366) References ---------- .. [1] A. H. Abdo, A. P. S. de Moura, "Clustering as a measure of the local topology of networks", arXiv:physics/0605235v4 """ was_directed = g.directed() if g.directed() and undirected: g.set_directed(False)  Tiago Peixoto committed Jul 15, 2008 239 240 241 242  if props == None: props = [] for i in xrange(0, max_depth): props.append(g.new_vertex_property("double"))  Tiago Peixoto committed Jan 11, 2009 243  try:  Tiago Peixoto committed Jan 18, 2009 244 245  _gt.extended_clustering(g._Graph__graph, [_prop("v", g, p) for p in props])  Tiago Peixoto committed Jan 11, 2009 246 247 248  finally: if was_directed and undirected: g.set_directed(True)  Tiago Peixoto committed Jul 15, 2008 249  return props  Tiago Peixoto committed Mar 09, 2009 250 251   Tiago Peixoto committed Apr 24, 2009 252 def motifs(g, k, p=1.0, motif_list=None, undirected=None, seed=0):  Tiago Peixoto committed Mar 09, 2009 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268  r""" Count the occurrence of k-size subgraphs (motifs). A tuple with two lists is returned: the list of motifs found, and the list with their respective counts. Parameters ---------- g : Graph Graph to be used. k : int number of vertices of the motifs p : float or float list, optional (default: 1.0) uniform fraction of the motifs to be sampled. If a float list is provided, it will be used as the fraction at each depth :math:[1,\dots,k] in the algorithm. See [wernicke_efficient_2006]_ for more details.  Tiago Peixoto committed Apr 24, 2009 269  motif_list : list of Graph objects, optional  Tiago Peixoto committed Mar 09, 2009 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290  If supplied, the algorithms will only search for the motifs in this list (or isomorphisms thereof) undirected : bool, optional Treat the graph as *undirected*, if graph is directed (this option has no effect if the graph is undirected). seed : int, optional (default: 0) Seed for the random number generator. It the value is 0, a random seed is used. Returns ------- motifs : list of Graph objects List of motifs of size k found in the Graph. Graphs are grouped according to their isomorphism class, and only one of each class appears in this list. The list is sorted according to in-degree sequence, out-degree-sequence, and number of edges (in this order). counts : list of ints The number of times the respective motif in the motifs list was counted See Also --------  Tiago Peixoto committed Apr 24, 2009 291  motif_significance: significance profile of motifs  Tiago Peixoto committed Mar 09, 2009 292 293 294 295 296 297 298 299 300 301 302 303 304 305  local_clustering: local clustering coefficient global_clustering: global clustering coefficient extended_clustering: extended (generalized) clustering coefficient Notes ----- This functions implements the ESU and RAND-ESU algorithms described in [wernicke_efficient_2006]_. If enabled during compilation, this algorithm runs in parallel. Examples -------- >>> g = gt.random_graph(1000, lambda: (5,5), seed=42)  Tiago Peixoto committed Apr 24, 2009 306  >>> motifs, counts = gt.motifs(g, 4, undirected=True))  Tiago Peixoto committed Mar 09, 2009 307  >>> print len(motifs)  Tiago Peixoto committed Apr 24, 2009 308  11  Tiago Peixoto committed Mar 09, 2009 309  >>> print counts  Tiago Peixoto committed Apr 24, 2009 310  [115708, 390659, 612, 696, 2872, 1556, 811, 4, 8, 6, 1]  Tiago Peixoto committed Mar 09, 2009 311 312 313 314 315 316 317 318 319 320 321 322 323 324  References ---------- .. [wernicke_efficient_2006] S. Wernicke, "Efficient detection of network motifs", IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB), Volume 3, Issue 4, Pages 347-359, 2006. """ if seed == 0: seed = random.randint(0, sys.maxint) sub_list = [] directed_motifs = g.is_directed() if undirected == None else not undirected  Tiago Peixoto committed Apr 24, 2009 325 326 327  if motif_list != None: directed_motifs = motif_list[0].is_directed() for m in motif_list:  Tiago Peixoto committed Mar 09, 2009 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375  if m.is_directed() != directed_motifs: raise ValueError("all motif graphs must be either directed or undirected") if m.num_vertices() != k: raise ValueError("all motifs must have the same number of vertices: " + k) sub_list.append(m._Graph__graph) if type(p) == float: pd = [1.0]*(k-1) pd.append(p) if type(p) == list: pd = [float(x) for x in p] hist = [] was_directed = g.is_directed() if g.is_directed() and not directed_motifs: g.set_directed(False) try: _gt.get_motifs(g._Graph__graph, k, sub_list, hist, pd, True, len(sub_list) == 0, seed) finally: if was_directed and not directed_motifs: g.set_directed(True) # assemble graphs temp = [] for m in sub_list: mg = Graph() mg._Graph__graph = m temp.append(mg) sub_list = temp list_hist = zip(sub_list, hist) # sort according to in-degree sequence list_hist.sort(lambda x,y: cmp(sorted([v.in_degree() for v in x[0].vertices()]), sorted([v.in_degree() for v in y[0].vertices()]))) # sort according to out-degree sequence list_hist.sort(lambda x,y: cmp(sorted([v.out_degree() for v in x[0].vertices()]), sorted([v.out_degree() for v in y[0].vertices()]))) # sort according to ascending number of edges list_hist.sort(lambda x,y: cmp(x[0].num_edges(), y[0].num_edges())) sub_list = [x[0] for x in list_hist] hist = [x[1] for x in list_hist] return sub_list, hist  Tiago Peixoto committed Apr 24, 2009 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510  def motif_significance(g, k, n_shuffles=10, p=1.0, motif_list=None, undirected=None, self_loops=False, parallel_edges=False, full_output=False, seed=0): r""" Obtain the motif significance profile, for subgraphs with k vertices. A tuple with two lists is returned: the list of motifs found, and their respective z-scores. Parameters ---------- g : Graph Graph to be used. k : int number of vertices of the motifs n_shuffles : int, optional (default: 10) number of shuffled networks to consider for the z-score p : float or float list, optional (default: 1.0) uniform fraction of the motifs to be sampled. If a float list is provided, it will be used as the fraction at each depth :math:[1,\dots,k] in the algorithm. See [wernicke_efficient_2006]_ for more details. motif_list : list of Graph objects, optional If supplied, the algorithms will only search for the motifs in this list (or isomorphisms thereof) undirected : bool, optional Treat the graph as *undirected*, if graph is directed (this option has no effect if the graph is undirected). self_loops : bool, optional (default: False) Whether or not the shuffled graphs are allowed to contain self-loops parallel_edges : bool, optional (default: False) Whether or not the shuffled graphs are allowed to contain parallel edges. full_output : bool, optional (default: False) If set to True, three additional lists are returned: the count of each motif, the average count of each motif in the shuffled networks, and the standard deviation of the average count of each motif in the shuffled networks. seed : int, optional (default: 0) Seed for the random number generator. It the value is 0, a random seed is used. Returns ------- motifs : list of Graph objects List of motifs of size k found in the Graph. Graphs are grouped according to their isomorphism class, and only one of each class appears in this list. The list is sorted according to in-degree sequence, out-degree-sequence, and number of edges (in this order). z-scores : list of floats The z-score of the respective motives. See below for the definition of a z-score. See Also -------- motifs: motif counting or sampling local_clustering: local clustering coefficient global_clustering: global clustering coefficient extended_clustering: extended (generalized) clustering coefficient Notes ----- The z-score :math:z_i of motif i is defined as .. math:: z_i = \frac{N_i - \left} {\sqrt{\left<(N^s_i)^2\right> - \left^2}}, where :math:N_i is the number of times motif $i$ found, and :math:N^s_i is the count of the same motif but on a shuffled network. It measures how many standard deviations is each motif count, in respect to a ensemble of randomly shuffled graphs with the same degree sequence. The z-scores values are not normalized. If enabled during compilation, this algorithm runs in parallel. Examples -------- >>> g = gt.random_graph(1000, lambda: (5,5), seed=42) >>> motifs, zscores = gt.motif_significance(g, 3) >>> print len(motifs) 11 >>> print zscores [0.23425857453240315, 0.23849227914686885, 0.46705666396159251, 0.26171196129510765, -0.28131244310816039, -0.29007872608538582, -0.56694670951384085, -0.5, -0.33333333333333337, -0.46852128566581813, -0.5] """ from itertools import izip from .. misc import random_rewire, isomorphism s_ms, counts = motifs(g, k, p, motif_list, undirected, seed) s_counts = [0]*len(s_ms) s_dev = [0]*len(s_ms) # get samples sg = g.copy() for i in xrange(0, n_shuffles): random_rewire(sg, self_loops=self_loops, parallel_edges=parallel_edges) m_temp, count_temp = motifs(sg, k, p, motif_list, undirected, seed) for j in xrange(0, len(m_temp)): found = False for l in xrange(0, len(s_ms)): if isomorphism(s_ms[l], m_temp[j]): found = True s_counts[l] += count_temp[j] s_dev[l] += count_temp[j]**2 if not found: s_ms.append(m_temp[j]) s_counts.append(count_temp[j]) s_dev.append(count_temp[j]**2) counts.append(0) s_counts = [ x/float(n_shuffles) for x in s_counts ] s_dev = [ sqrt(x[0]/float(n_shuffles) - x[1]**2) \ for x in izip(s_dev,s_counts) ] list_hist = zip(s_ms, s_counts, s_dev) # sort according to in-degree sequence list_hist.sort(lambda x,y: cmp(sorted([v.in_degree() for v in x[0].vertices()]), sorted([v.in_degree() for v in y[0].vertices()]))) # sort according to out-degree sequence list_hist.sort(lambda x,y: cmp(sorted([v.out_degree() for v in x[0].vertices()]), sorted([v.out_degree() for v in y[0].vertices()]))) # sort according to ascending number of edges list_hist.sort(lambda x,y: cmp(x[0].num_edges(), y[0].num_edges())) s_ms, s_counts, s_dev = zip(*list_hist) zscore = [(x[0] - x[1])/x[2] for x in izip(counts, s_counts, s_dev)] if full_output: return s_ms, zscore, counts, s_counts, s_dev else: return s_ms, zscore