inference.rst 54.3 KB
 Tiago Peixoto committed Jul 06, 2016 1 2 .. _inference-howto:  Tiago Peixoto committed Jul 06, 2016 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 Inferring network structure =========================== graph-tool includes algorithms to identify the large-scale structure of networks in the :mod:~graph_tool.inference submodule. Here we explain the basic functionality with self-contained examples. Background: Nonparametric statistical inference ----------------------------------------------- A common task when analyzing networks is to characterize their structures in simple terms, often by dividing the nodes into modules or "communities". A principled approach to perform this task is to formulate generative models _ that include the idea of "modules" in their descriptions, which then can be detected by inferring _ the model parameters from data. More precisely, given the partition :math:\boldsymbol b = \{b_i\} of the network into :math:B groups, where :math:b_i\in[0,B-1] is the group membership of node :math:i,  Tiago Peixoto committed Oct 14, 2016 24 25 we define a model that generates a network :math:\boldsymbol G with a probability  Tiago Peixoto committed Jul 06, 2016 26 27 28 29  .. math:: :label: model-likelihood  Tiago Peixoto committed Oct 14, 2016 30  P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)  Tiago Peixoto committed Jul 06, 2016 31   Tiago Peixoto committed Oct 14, 2016 32 33 where :math:\boldsymbol\theta are additional model parameters. Therefore, if we observe a network :math:\boldsymbol G, the likelihood that it was generated by a  Tiago Peixoto committed Jul 06, 2016 34 35 36 37 38 39 given partition :math:\boldsymbol b is obtained via the Bayesian _ posterior .. math:: :label: model-posterior-sum  Tiago Peixoto committed Oct 14, 2016 40  P(\boldsymbol b | \boldsymbol G) = \frac{\sum_{\boldsymbol\theta}P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)P(\boldsymbol\theta, \boldsymbol b)}{P(\boldsymbol G)}  Tiago Peixoto committed Jul 06, 2016 41   Tiago Peixoto committed Oct 14, 2016 42 where :math:P(\boldsymbol\theta, \boldsymbol b) is the prior likelihood of the  Tiago Peixoto committed Jul 06, 2016 43 44 45 46 47 model parameters, and .. math:: :label: model-evidence  Tiago Peixoto committed Oct 14, 2016 48  P(\boldsymbol G) = \sum_{\boldsymbol\theta,\boldsymbol b}P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)P(\boldsymbol\theta, \boldsymbol b)  Tiago Peixoto committed Jul 06, 2016 49 50 51  is called the model evidence. The particular types of model that will be considered here have "hard constraints", such that there is only one  Tiago Peixoto committed Oct 14, 2016 52 choice for the remaining parameters :math:\boldsymbol\theta that is compatible  Tiago Peixoto committed Jul 06, 2016 53 54 55 56 57 with the generated network, such that Eq. :eq:model-posterior-sum simplifies to .. math:: :label: model-posterior  Tiago Peixoto committed Oct 14, 2016 58  P(\boldsymbol b | \boldsymbol G) = \frac{P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)P(\boldsymbol\theta, \boldsymbol b)}{P(\boldsymbol G)}  Tiago Peixoto committed Jul 06, 2016 59   Tiago Peixoto committed Oct 14, 2016 60 61 with :math:\boldsymbol\theta above being the only choice compatible with :math:\boldsymbol G and :math:\boldsymbol b. The inference procedures considered  Tiago Peixoto committed Jul 06, 2016 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 here will consist in either finding a network partition that maximizes Eq. :eq:model-posterior, or sampling different partitions according its posterior probability. As we will show below, this approach will also enable the comparison of different models according to statistical evidence (a.k.a. model selection). Minimum description length (MDL) ++++++++++++++++++++++++++++++++ We note that Eq. :eq:model-posterior can be written as .. math::  Tiago Peixoto committed Oct 14, 2016 77  P(\boldsymbol b | \boldsymbol G) = \frac{\exp(-\Sigma)}{P(\boldsymbol G)}  Tiago Peixoto committed Jul 06, 2016 78 79 80 81 82 83  where .. math:: :label: model-dl  Tiago Peixoto committed Oct 14, 2016 84  \Sigma = -\ln P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b) - \ln P(\boldsymbol\theta, \boldsymbol b)  Tiago Peixoto committed Jul 06, 2016 85   Tiago Peixoto committed Oct 14, 2016 86 is called the **description length** of the network :math:\boldsymbol G. It  Tiago Peixoto committed Jul 06, 2016 87 88 89 90 91 measures the amount of information _ required to describe the data, if we encode _ it using the particular parametrization of the generative model given by  Tiago Peixoto committed Oct 14, 2016 92 :math:\boldsymbol\theta and :math:\boldsymbol b, as well as the parameters  Tiago Peixoto committed Jul 06, 2016 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 themselves. Therefore, if we choose to maximize the posterior likelihood of Eq. :eq:model-posterior it will be fully equivalent to the so-called minimum description length _ method. This approach corresponds to an implementation of Occam's razor _, where the simplest model is selected, among all possibilities with the same explanatory power. The selection is based on the statistical evidence available, and therefore will not overfit _, i.e. mistake stochastic fluctuations for actual structure. The stochastic block model (SBM) -------------------------------- The stochastic block model _ is arguably the simplest generative process based on the notion of groups of nodes [holland-stochastic-1983]_. The microcanonical _ formulation  Tiago Peixoto committed Oct 14, 2016 113 [peixoto-nonparametric-2016]_ of the basic or "traditional" version takes  Tiago Peixoto committed Jul 06, 2016 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 as parameters the partition of the nodes into groups :math:\boldsymbol b and a :math:B\times B matrix of edge counts :math:\boldsymbol e, where :math:e_{rs} is the number of edges between groups :math:r and :math:s. Given these constraints, the edges are then placed randomly. Hence, nodes that belong to the same group possess the same probability of being connected with other nodes of the network. An example of a possible parametrization is given in the following figure. .. testcode:: sbm-example :hide: import os try: os.chdir("demos/inference") except FileNotFoundError: pass g = gt.load_graph("blockmodel-example.gt.gz") gt.graph_draw(g, pos=g.vp.pos, vertex_size=10, vertex_fill_color=g.vp.bo, vertex_color="#333333", edge_gradient=g.new_ep("vector", val=[0]), output="sbm-example.svg") ers = g.gp.w from pylab import * figure() matshow(log(ers)) xlabel("Group $r$") ylabel("Group $s$") gca().xaxis.set_label_position("top") savefig("sbm-example-ers.svg") .. table:: :class: figure +----------------------------------+------------------------------+ |.. figure:: sbm-example-ers.svg |.. figure:: sbm-example.svg | | :width: 300px | :width: 300px | | :align: center | :align: center | | | | | Matrix of edge counts | Generated network. | | :math:\boldsymbol e between | | | groups. | | +----------------------------------+------------------------------+ .. note:: We emphasize that no constraints are imposed on what kind of modular structure is allowed. Hence, we can detect the putatively typical pattern of "community structure" _, i.e. when nodes are connected mostly to other nodes of the same group, if it happens to be the most likely network description, but we can also detect a large multiplicity of other patterns, such as bipartiteness _, core-periphery, and many others, all under the same inference framework. Although quite general, the traditional model assumes that the edges are placed randomly inside each group, and as such the nodes that belong to the same group have very similar degrees. As it turns out, this is often a poor model for many networks, which possess highly heterogeneous degree distributions. A better model for such networks is called the degree-corrected stochastic block model [karrer-stochastic-2011]_, and it is defined just like the traditional model, with the addition of the degree sequence :math:\boldsymbol k = \{k_i\} of the graph as an additional set of parameters (assuming again a microcanonical  Tiago Peixoto committed Oct 14, 2016 185 formulation [peixoto-nonparametric-2016]_).  Tiago Peixoto committed Jul 06, 2016 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308  The nested stochastic block model +++++++++++++++++++++++++++++++++ The regular SBM has a drawback when applied to very large networks. Namely, it cannot be used to find relatively small groups in very large networks: The maximum number of groups that can be found scales as :math:B_{\text{max}}\sim\sqrt{N}, where :math:N is the number of nodes in the network, if Bayesian inference is performed [peixoto-parsimonious-2013]_. In order to circumvent this, we need to replace the noninformative priors used by a hierarchy of priors and hyperpriors, which amounts to a nested SBM, where the groups themselves are clustered into groups, and the matrix :math:e of edge counts are generated from another SBM, and so on recursively [peixoto-hierarchical-2014]_. .. figure:: nested-diagram.* :width: 400px :align: center Example of a nested SBM with three levels. In addition to being able to find small groups in large networks, this model also provides a multilevel hierarchical description of the network, that describes its structure at multiple scales. Inferring the best partition ---------------------------- The simplest and most efficient approach is to find the best partition of the network by maximizing Eq. :eq:model-posterior according to some version of the model. This is obtained via the functions :func:~graph_tool.inference.minimize_blockmodel_dl or :func:~graph_tool.inference.minimize_nested_blockmodel_dl, which employs an agglomerative multilevel Markov chain Monte Carlo (MCMC) _ algorithm [peixoto-efficient-2014]_. We focus first on the non-nested model, and we illustrate its use with a network of American football teams, which we load from the :mod:~graph_tool.collection module: .. testsetup:: football import os try: os.chdir("demos/inference") except FileNotFoundError: pass .. testcode:: football g = gt.collection.data["football"] print(g) which yields .. testoutput:: football we then fit the traditional model by calling .. testcode:: football state = gt.minimize_blockmodel_dl(g, deg_corr=False) This returns a :class:~graph_tool.inference.BlockState object that includes the inference results. .. note:: The inference algorithm used is stochastic by nature, and may return a slightly different answer each time it is run. This may be due to the fact that there are alternative partitions with similar likelihoods, or that the optimum is difficult to find. Note that the inference problem here is, in general, NP-Hard _, hence there is no efficient algorithm that is guaranteed to always find the best answer. Because of this, typically one would call the algorithm many times, and select the partition with the largest posterior likelihood of Eq. :eq:model-posterior, or equivalently, the minimum description length of Eq. :eq:model-dl. The description length of a fit can be obtained with the :meth:~graph_tool.inference.BlockState.entropy method. See also :ref:sec_model_selection below. We may perform a drawing of the partition obtained via the :mod:~graph_tool.inference.BlockState.draw method, that functions as a convenience wrapper to the :func:~graph_tool.draw.graph_draw function .. testcode:: football state.draw(pos=g.vp.pos, output="football-sbm-fit.svg") which yields the following image. .. figure:: football-sbm-fit.* :align: center :width: 400px Stochastic block model inference of a network of American college football teams. The colors correspond to inferred group membership of the nodes. We can obtain the group memberships as a :class:~graph_tool.PropertyMap on the vertices via the :mod:~graph_tool.inference.BlockState.get_blocks method: .. testcode:: football b = state.get_blocks() r = b[10] # group membership of vertex 10 print(r) which yields: .. testoutput:: football  Tiago Peixoto committed Sep 22, 2016 309  10  Tiago Peixoto committed Jul 06, 2016 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396  We may also access the matrix of edge counts between groups via :mod:~graph_tool.inference.BlockState.get_matrix .. testcode:: football e = state.get_matrix() matshow(e.todense()) savefig("football-edge-counts.svg") .. figure:: football-edge-counts.* :align: center Matrix of edge counts between groups. We may obtain the same matrix of edge counts as a graph, which has internal edge and vertex property maps with the edge and vertex counts, respectively: .. testcode:: football bg = state.get_bg() ers = bg.ep.count # edge counts nr = bg.vp.count # node counts .. _sec_model_selection: Hierarchical partitions +++++++++++++++++++++++ The inference of the nested family of SBMs is done in a similar manner, but we must use instead the :func:~graph_tool.inference.minimize_nested_blockmodel_dl function. We illustrate its use with the neural network of the C. elegans _ worm: .. testcode:: celegans g = gt.collection.data["celegansneural"] print(g) which has 297 vertices and 2359 edges. .. testoutput:: celegans A hierarchical fit of the degree-corrected model is performed as follows. .. testcode:: celegans state = gt.minimize_nested_blockmodel_dl(g, deg_corr=True) The object returned is an instance of a :class:~graph_tool.inference.NestedBlockState class, which encapsulates the results. We can again draw the resulting hierarchical clustering using the :meth:~graph_tool.inference.NestedBlockState.draw method: .. testcode:: celegans state.draw(output="celegans-hsbm-fit.svg") .. figure:: celegans-hsbm-fit.* :align: center Most likely hierarchical partition of the neural network of the C. elegans worm according to the nested degree-corrected SBM. .. note:: If the output parameter to :meth:~graph_tool.inference.NestedBlockState.draw is omitted, an interactive visualization is performed, where the user can re-order the hierarchy nodes using the mouse and pressing the r key. A summary of the inferred hierarchy can be obtained with the :meth:~graph_tool.inference.NestedBlockState.print_summary method, which shows the number of nodes and groups in all levels: .. testcode:: celegans state.print_summary() .. testoutput:: celegans  Tiago Peixoto committed Sep 22, 2016 397 398 399 400  l: 0, N: 297, B: 13 l: 1, N: 13, B: 5 l: 2, N: 5, B: 2 l: 3, N: 2, B: 1  Tiago Peixoto committed Jul 06, 2016 401 402  The hierarchical levels themselves are represented by individual  Tiago Peixoto committed Jul 20, 2016 403 :meth:~graph_tool.inference.BlockState instances obtained via the  Tiago Peixoto committed Jul 06, 2016 404 405 406 407 408 409 410 411 412 413 :meth:~graph_tool.inference.NestedBlockState.get_levels() method: .. testcode:: celegans levels = state.get_levels() for s in levels: print(s) .. testoutput:: celegans  Tiago Peixoto committed Sep 22, 2016 414 415 416 417  , at 0x...> , at 0x...> , at 0x...> , at 0x...>  Tiago Peixoto committed Jul 06, 2016 418 419 420 421 422  This means that we can inspect the hierarchical partition just as before: .. testcode:: celegans  Tiago Peixoto committed Sep 22, 2016 423  r = levels[0].get_blocks()[46] # group membership of node 46 in level 0  Tiago Peixoto committed Jul 06, 2016 424  print(r)  Tiago Peixoto committed Sep 22, 2016 425  r = levels[0].get_blocks()[r] # group membership of node 46 in level 1  Tiago Peixoto committed Jul 06, 2016 426  print(r)  Tiago Peixoto committed Sep 22, 2016 427  r = levels[0].get_blocks()[r] # group membership of node 46 in level 2  Tiago Peixoto committed Jul 06, 2016 428 429 430 431  print(r) .. testoutput:: celegans  Tiago Peixoto committed Jul 18, 2016 432 433  2 1  Tiago Peixoto committed Sep 22, 2016 434  0  Tiago Peixoto committed Jul 06, 2016 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456  Model selection +++++++++++++++ As mentioned above, one can select the best model according to the choice that yields the smallest description length. For instance, in case of the C. elegans network we have .. testcode:: model-selection g = gt.collection.data["celegansneural"] state_ndc = gt.minimize_nested_blockmodel_dl(g, deg_corr=False) state_dc = gt.minimize_nested_blockmodel_dl(g, deg_corr=True) print("Non-degree-corrected DL:\t", state_ndc.entropy()) print("Degree-corrected DL:\t", state_dc.entropy()) .. testoutput:: model-selection :options: +NORMALIZE_WHITESPACE  Tiago Peixoto committed Sep 22, 2016 457 458  Non-degree-corrected DL: 8568.61212614 Degree-corrected DL: 8246.48662192  Tiago Peixoto committed Jul 06, 2016 459 460 461  Since it yields the smallest description length, the degree-corrected fit should be preferred. The statistical significance of the choice can  Tiago Peixoto committed Oct 14, 2016 462 463 be accessed by inspecting the posterior odds ratio [peixoto-nonparametric-2016]_  Tiago Peixoto committed Jul 06, 2016 464 465 466  .. math::  Tiago Peixoto committed Oct 14, 2016 467 468  \Lambda &= \frac{P(\boldsymbol b, \mathcal{H}_\text{NDC} | \boldsymbol G)}{P(\boldsymbol b, \mathcal{H}_\text{DC} | \boldsymbol G)} \\ &= \frac{P(\boldsymbol G, \boldsymbol b | \mathcal{H}_\text{NDC})}{P(\boldsymbol G, \boldsymbol b | \mathcal{H}_\text{DC})}\times\frac{P(\mathcal{H}_\text{NDC})}{P(\mathcal{H}_\text{DC})} \\  Tiago Peixoto committed Jul 06, 2016 469 470 471 472  &= \exp(-\Delta\Sigma) where :math:\mathcal{H}_\text{NDC} and :math:\mathcal{H}_\text{DC} correspond to the non-degree-corrected and degree-corrected model  Tiago Peixoto committed Sep 22, 2016 473 474 475 hypotheses (assumed to be equally likely a priori), respectively, and :math:\Delta\Sigma is the difference of the description length of both fits. In our particular case, we have  Tiago Peixoto committed Jul 06, 2016 476 477 478 479 480 481 482 483  .. testcode:: model-selection print("ln Λ: ", state_dc.entropy() - state_ndc.entropy()) .. testoutput:: model-selection :options: +NORMALIZE_WHITESPACE  Tiago Peixoto committed Sep 22, 2016 484  ln Λ: -322.125504215  Tiago Peixoto committed Jul 06, 2016 485 486 487 488  The precise threshold that should be used to decide when to reject a hypothesis _ is subjective and context-dependent, but the value above implies that the  Tiago Peixoto committed Sep 22, 2016 489 particular degree-corrected fit is around :math:e^{322} \sim 10^{140}  Tiago Peixoto committed Jul 06, 2016 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 times more likely than the non-degree corrected one, and hence it can be safely concluded that it provides a substantially better fit. Although it is often true that the degree-corrected model provides a better fit for many empirical networks, there are also exceptions. For example, for the American football network above, we have: .. testcode:: model-selection g = gt.collection.data["football"] state_ndc = gt.minimize_nested_blockmodel_dl(g, deg_corr=False) state_dc = gt.minimize_nested_blockmodel_dl(g, deg_corr=True) print("Non-degree-corrected DL:\t", state_ndc.entropy()) print("Degree-corrected DL:\t", state_dc.entropy()) print("ln Λ:\t\t\t", state_ndc.entropy() - state_dc.entropy()) .. testoutput:: model-selection :options: +NORMALIZE_WHITESPACE  Tiago Peixoto committed Sep 22, 2016 511 512 513  Non-degree-corrected DL: 1757.84382615 Degree-corrected DL: 1787.60777164 ln Λ: -29.7639454931  Tiago Peixoto committed Jul 06, 2016 514   Tiago Peixoto committed Sep 22, 2016 515 516 Hence, with a posterior odds ratio of :math:\Lambda \sim e^{-29} \sim 10^{-13} in favor of the non-degree-corrected model, it seems like the  Tiago Peixoto committed Jul 06, 2016 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 degree-corrected variant is an unnecessarily complex description for this network. Averaging over models --------------------- When analyzing empirical networks, one should be open to the possibility that there will be more than one fit of the SBM with similar posterior likelihoods. In such situations, one should instead sample partitions from the posterior likelihood, instead of simply finding its maximum. One can then compute quantities that are averaged over the different model fits, weighted according to their posterior likelihoods. Full support for model averaging is implemented in graph-tool via an efficient Markov chain Monte Carlo (MCMC) _ algorithm [peixoto-efficient-2014]_. It works by attempting to move nodes into different groups with specific probabilities, and accepting or rejecting _ such moves such that, after a sufficiently long time, the partitions will be observed with the desired posterior probability. The algorithm is so designed, that its run-time is independent on the number of groups being used in the model, and hence is suitable for use on very large networks. In order to perform such moves, one needs again to operate with :class:~graph_tool.inference.BlockState or :class:~graph_tool.inference.NestedBlockState instances, and calling their :meth:~graph_tool.inference.BlockState.mcmc_sweep methods. For example, the following will perform 1000 sweeps of the algorithm with the network of characters in the novel Les Misérables, starting from a random partition into 20 groups .. testcode:: model-averaging g = gt.collection.data["lesmis"] state = gt.BlockState(g, B=20) # This automatically initializes the state # with a random partition into B=20 # nonempty groups; The user could # also pass an arbitrary initial # partition using the 'b' parameter. # If we work with the above state object, we will be restricted to # partitions into at most B=20 groups. But since we want to consider # an arbitrary number of groups in the range [1, N], we transform it # into a state with B=N groups (where N-20 will be empty). state = state.copy(B=g.num_vertices()) # Now we run 1,000 sweeps of the MCMC dS, nmoves = state.mcmc_sweep(niter=1000) print("Change in description length:", dS) print("Number of accepted vertex moves:", nmoves) .. testoutput:: model-averaging  Tiago Peixoto committed Sep 22, 2016 577  Change in description length: -360.18357903823386  Tiago Peixoto committed Jul 27, 2016 578  Number of accepted vertex moves: 4743  Tiago Peixoto committed Jul 06, 2016 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600  .. note:: Starting from a random partition is rarely the best option, since it may take a long time for it to equilibrate; It was done above simply as an illustration on how to initialize :class:~graph_tool.inference.BlockState by hand. Instead, a much better option in practice is to start from the "ground state" obtained with :func:~graph_tool.inference.minimize_blockmodel_dl, e.g. .. testcode:: model-averaging state = gt.minimize_blockmodel_dl(g) state = state.copy(B=g.num_vertices()) dS, nmoves = state.mcmc_sweep(niter=1000) print("Change in description length:", dS) print("Number of accepted vertex moves:", nmoves) .. testoutput:: model-averaging  Tiago Peixoto committed Sep 22, 2016 601 602  Change in description length: 0.23920882820149814 Number of accepted vertex moves: 4016  Tiago Peixoto committed Jul 06, 2016 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622  Although the above is sufficient to implement model averaging, there is a convenience function called :func:~graph_tool.inference.mcmc_equilibrate that is intend to simplify the detection of equilibration, by keeping track of the maximum and minimum values of description length encountered and how many sweeps have been made without a "record breaking" event. For example, .. testcode:: model-averaging # We will accept equilibration if 10 sweeps are completed without a # record breaking event, 2 consecutive times. gt.mcmc_equilibrate(state, wait=10, nbreaks=2, mcmc_args=dict(niter=10), verbose=True) will output: .. testoutput:: model-averaging :options: +NORMALIZE_WHITESPACE  Tiago Peixoto committed Sep 22, 2016 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657  niter: 1 count: 0 breaks: 0 min_S: 699.58882 max_S: 713.12054 S: 713.12054 ΔS: 13.5317 moves: 31 niter: 2 count: 1 breaks: 0 min_S: 699.58882 max_S: 713.12054 S: 711.03345 ΔS: -2.08709 moves: 25 niter: 3 count: 0 breaks: 0 min_S: 699.58882 max_S: 715.72860 S: 715.72860 ΔS: 4.69514 moves: 37 niter: 4 count: 1 breaks: 0 min_S: 699.58882 max_S: 715.72860 S: 704.76394 ΔS: -10.9647 moves: 36 niter: 5 count: 2 breaks: 0 min_S: 699.58882 max_S: 715.72860 S: 706.55192 ΔS: 1.78798 moves: 27 niter: 6 count: 3 breaks: 0 min_S: 699.58882 max_S: 715.72860 S: 706.97865 ΔS: 0.426724 moves: 30 niter: 7 count: 4 breaks: 0 min_S: 699.58882 max_S: 715.72860 S: 706.41383 ΔS: -0.564821 moves: 33 niter: 8 count: 0 breaks: 0 min_S: 699.58882 max_S: 718.47291 S: 718.47291 ΔS: 12.0591 moves: 43 niter: 9 count: 1 breaks: 0 min_S: 699.58882 max_S: 718.47291 S: 709.59053 ΔS: -8.88238 moves: 29 niter: 10 count: 2 breaks: 0 min_S: 699.58882 max_S: 718.47291 S: 700.39505 ΔS: -9.19548 moves: 27 niter: 11 count: 3 breaks: 0 min_S: 699.58882 max_S: 718.47291 S: 710.45317 ΔS: 10.0581 moves: 44 niter: 12 count: 0 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 719.51868 ΔS: 9.06551 moves: 38 niter: 13 count: 1 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 711.66691 ΔS: -7.85177 moves: 41 niter: 14 count: 2 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 701.51609 ΔS: -10.1508 moves: 34 niter: 15 count: 3 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 705.78796 ΔS: 4.27188 moves: 41 niter: 16 count: 4 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 715.67960 ΔS: 9.89164 moves: 33 niter: 17 count: 5 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 703.65838 ΔS: -12.0212 moves: 33 niter: 18 count: 6 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 709.55912 ΔS: 5.90074 moves: 28 niter: 19 count: 7 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 718.42521 ΔS: 8.86609 moves: 28 niter: 20 count: 8 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 711.13784 ΔS: -7.28737 moves: 48 niter: 21 count: 9 breaks: 0 min_S: 699.58882 max_S: 719.51868 S: 706.63047 ΔS: -4.50737 moves: 28 niter: 22 count: 0 breaks: 1 min_S: 707.03211 max_S: 707.03211 S: 707.03211 ΔS: 0.401639 moves: 57 niter: 23 count: 0 breaks: 1 min_S: 707.03211 max_S: 717.50359 S: 717.50359 ΔS: 10.4715 moves: 31 niter: 24 count: 0 breaks: 1 min_S: 707.03211 max_S: 726.72811 S: 726.72811 ΔS: 9.22451 moves: 63 niter: 25 count: 0 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 701.53898 ΔS: -25.1891 moves: 26 niter: 26 count: 1 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 710.04615 ΔS: 8.50718 moves: 25 niter: 27 count: 2 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 710.59565 ΔS: 0.549493 moves: 31 niter: 28 count: 3 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 713.53473 ΔS: 2.93908 moves: 19 niter: 29 count: 4 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 707.57709 ΔS: -5.95763 moves: 34 niter: 30 count: 5 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 712.49104 ΔS: 4.91395 moves: 24 niter: 31 count: 6 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 716.40137 ΔS: 3.91032 moves: 33 niter: 32 count: 7 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 717.30606 ΔS: 0.904690 moves: 54 niter: 33 count: 8 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 712.74418 ΔS: -4.56187 moves: 43 niter: 34 count: 9 breaks: 1 min_S: 701.53898 max_S: 726.72811 S: 711.01120 ΔS: -1.73298 moves: 57 niter: 35 count: 10 breaks: 2 min_S: 701.53898 max_S: 726.72811 S: 717.58446 ΔS: 6.57326 moves: 47  Tiago Peixoto committed Jul 06, 2016 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726  Note that the value of wait above was made purposefully low so that the output would not be overly long. The most appropriate value requires experimentation, but a typically good value is wait=1000. The function :func:~graph_tool.inference.mcmc_equilibrate accepts a callback argument that takes an optional function to be invoked after each call to :meth:~graph_tool.inference.BlockState.mcmc_sweep. This function should accept a single parameter which will contain the actual :class:~graph_tool.inference.BlockState instance. We will use this in the example below to collect the posterior vertex marginals, i.e. the posterior probability that a node belongs to a given group: .. testcode:: model-averaging # We will first equilibrate the Markov chain gt.mcmc_equilibrate(state, wait=1000, mcmc_args=dict(niter=10)) pv = None def collect_marginals(s): global pv pv = s.collect_vertex_marginals(pv) # Now we collect the marginals for exactly 100,000 sweeps gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10), callback=collect_marginals) # Now the node marginals are stored in property map pv. We can # visualize them as pie charts on the nodes: state.draw(pos=g.vp.pos, vertex_shape="pie", vertex_pie_fractions=pv, edge_gradient=None, output="lesmis-sbm-marginals.svg") .. figure:: lesmis-sbm-marginals.* :align: center :width: 450px Marginal probabilities of group memberships of the network of characters in the novel Les Misérables, according to the degree-corrected SBM. The pie fractions _ on the nodes correspond to the probability of being in group associated with the respective color. We can also obtain a marginal probability on the number of groups itself, as follows. .. testcode:: model-averaging h = np.zeros(g.num_vertices() + 1) def collect_num_groups(s): B = s.get_nonempty_B() h[B] += 1 # Now we collect the marginal distribution for exactly 100,000 sweeps gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10), callback=collect_num_groups) .. testcode:: model-averaging :hide: figure() Bs = np.arange(len(h)) idx = h > 0 bar(Bs[idx] - .5, h[idx] / h.sum(), width=1, color="#ccb974") gca().set_xticks([6,7,8,9]) xlabel("$B$")  Tiago Peixoto committed Oct 14, 2016 727  ylabel(r"$P(B|\boldsymbol G)$")  Tiago Peixoto committed Jul 06, 2016 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  savefig("lesmis-B-posterior.svg") .. figure:: lesmis-B-posterior.* :align: center Marginal posterior likelihood of the number of nonempty groups for the network of characters in the novel Les Misérables, according to the degree-corrected SBM. Hierarchical partitions +++++++++++++++++++++++ We can also perform model averaging using the nested SBM, which will give us a distribution over hierarchies. The whole procedure is fairly analogous, but now we make use of :class:~graph_tool.inference.NestedBlockState instances. .. note:: When using :class:~graph_tool.inference.NestedBlockState instances to perform model averaging, they need to be constructed with the option sampling=True. Here we perform the sampling of hierarchical partitions using the same network as above. .. testcode:: nested-model-averaging g = gt.collection.data["lesmis"] state = gt.minimize_nested_blockmodel_dl(g) # Initialize he Markov # chain from the "ground # state" # Before doing model averaging, the need to create a NestedBlockState # by passing sampling = True. # We also want to increase the maximum hierarchy depth to L = 10 # We can do both of the above by copying. bs = state.get_bs() # Get hierarchical partition. bs += [np.zeros(1)] * (10 - len(bs)) # Augment it to L = 10 with # single-group levels. state = state.copy(bs=bs, sampling=True) # Now we run 1000 sweeps of the MCMC dS, nmoves = state.mcmc_sweep(niter=1000) print("Change in description length:", dS) print("Number of accepted vertex moves:", nmoves) .. testoutput:: nested-model-averaging  Tiago Peixoto committed Sep 22, 2016 785 786  Change in description length: 6.1062707... Number of accepted vertex moves: 8488  Tiago Peixoto committed Jul 06, 2016 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819  Similarly to the the non-nested case, we can use :func:~graph_tool.inference.mcmc_equilibrate to do most of the boring work, and we can now obtain vertex marginals on all hierarchical levels: .. testcode:: nested-model-averaging # We will first equilibrate the Markov chain gt.mcmc_equilibrate(state, wait=1000, mcmc_args=dict(niter=10)) pv = [None] * len(state.get_levels()) def collect_marginals(s): global pv pv = [sl.collect_vertex_marginals(pv[l]) for l, sl in enumerate(s.get_levels())] # Now we collect the marginals for exactly 100,000 sweeps gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10), callback=collect_marginals) # Now the node marginals for all levels are stored in property map # list pv. We can visualize the first level as pie charts on the nodes: state_0 = state.get_levels()[0] state_0.draw(pos=g.vp.pos, vertex_shape="pie", vertex_pie_fractions=pv[0], edge_gradient=None, output="lesmis-nested-sbm-marginals.svg") .. figure:: lesmis-nested-sbm-marginals.* :align: center :width: 450px Marginal probabilities of group memberships of the network of characters in the novel Les Misérables, according to the nested  Tiago Peixoto committed Jul 20, 2016 820 821 822  degree-corrected SBM. The pie fractions on the nodes correspond to the probability of being in group associated with the respective color.  Tiago Peixoto committed Jul 06, 2016 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843  We can also obtain a marginal probability of the number of groups itself, as follows. .. testcode:: nested-model-averaging h = [np.zeros(g.num_vertices() + 1) for s in state.get_levels()] def collect_num_groups(s): for l, sl in enumerate(s.get_levels()): B = sl.get_nonempty_B() h[l][B] += 1 # Now we collect the marginal distribution for exactly 100,000 sweeps gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10), callback=collect_num_groups) .. testcode:: nested-model-averaging :hide: figure()  Tiago Peixoto committed Oct 14, 2016 844  f, ax = plt.subplots(1, 5, figsize=(10, 3))  Tiago Peixoto committed Sep 22, 2016 845  for i, h_ in enumerate(h[:5]):  Tiago Peixoto committed Jul 06, 2016 846 847  Bs = np.arange(len(h_)) idx = h_ > 0  Tiago Peixoto committed Sep 22, 2016 848 849 850  ax[i].bar(Bs[idx] - .5, h_[idx] / h_.sum(), width=1, color="#ccb974") ax[i].set_xticks(Bs[idx]) ax[i].set_xlabel("$B_{%d}$" % i)  Tiago Peixoto committed Oct 14, 2016 851  ax[i].set_ylabel(r"$P(B_{%d}|\boldsymbol G)$" % i)  Tiago Peixoto committed Jul 06, 2016 852  locator = MaxNLocator(prune='both', nbins=5)  Tiago Peixoto committed Sep 22, 2016 853  ax[i].yaxis.set_major_locator(locator)  Tiago Peixoto committed Jul 06, 2016 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  tight_layout() savefig("lesmis-nested-B-posterior.svg") .. figure:: lesmis-nested-B-posterior.* :align: center Marginal posterior likelihood of the number of nonempty groups :math:B_l at each hierarchy level :math:l for the network of characters in the novel Les Misérables, according to the nested degree-corrected SBM. Below we obtain some hierarchical partitions sampled from the posterior distribution. .. testcode:: nested-model-averaging for i in range(10): state.mcmc_sweep(niter=1000) state.draw(output="lesmis-partition-sample-%i.svg" % i, empty_branches=False) .. image:: lesmis-partition-sample-0.svg :width: 200px .. image:: lesmis-partition-sample-1.svg :width: 200px .. image:: lesmis-partition-sample-2.svg :width: 200px .. image:: lesmis-partition-sample-3.svg :width: 200px .. image:: lesmis-partition-sample-4.svg :width: 200px .. image:: lesmis-partition-sample-5.svg :width: 200px .. image:: lesmis-partition-sample-6.svg :width: 200px .. image:: lesmis-partition-sample-7.svg :width: 200px .. image:: lesmis-partition-sample-8.svg :width: 200px .. image:: lesmis-partition-sample-9.svg :width: 200px Model class selection +++++++++++++++++++++ When averaging over partitions, we may be interested in evaluating which  Tiago Peixoto committed Oct 14, 2016 899 900 901 **model class** provides a better fit of the data, considering all possible parameter choices. This is done by evaluating the model evidence [peixoto-nonparametric-2016]_  Tiago Peixoto committed Jul 06, 2016 902 903 904  .. math::  Tiago Peixoto committed Oct 14, 2016 905  P(\boldsymbol G) = \sum_{\boldsymbol\theta,\boldsymbol b}P(\boldsymbol G,\boldsymbol\theta, \boldsymbol b) = \sum_{\boldsymbol b}P(\boldsymbol G,\boldsymbol b).  Tiago Peixoto committed Jul 06, 2016 906 907 908 909 910 911 912 913 914 915 916  This quantity is analogous to a partition function _ in statistical physics, which we can write more conveniently as a negative free energy _ by taking its logarithm .. math:: :label: free-energy  Tiago Peixoto committed Oct 14, 2016 917  \ln P(\boldsymbol G) = \underbrace{\sum_{\boldsymbol b}q(\boldsymbol b)\ln P(\boldsymbol G,\boldsymbol b)}_{-\left<\Sigma\right>}\;  Tiago Peixoto committed Jul 06, 2016 918 919 920 921 922 923  \underbrace{- \sum_{\boldsymbol b}q(\boldsymbol b)\ln q(\boldsymbol b)}_{\mathcal{S}} where .. math::  Tiago Peixoto committed Oct 14, 2016 924  q(\boldsymbol b) = \frac{P(\boldsymbol G,\boldsymbol b)}{\sum_{\boldsymbol b'}P(\boldsymbol G,\boldsymbol b')}  Tiago Peixoto committed Jul 06, 2016 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962  is the posterior likelihood of partition :math:\boldsymbol b. The first term of Eq. :eq:free-energy (the "negative energy") is minus the average of description length :math:\left<\Sigma\right>, weighted according to the posterior distribution. The second term :math:\mathcal{S} is the entropy _ of the posterior distribution, and measures, in a sense, the "quality of fit" of the model: If the posterior is very "peaked", i.e. dominated by a single partition with a very large likelihood, the entropy will tend to zero. However, if there are many partitions with similar likelihoods --- meaning that there is no single partition that describes the network uniquely well --- it will take a large value instead. Since the MCMC algorithm samples partitions from the distribution :math:q(\boldsymbol b), it can be used to compute :math:\left<\Sigma\right> easily, simply by averaging the description length values encountered by sampling from the posterior distribution many times. The computation of the posterior entropy :math:\mathcal{S}, however, is significantly more difficult, since it involves measuring the precise value of :math:q(\boldsymbol b). A direct "brute force" computation of :math:\mathcal{S} is implemented via :meth:~graph_tool.inference.BlockState.collect_partition_histogram and :func:~graph_tool.inference.microstate_entropy, however this is only feasible for very small networks. For larger networks, we are forced to perform approximations. The simplest is a "mean field" one, where we assume the posterior factorizes as .. math:: q(\boldsymbol b) \approx \prod_i{q_i(b_i)} where .. math::  Tiago Peixoto committed Oct 14, 2016 963  q_i(r) = P(b_i = r | \boldsymbol G)  Tiago Peixoto committed Jul 06, 2016 964 965 966 967 968 969 970 971 972 973 974 975  is the marginal group membership distribution of node :math:i. This yields an entropy value given by .. math:: S \approx -\sum_i\sum_rq_i(r)\ln q_i(r). This approximation should be seen as an upper bound, since any existing correlation between the nodes (which are ignored here) will yield smaller entropy values.  Tiago Peixoto committed Jul 20, 2016 976 A more accurate assumption is called the Bethe approximation  Tiago Peixoto committed Jul 06, 2016 977 978 979 980 981 982 983 984 985 986 987 988 989 [mezard-information-2009]_, and takes into account the correlation between adjacent nodes in the network, .. math:: q(\boldsymbol b) \approx \prod_{i_, :math:k_i is the degree of node :math:i, and .. math::  Tiago Peixoto committed Oct 14, 2016 990  q_{ij}(r, s) = P(b_i = r, b_j = s|\boldsymbol G)  Tiago Peixoto committed Jul 06, 2016 991 992 993 994 995 996 997 998 999 1000  is the joint group membership distribution of nodes :math:i and :math:j (a.k.a. the edge marginals`). This yields an entropy value given by .. math:: S \approx -\sum_{i