inference.rst 56.6 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  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  Tiago Peixoto committed Nov 01, 2016 237  gt.seed_rng(3)  Tiago Peixoto committed Jul 06, 2016 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309  .. 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 Nov 01, 2016 310  3  Tiago Peixoto committed Jul 06, 2016 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397  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 398 399 400 401  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 402 403  The hierarchical levels themselves are represented by individual  Tiago Peixoto committed Jul 20, 2016 404 :meth:~graph_tool.inference.BlockState instances obtained via the  Tiago Peixoto committed Jul 06, 2016 405 406 407 408 409 410 411 412 413 414 :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 415  , at 0x...>  Tiago Peixoto committed Nov 01, 2016 416 417  , at 0x...> , at 0x...>  Tiago Peixoto committed Sep 22, 2016 418  , at 0x...>  Tiago Peixoto committed Jul 06, 2016 419 420 421 422 423  This means that we can inspect the hierarchical partition just as before: .. testcode:: celegans  Tiago Peixoto committed Sep 22, 2016 424  r = levels[0].get_blocks()[46] # group membership of node 46 in level 0  Tiago Peixoto committed Jul 06, 2016 425  print(r)  Tiago Peixoto committed Sep 22, 2016 426  r = levels[0].get_blocks()[r] # group membership of node 46 in level 1  Tiago Peixoto committed Jul 06, 2016 427  print(r)  Tiago Peixoto committed Sep 22, 2016 428  r = levels[0].get_blocks()[r] # group membership of node 46 in level 2  Tiago Peixoto committed Jul 06, 2016 429 430 431 432  print(r) .. testoutput:: celegans  Tiago Peixoto committed Jul 18, 2016 433 434  2 1  Tiago Peixoto committed Sep 22, 2016 435  0  Tiago Peixoto committed Jul 06, 2016 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457  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 Nov 01, 2016 458 459  Non-degree-corrected DL: 8507.97432099 Degree-corrected DL: 8228.11609772  Tiago Peixoto committed Jul 06, 2016 460 461 462  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 463 464 be accessed by inspecting the posterior odds ratio [peixoto-nonparametric-2016]_  Tiago Peixoto committed Jul 06, 2016 465 466 467  .. math::  Tiago Peixoto committed Oct 14, 2016 468 469  \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 470 471 472 473  &= \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 474 475 476 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 477 478 479 480 481 482 483 484  .. testcode:: model-selection print("ln Λ: ", state_dc.entropy() - state_ndc.entropy()) .. testoutput:: model-selection :options: +NORMALIZE_WHITESPACE  Tiago Peixoto committed Nov 01, 2016 485  ln Λ: -279.858223272  Tiago Peixoto committed Jul 06, 2016 486 487 488 489  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 Nov 01, 2016 490 particular degree-corrected fit is around :math:e^{280} \sim 10^{121}  Tiago Peixoto committed Jul 06, 2016 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 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 Nov 01, 2016 512 513 514  Non-degree-corrected DL: 1751.86962605 Degree-corrected DL: 1787.64676873 ln Λ: -35.7771426724  Tiago Peixoto committed Jul 06, 2016 515   Tiago Peixoto committed Nov 01, 2016 516 517 Hence, with a posterior odds ratio of :math:\Lambda \sim e^{-36} \sim 10^{-16} in favor of the non-degree-corrected model, it seems like the  Tiago Peixoto committed Jul 06, 2016 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 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 Nov 01, 2016 578 579  Change in description length: -355.3963421220926 Number of accepted vertex moves: 4561  Tiago Peixoto committed Jul 06, 2016 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601  .. 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 Nov 01, 2016 602 603  Change in description length: 7.3423409719804855 Number of accepted vertex moves: 3939  Tiago Peixoto committed Jul 06, 2016 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623  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 Nov 01, 2016 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677  niter: 1 count: 0 breaks: 0 min_S: 709.95524 max_S: 726.36140 S: 726.36140 ΔS: 16.4062 moves: 57 niter: 2 count: 1 breaks: 0 min_S: 709.95524 max_S: 726.36140 S: 721.68682 ΔS: -4.67459 moves: 67 niter: 3 count: 0 breaks: 0 min_S: 709.37313 max_S: 726.36140 S: 709.37313 ΔS: -12.3137 moves: 47 niter: 4 count: 1 breaks: 0 min_S: 709.37313 max_S: 726.36140 S: 711.61100 ΔS: 2.23787 moves: 57 niter: 5 count: 2 breaks: 0 min_S: 709.37313 max_S: 726.36140 S: 716.08147 ΔS: 4.47047 moves: 28 niter: 6 count: 3 breaks: 0 min_S: 709.37313 max_S: 726.36140 S: 712.93940 ΔS: -3.14207 moves: 47 niter: 7 count: 4 breaks: 0 min_S: 709.37313 max_S: 726.36140 S: 712.38780 ΔS: -0.551596 moves: 46 niter: 8 count: 5 breaks: 0 min_S: 709.37313 max_S: 726.36140 S: 718.00449 ΔS: 5.61668 moves: 40 niter: 9 count: 0 breaks: 0 min_S: 709.37313 max_S: 731.89940 S: 731.89940 ΔS: 13.8949 moves: 50 niter: 10 count: 0 breaks: 0 min_S: 707.07048 max_S: 731.89940 S: 707.07048 ΔS: -24.8289 moves: 45 niter: 11 count: 1 breaks: 0 min_S: 707.07048 max_S: 731.89940 S: 711.91030 ΔS: 4.83982 moves: 31 niter: 12 count: 2 breaks: 0 min_S: 707.07048 max_S: 731.89940 S: 726.56358 ΔS: 14.6533 moves: 56 niter: 13 count: 3 breaks: 0 min_S: 707.07048 max_S: 731.89940 S: 731.77165 ΔS: 5.20807 moves: 72 niter: 14 count: 4 breaks: 0 min_S: 707.07048 max_S: 731.89940 S: 707.08606 ΔS: -24.6856 moves: 57 niter: 15 count: 0 breaks: 0 min_S: 707.07048 max_S: 735.85102 S: 735.85102 ΔS: 28.7650 moves: 65 niter: 16 count: 1 breaks: 0 min_S: 707.07048 max_S: 735.85102 S: 707.29116 ΔS: -28.5599 moves: 43 niter: 17 count: 0 breaks: 0 min_S: 702.18860 max_S: 735.85102 S: 702.18860 ΔS: -5.10256 moves: 39 niter: 18 count: 1 breaks: 0 min_S: 702.18860 max_S: 735.85102 S: 716.40444 ΔS: 14.2158 moves: 55 niter: 19 count: 2 breaks: 0 min_S: 702.18860 max_S: 735.85102 S: 703.51896 ΔS: -12.8855 moves: 32 niter: 20 count: 3 breaks: 0 min_S: 702.18860 max_S: 735.85102 S: 714.30455 ΔS: 10.7856 moves: 34 niter: 21 count: 4 breaks: 0 min_S: 702.18860 max_S: 735.85102 S: 707.26722 ΔS: -7.03733 moves: 25 niter: 22 count: 5 breaks: 0 min_S: 702.18860 max_S: 735.85102 S: 730.23976 ΔS: 22.9725 moves: 21 niter: 23 count: 6 breaks: 0 min_S: 702.18860 max_S: 735.85102 S: 730.56562 ΔS: 0.325858 moves: 59 niter: 24 count: 0 breaks: 0 min_S: 702.18860 max_S: 738.45136 S: 738.45136 ΔS: 7.88574 moves: 60 niter: 25 count: 0 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 740.29015 ΔS: 1.83879 moves: 88 niter: 26 count: 1 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 720.86367 ΔS: -19.4265 moves: 68 niter: 27 count: 2 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 723.60308 ΔS: 2.73941 moves: 48 niter: 28 count: 3 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 732.81310 ΔS: 9.21002 moves: 44 niter: 29 count: 4 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 729.62283 ΔS: -3.19028 moves: 62 niter: 30 count: 5 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 730.15676 ΔS: 0.533935 moves: 59 niter: 31 count: 6 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 728.27350 ΔS: -1.88326 moves: 65 niter: 32 count: 7 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 732.19406 ΔS: 3.92056 moves: 57 niter: 33 count: 8 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 730.53906 ΔS: -1.65500 moves: 72 niter: 34 count: 9 breaks: 0 min_S: 702.18860 max_S: 740.29015 S: 725.59638 ΔS: -4.94268 moves: 72 niter: 35 count: 0 breaks: 1 min_S: 733.07687 max_S: 733.07687 S: 733.07687 ΔS: 7.48049 moves: 54 niter: 36 count: 0 breaks: 1 min_S: 728.56326 max_S: 733.07687 S: 728.56326 ΔS: -4.51361 moves: 57 niter: 37 count: 0 breaks: 1 min_S: 728.56326 max_S: 755.55140 S: 755.55140 ΔS: 26.9881 moves: 83 niter: 38 count: 0 breaks: 1 min_S: 728.56326 max_S: 761.09434 S: 761.09434 ΔS: 5.54294 moves: 96 niter: 39 count: 0 breaks: 1 min_S: 713.60740 max_S: 761.09434 S: 713.60740 ΔS: -47.4869 moves: 71 niter: 40 count: 1 breaks: 1 min_S: 713.60740 max_S: 761.09434 S: 713.98904 ΔS: 0.381637 moves: 67 niter: 41 count: 2 breaks: 1 min_S: 713.60740 max_S: 761.09434 S: 729.22460 ΔS: 15.2356 moves: 68 niter: 42 count: 3 breaks: 1 min_S: 713.60740 max_S: 761.09434 S: 724.70143 ΔS: -4.52317 moves: 69 niter: 43 count: 0 breaks: 1 min_S: 703.51896 max_S: 761.09434 S: 703.51896 ΔS: -21.1825 moves: 40 niter: 44 count: 0 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 702.85027 ΔS: -0.668696 moves: 33 niter: 45 count: 1 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 722.46508 ΔS: 19.6148 moves: 49 niter: 46 count: 2 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 714.77930 ΔS: -7.68578 moves: 62 niter: 47 count: 3 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 722.04551 ΔS: 7.26621 moves: 55 niter: 48 count: 4 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 708.96879 ΔS: -13.0767 moves: 37 niter: 49 count: 5 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 714.84009 ΔS: 5.87130 moves: 37 niter: 50 count: 6 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 718.28558 ΔS: 3.44549 moves: 55 niter: 51 count: 7 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 720.86398 ΔS: 2.57840 moves: 44 niter: 52 count: 8 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 710.93672 ΔS: -9.92726 moves: 45 niter: 53 count: 9 breaks: 1 min_S: 702.85027 max_S: 761.09434 S: 735.06773 ΔS: 24.1310 moves: 28 niter: 54 count: 10 breaks: 2 min_S: 702.85027 max_S: 761.09434 S: 738.16756 ΔS: 3.09983 moves: 115  Tiago Peixoto committed Jul 06, 2016 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746  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 747  ylabel(r"$P(B|\boldsymbol G)$")  Tiago Peixoto committed Jul 06, 2016 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  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 Nov 01, 2016 805 806  Change in description length: 6.222068... Number of accepted vertex moves: 7615  Tiago Peixoto committed Jul 06, 2016 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839  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 840 841 842  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 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863  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 864  f, ax = plt.subplots(1, 5, figsize=(10, 3))  Tiago Peixoto committed Sep 22, 2016 865  for i, h_ in enumerate(h[:5]):  Tiago Peixoto committed Jul 06, 2016 866 867  Bs = np.arange(len(h_)) idx = h_ > 0  Tiago Peixoto committed Sep 22, 2016 868 869 870  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 871  ax[i].set_ylabel(r"$P(B_{%d}|\boldsymbol G)$" % i)  Tiago Peixoto committed Jul 06, 2016 872  locator = MaxNLocator(prune='both', nbins=5)  Tiago Peixoto committed Sep 22, 2016 873  ax[i].yaxis.set_major_locator(locator)  Tiago Peixoto committed Jul 06, 2016 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  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 919 920 921 **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 922 923 924  .. math::  Tiago Peixoto committed Oct 14, 2016 925  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 926 927 928 929 930 931 932 933 934 935 936  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 937  \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 938 939 940 941 942 943  \underbrace{- \sum_{\boldsymbol b}q(\boldsymbol b)\ln q(\boldsymbol b)}_{\mathcal{S}} where .. math::  Tiago Peixoto committed Oct 14, 2016 944  q(\boldsymbol b) = \frac{P(\boldsymbol G,\boldsymbol b)}{\sum_{\boldsymbol b'}P(\boldsymbol G,\boldsymbol b')}  Tiago Peixoto committed Jul 06, 2016 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982  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 983  q_i(r) = P(b_i = r | \boldsymbol G)  Tiago Peixoto committed Jul 06, 2016 984 985 986 987 988 989 990 991 992 993 994 995  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 996 A more accurate assumption is called the Bethe approximation  Tiago Peixoto committed Jul 06, 2016 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 [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 1010  q_{ij}(r, s) = P(b_i = r, b_j = s|\boldsymbol G)  Tiago Peixoto committed Jul 06, 2016 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057  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_{i0 only the mean-field approximation is applicable, since the adjacency matrix of the higher layers is not constant. We show below the approach for the same network, using the nested model. .. testcode:: model-evidence g = gt.collection.data["lesmis"] L = 10 for deg_corr in [True, False]: state = gt.minimize_nested_blockmodel_dl(g, deg_corr=deg_corr) # Initialize the Markov # chain from the "ground # state" bs = state.get_bs() # Get hierarchical partition. bs += [np.zeros(1)] * (L - len(bs)) # Augment it to L = 10 with # single-group levels. state = state.copy(bs=bs, sampling=True) dls = [] # description length history vm = [None] * len(state.get_levels()) # vertex marginals em = None # edge marginals def collect_marginals(s): global vm, em levels = s.get_levels() vm = [sl.collect_vertex_marginals(vm[l]) for l, sl in enumerate(levels)] em = levels[0].collect_edge_marginals(em) dls.append(s.entropy())  Tiago Peixoto committed Jul 20, 2016 1122 1123  # Now we collect the marginal distributions for exactly 200,000 sweeps gt.mcmc_equilibrate(state, force_niter=20000, mcmc_args=dict(niter=10),  Tiago Peixoto committed Jul 06, 2016 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135  callback=collect_marginals) S_mf = [gt.mf_entropy(sl.g, vm[l]) for l, sl in enumerate(state.get_levels())] S_bethe = gt.bethe_entropy(g, em)[0] L = -mean(dls) print("Model evidence for deg_corr = %s:" % deg_corr, L + sum(S_mf), "(mean field),", L + S_bethe + sum(S_mf[1:]), "(Bethe)") .. testoutput:: model-evidence  Tiago Peixoto committed Nov 01, 2016 1136 1137  Model evidence for deg_corr = True: -358.493559653 (mean field), -649.40897099 (Bethe) Model evidence for deg_corr = False: -372.104532802 (mean field), -561.973406506 (Bethe)  Tiago Peixoto committed Jul 06, 2016 1138   Tiago Peixoto committed Jul 20, 2016 1139 1140 1141 1142 1143 1144 The results are similar: If we consider the most accurate approximation, the non-degree-corrected model possesses the largest evidence. Note also that we observe a better evidence for the nested models themselves, when comparing to the evidences for the non-nested model --- which is not quite surprising, since the non-nested model is a special case of the nested one.  Tiago Peixoto committed Jul 06, 2016 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202  Edge layers and covariates -------------------------- In many situations, the edges of the network may posses discrete covariates on them, or they may be distributed in discrete "layers". Extensions to the SBM may be defined for such data, and they can be inferred using the exact same interface shown above, except one should use the :class:~graph_tool.inference.LayeredBlockState class, instead of :class:~graph_tool.inference.BlockState. This class takes two additional parameters: the ec parameter, that must correspond to an edge :class:~graph_tool.PropertyMap with the layer/covariate values on the edges, and the Boolean layers parameter, which if True specifies a layered model, otherwise one with edge covariates. If we use :func:~graph_tool.inference.minimize_blockmodel_dl, this can be achieved simply by passing the option layers=True as well as the appropriate value of state_args, which will be propagated to :class:~graph_tool.inference.LayeredBlockState's constructor. For example, consider again the Les Misérables network, where we consider the number of co-appearances between characters as edge covariates. .. testsetup:: layered-model import os try: os.chdir("demos/inference") except FileNotFoundError: pass .. testcode:: layered-model g = gt.collection.data["lesmis"] # Note the different meaning of the two 'layers' parameters below: The # first enables the use of LayeredBlockState, and the second selects # the 'edge covariates' version. state = gt.minimize_blockmodel_dl(g, deg_corr=False, layers=True, state_args=dict(ec=g.ep.value, layers=False)) state.draw(pos=g.vp.pos, edge_color=g.ep.value, edge_gradient=None, output="lesmis-sbm-edge-cov.svg") .. figure:: lesmis-sbm-edge-cov.* :align: center :width: 350px Best fit of the non-degree-corrected SBM with edge covariates for the network of characters in the novel Les Misérables, using the number of co-appearances as edge covariates. The edge colors correspond to the edge covariates. In the case of the nested model, we still should use the :class:~graph_tool.inference.NestedBlockState class, but it must be  Tiago Peixoto committed Aug 05, 2016 1203 initialized with the parameter base_type = LayeredBlockState. But if  Tiago Peixoto committed Jul 06, 2016 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 we use :func:~graph_tool.inference.minimize_nested_blockmodel_dl, it works identically to the above: .. testcode:: layered-model state = gt.minimize_nested_blockmodel_dl(g, deg_corr=False, layers=True, state_args=dict(ec=g.ep.value, layers=False)) state.draw(eprops=dict(color=g.ep.value, gradient=None), output="lesmis-nested-sbm-edge-cov.svg") .. figure:: lesmis-nested-sbm-edge-cov.* :align: center :width: 350px Best fit of the nested non-degree-corrected SBM with edge covariates for the network of characters in the novel Les Misérables, using the number of co-appearances as edge covariates. The edge colors correspond to the edge covariates. It is possible to perform model averaging of all layered variants exactly like for the regular SBMs as was shown above. Predicting spurious and missing edges ------------------------------------- An important application of generative models is to be able to generalize from observations and make predictions that go beyond what is seen in the data. This is particularly useful when the network we observe is incomplete, or contains errors, i.e. some of the edges are either missing or are outcomes of mistakes in measurement. In this situation, the fit we make of the observed network can help us predict missing or spurious edges in the network [clauset-hierarchical-2008]_ [guimera-missing-2009]_.  Tiago Peixoto committed Oct 14, 2016 1239 1240 We do so by dividing the edges into two sets :math:\boldsymbol G and :math:\delta \boldsymbol G, where the former corresponds to the observed network and the latter  Tiago Peixoto committed Jul 06, 2016 1241 either to the missing or spurious edges. In the case of missing edges,  Tiago Peixoto committed Oct 14, 2016 1242 we may compute the posterior of :math:\delta \boldsymbol G as  Tiago Peixoto committed Jul 06, 2016 1243 1244 1245 1246  .. math:: :label: posterior-missing  Tiago Peixoto committed Oct 14, 2016 1247  P(\delta \boldsymbol G | \boldsymbol G) = \frac{\sum_{\boldsymbol b}P(\boldsymbol G+\delta \boldsymbol G | \boldsymbol b)P(\boldsymbol b | \boldsymbol G)}{P_{\delta}(\boldsymbol G)}  Tiago Peixoto committed Jul 06, 2016 1248 1249 1250 1251 1252  where .. math::  Tiago Peixoto committed Oct 14, 2016 1253  P_{\delta}(\boldsymbol G) = \sum_{\delta \boldsymbol G}\sum_{\boldsymbol b}P(\boldsymbol G+\delta \boldsymbol G | \boldsymbol b)P(\boldsymbol b | \boldsymbol G)  Tiago Peixoto committed Jul 06, 2016 1254   Tiago Peixoto committed Oct 14, 2016 1255 1256 1257 is a normalization constant. Although the value of :math:P_{\delta}(\boldsymbol G) is difficult to obtain in general (since we need to perform a sum over all possible spurious/missing edges), the numerator of  Tiago Peixoto committed Jul 06, 2016 1258 1259 1260 Eq. :eq:posterior-missing can be computed by sampling partitions from the posterior, and then inserting or deleting edges from the graph and computing the new likelihood. This means that we can easily compare  Tiago Peixoto committed Oct 14, 2016 1261 alternative predictive hypotheses :math:\{\delta \boldsymbol G_i\} via their  Tiago Peixoto committed Jul 06, 2016 1262 1263 1264 1265 likelihood ratios .. math::  Tiago Peixoto committed Oct 14, 2016 1266 1267 1268  \lambda_i = \frac{P(\delta \boldsymbol G_i | \boldsymbol G)}{\sum_j P(\delta \boldsymbol G_j | \boldsymbol G)} = \frac{\sum_{\boldsymbol b}P(\boldsymbol G+\delta \boldsymbol G_i | \boldsymbol b)P(\boldsymbol b | \boldsymbol G)} {\sum_j \sum_{\boldsymbol b}P(\boldsymbol G+\delta \boldsymbol G_j | \boldsymbol b)P(\boldsymbol b | \boldsymbol G)}  Tiago Peixoto committed Jul 06, 2016 1269   Tiago Peixoto committed Oct 14, 2016 1270 which do not depend on the value of :math:P_{\delta}(\boldsymbol G).  Tiago Peixoto committed Jul 06, 2016 1271   Tiago Peixoto committed Oct 14, 2016 1272 1273 The values :math:P(\boldsymbol G+\delta \boldsymbol G | \boldsymbol b) can be computed with  Tiago Peixoto committed Jul 06, 2016 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 :meth:~graph_tool.inference.BlockState.get_edges_prob. Hence, we can compute spurious/missing edge probabilities just as if we were collecting marginal distributions when doing model averaging. Below is an example for predicting the two following edges in the football network, using the nested model (for which we need to replace :math:\boldsymbol b by :math:\{\boldsymbol b_l\} in the equations above). .. testcode:: missing-edges :hide: g = gt.collection.data["football"].copy() color = g.new_vp("string", val="#cccccc") ecolor = g.new_ep("string", val="#cccccc") e = g.add_edge(101, 102) ecolor[e] = "#a40000" e = g.add_edge(17, 56) ecolor[e] = "#a40000" eorder = g.edge_index.copy("int") gt.graph_draw(g, pos=g.vp.pos, vertex_color=color, vertex_fill_color=color, edge_color=ecolor, eorder=eorder, output="football_missing.svg") .. figure:: football_missing.* :align: center :width: 350px Two non-existing edges in the football network (in red): :math:(101,102) in the middle, and :math:(17,56) in the upper right region of the figure. .. testcode:: missing-edges g = gt.collection.data["football"] missing_edges = [(101, 102), (17, 56)] L = 10 state = gt.minimize_nested_blockmodel_dl(g, deg_corr=True) bs = state.get_bs() # Get hierarchical partition. bs += [np.zeros(1)] * (L - len(bs)) # Augment it to L = 10 with # single-group levels. state = state.copy(bs=bs, sampling=True) probs = ([], []) def collect_edge_probs(s):  Tiago Peixoto committed Oct 14, 2016 1327 1328  p1 = s.get_edges_prob([missing_edges[0]], entropy_args=dict(partition_dl=False)) p2 = s.get_edges_prob([missing_edges[1]], entropy_args=dict(partition_dl=False))  Tiago Peixoto committed Jul 06, 2016 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356  probs[0].append(p1) probs[1].append(p2) # Now we collect the probabilities for exactly 10,000 sweeps gt.mcmc_equilibrate(state, force_niter=1000, mcmc_args=dict(niter=10), callback=collect_edge_probs) def get_avg(p): p = np.array(p) pmax = p.max() p -= pmax return pmax + log(exp(p).mean()) p1 = get_avg(probs[0]) p2 = get_avg(probs[1]) p_sum = get_avg([p1, p2]) + log(2) l1 = p1 - p_sum l2 = p2 - p_sum print("likelihood-ratio for %s: %g" % (missing_edges[0], exp(l1))) print("likelihood-ratio for %s: %g" % (missing_edges[1], exp(l2))) .. testoutput:: missing-edges  Tiago Peixoto committed Nov 01, 2016 1357 1358  likelihood-ratio for (101, 102): 0.372308 likelihood-ratio for (17, 56): 0.627692  Tiago Peixoto committed Jul 06, 2016 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376  From which we can conclude that edge :math:(17, 56) is around twice as likely as :math:(101, 102) to be a missing edge. The prediction using the non-nested model can be performed in an entirely analogous fashion. References ---------- .. [holland-stochastic-1983] Paul W. Holland, Kathryn Blackmond Laskey, Samuel Leinhardt, "Stochastic blockmodels: First steps", Social Networks Volume 5, Issue 2, Pages 109-137 (1983), :doi:10.1016/0378-8733(83)90021-7 .. [karrer-stochastic-2011] Brian Karrer, M. E. J. Newman "Stochastic blockmodels and community structure in networks", Phys. Rev. E 83, 016107 (2011), :doi:10.1103/PhysRevE.83.016107, :arxiv:1008.3926  Tiago Peixoto committed Oct 14, 2016 1377 1378 1379 .. [peixoto-nonparametric-2016] Tiago P. Peixoto, "Nonparametric Bayesian inference of the microcanonical stochastic block model" :arxiv:1610.02703  Tiago Peixoto committed Jul 06, 2016 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412  .. [peixoto-parsimonious-2013] Tiago P. Peixoto, "Parsimonious module inference in large networks", Phys. Rev. Lett. 110, 148701 (2013), :doi:10.1103/PhysRevLett.110.148701, :arxiv:1212.4794. .. [peixoto-hierarchical-2014] Tiago P. Peixoto, "Hierarchical block structures and high-resolution model selection in large networks", Phys. Rev. X 4, 011047 (2014), :doi:10.1103/PhysRevX.4.011047, :arxiv:1310.4377. .. [peixoto-model-2016] Tiago P. Peixoto, "Model selection and hypothesis testing for large-scale network models with overlapping groups", Phys. Rev. X 5, 011033 (2016), :doi:10.1103/PhysRevX.5.011033, :arxiv:1409.3059. .. [peixoto-efficient-2014] Tiago P. Peixoto, "Efficient Monte Carlo and greedy heuristic for the inference of stochastic block models", Phys. Rev. E 89, 012804 (2014), :doi:10.1103/PhysRevE.89.012804, :arxiv:1310.4378 .. [clauset-hierarchical-2008] Aaron Clauset, Cristopher Moore, M. E. J. Newman, "Hierarchical structure and the prediction of missing links in networks", Nature 453, 98-101 (2008), :doi:10.1038/nature06830 .. [guimera-missing-2009] Roger Guimerà, Marta Sales-Pardo, "Missing and spurious interactions and the reconstruction of complex networks", PNAS vol. 106 no. 52 (2009), :doi:10.1073/pnas.0908366106 .. [mezard-information-2009] Marc Mézard, Andrea Montanari, "Information, Physics, and Computation", Oxford Univ Press, 2009. :DOI:10.1093/acprof:oso/9780198570837.001.0001