inference.rst 68.7 KB
 Tiago Peixoto committed Jul 06, 2016 1 2 .. _inference-howto:  Tiago Peixoto committed Sep 20, 2017 3 4 Inferring modular network structure ===================================  Tiago Peixoto committed Jul 06, 2016 5 6 7  graph-tool includes algorithms to identify the large-scale structure of networks in the :mod:~graph_tool.inference submodule. Here we  Tiago Peixoto committed Jun 09, 2017 8 9 10 explain the basic functionality with self-contained examples. For a more thorough theoretical introduction to the methods described here, the reader is referred to [peixoto-bayesian-2017]_.  Tiago Peixoto committed Jul 06, 2016 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25  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 26 27 we define a model that generates a network :math:\boldsymbol G with a probability  Tiago Peixoto committed Jul 06, 2016 28 29 30 31  .. math:: :label: model-likelihood  Tiago Peixoto committed Oct 14, 2016 32  P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)  Tiago Peixoto committed Jul 06, 2016 33   Tiago Peixoto committed Sep 20, 2017 34 35 36 37 38 where :math:\boldsymbol\theta are additional model parameters that control how the node partition affects the structure of the network. Therefore, if we observe a network :math:\boldsymbol G, the likelihood that it was generated by a given partition :math:\boldsymbol b is obtained via the Bayesian  Tiago Peixoto committed Jul 06, 2016 39 40 41 42 43 _ posterior .. math:: :label: model-posterior-sum  Tiago Peixoto committed Oct 14, 2016 44  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 45   Tiago Peixoto committed Aug 15, 2017 46 47 where :math:P(\boldsymbol\theta, \boldsymbol b) is the prior probability _ of the  Tiago Peixoto committed Jul 06, 2016 48 49 50 51 52 model parameters, and .. math:: :label: model-evidence  Tiago Peixoto committed Oct 14, 2016 53  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 54   Tiago Peixoto committed Sep 20, 2017 55 56 57 58 59 is called the evidence. The particular types of model that will be considered here have "hard constraints", such that there is only one choice for the remaining parameters :math:\boldsymbol\theta that is compatible with the generated network, such that Eq. :eq:model-posterior-sum simplifies to  Tiago Peixoto committed Jul 06, 2016 60 61 62 63  .. math:: :label: model-posterior  Tiago Peixoto committed Oct 14, 2016 64  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 65   Tiago Peixoto committed Oct 14, 2016 66 67 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 68 69 70 71 here will consist in either finding a network partition that maximizes Eq. :eq:model-posterior, or sampling different partitions according its posterior probability.  Tiago Peixoto committed Aug 15, 2017 72 As we will show below, this approach also enables the comparison of  Tiago Peixoto committed Jul 06, 2016 73 74 75 76 77 78 79 80 81 82 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 83  P(\boldsymbol b | \boldsymbol G) = \frac{\exp(-\Sigma)}{P(\boldsymbol G)}  Tiago Peixoto committed Jul 06, 2016 84 85 86 87 88 89  where .. math:: :label: model-dl  Tiago Peixoto committed Oct 14, 2016 90  \Sigma = -\ln P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b) - \ln P(\boldsymbol\theta, \boldsymbol b)  Tiago Peixoto committed Jul 06, 2016 91   Tiago Peixoto committed Sep 20, 2017 92 93 is called the **description length** of the network :math:\boldsymbol G. It measures the amount of information  Tiago Peixoto committed Jul 06, 2016 94 95 96 97 _ required to describe the data, if we encode _ it using the particular parametrization of the generative model given by  Tiago Peixoto committed Sep 20, 2017 98 99 100 101 :math:\boldsymbol\theta and :math:\boldsymbol b, as well as the parameters themselves. Therefore, if we choose to maximize the posterior distribution of Eq. :eq:model-posterior it will be fully equivalent to the so-called minimum description length  Tiago Peixoto committed Jul 06, 2016 102 103 104 105 106 107 108 _ 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  Tiago Peixoto committed Sep 20, 2017 109 110 111 112 fluctuations for actual structure. In particular this means that we will not find modules in networks if they could have arisen simply because of stochastic fluctuations, as they do in fully random graphs [guimera-modularity-2004]_.  Tiago Peixoto committed Jul 06, 2016 113 114 115 116 117 118 119 120 121  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 Jan 28, 2017 122 [peixoto-nonparametric-2017]_ of the basic or "traditional" version takes  Tiago Peixoto committed Jul 06, 2016 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 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  Tiago Peixoto committed Aug 15, 2017 175 176 177  modular structure is allowed, as the matrix of edge counts :math:e is unconstrained. Hence, we can detect the putatively typical pattern of "community structure"  Tiago Peixoto committed Jul 06, 2016 178 179 180 181 182 183 184 185 186  _, 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  Tiago Peixoto committed Aug 15, 2017 187 188 189 190 191 192 193 194 195 placed randomly inside each group, and because of this the nodes that belong to the same group tend to 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 formulation [peixoto-nonparametric-2017]_).  Tiago Peixoto committed Jul 06, 2016 196 197 198 199 200  The nested stochastic block model +++++++++++++++++++++++++++++++++  Tiago Peixoto committed Aug 15, 2017 201 202 203 204 205 The regular SBM has a drawback when applied to large networks. Namely, it cannot be used to find relatively small groups, as the maximum number of groups that can be found scales as :math:B_{\text{max}}=O(\sqrt{N}), where :math:N is the number of nodes in the network, if Bayesian inference is performed  Tiago Peixoto committed Jul 06, 2016 206 207 208 209 210 [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  Tiago Peixoto committed Aug 15, 2017 211 [peixoto-hierarchical-2014]_, as illustrated below.  Tiago Peixoto committed Jul 06, 2016 212 213 214 215 216 217 218  .. figure:: nested-diagram.* :width: 400px :align: center Example of a nested SBM with three levels.  Tiago Peixoto committed Jun 09, 2017 219 220 221 With this model, the maximum number of groups that can be inferred scales as :math:B_{\text{max}}=O(N/\log(N)). In addition to being able to find small groups in large networks, this model also provides a  Tiago Peixoto committed Sep 20, 2017 222 223 224 multilevel hierarchical description of the network. With such a description, we can uncover structural patterns at multiple scales, representing different levels of coarse-graining.  Tiago Peixoto committed Jul 06, 2016 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248  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 Aug 15, 2017 249  gt.seed_rng(7)  Tiago Peixoto committed Jul 06, 2016 250 251 252 253 254 255 256 257 258 259 260 261  .. testcode:: football g = gt.collection.data["football"] print(g) which yields .. testoutput:: football  Tiago Peixoto committed Aug 15, 2017 262 we then fit the degree-corrected model by calling  Tiago Peixoto committed Jul 06, 2016 263 264 265  .. testcode:: football  Tiago Peixoto committed Aug 15, 2017 266  state = gt.minimize_blockmodel_dl(g)  Tiago Peixoto committed Jul 06, 2016 267 268 269 270 271 272 273  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  Tiago Peixoto committed Aug 15, 2017 274 275 276 277  a different answer each time it is run. This may be due to the fact that there are alternative partitions with similar probabilities, or that the optimum is difficult to find. Note that the inference problem here is, in general, NP-Hard  Tiago Peixoto committed Jul 06, 2016 278 279 280 281 282  _, 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,  Tiago Peixoto committed Aug 15, 2017 283  and select the partition with the largest posterior probability of  Tiago Peixoto committed Jul 06, 2016 284 285 286  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  Tiago Peixoto committed Aug 15, 2017 287  method. See also Sec. :ref:sec_model_selection below.  Tiago Peixoto committed Jul 06, 2016 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321  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 322  3  Tiago Peixoto committed Jul 06, 2016 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345  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()  Tiago Peixoto committed Jun 09, 2017 346 347  ers = state.mrs # edge counts nr = state.wr # node counts  Tiago Peixoto committed Jul 06, 2016 348 349 350 351 352 353 354 355 356 357 358 359  .. _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:  Tiago Peixoto committed Feb 05, 2017 360 361 .. testsetup:: celegans  Tiago Peixoto committed Mar 11, 2018 362  gt.seed_rng(47)  Tiago Peixoto committed Feb 05, 2017 363   Tiago Peixoto committed Jul 06, 2016 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 .. 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  Tiago Peixoto committed Aug 15, 2017 379  state = gt.minimize_nested_blockmodel_dl(g)  Tiago Peixoto committed Jul 06, 2016 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413  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 Mar 11, 2018 414 415 416 417  l: 0, N: 297, B: 17 l: 1, N: 17, B: 9 l: 2, N: 9, B: 3 l: 3, N: 3, B: 1  Tiago Peixoto committed Jul 06, 2016 418 419  The hierarchical levels themselves are represented by individual  Tiago Peixoto committed Jul 20, 2016 420 :meth:~graph_tool.inference.BlockState instances obtained via the  Tiago Peixoto committed Jul 06, 2016 421 422 423 424 425 426 427 428 429 430 :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 Mar 11, 2018 431 432 433 434  , at 0x...> , at 0x...> , at 0x...> , at 0x...>  Tiago Peixoto committed Jul 06, 2016 435 436 437 438 439  This means that we can inspect the hierarchical partition just as before: .. testcode:: celegans  Tiago Peixoto committed Sep 22, 2016 440  r = levels[0].get_blocks()[46] # group membership of node 46 in level 0  Tiago Peixoto committed Jul 06, 2016 441  print(r)  Tiago Peixoto committed Sep 22, 2016 442  r = levels[0].get_blocks()[r] # group membership of node 46 in level 1  Tiago Peixoto committed Jul 06, 2016 443  print(r)  Tiago Peixoto committed Sep 22, 2016 444  r = levels[0].get_blocks()[r] # group membership of node 46 in level 2  Tiago Peixoto committed Jul 06, 2016 445 446 447 448  print(r) .. testoutput:: celegans  Tiago Peixoto committed Mar 11, 2018 449  7  Tiago Peixoto committed Mar 10, 2018 450  0  Tiago Peixoto committed Jun 09, 2017 451  0  Tiago Peixoto committed Jul 06, 2016 452   Tiago Peixoto committed Aug 15, 2017 453 .. _model_selection:  Tiago Peixoto committed Jul 06, 2016 454 455 456 457 458 459 460 461  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  Tiago Peixoto committed Sep 22, 2017 462 463 .. testsetup:: model-selection  Tiago Peixoto committed Sep 23, 2017 464  gt.seed_rng(43)  Tiago Peixoto committed Sep 22, 2017 465   Tiago Peixoto committed Jul 06, 2016 466 467 468 469 470 471 472 473 474 475 476 477 478 .. 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 Mar 11, 2018 479 480  Non-degree-corrected DL: 8456.994339... Degree-corrected DL: 8233.850036...  Tiago Peixoto committed Feb 05, 2017 481   Tiago Peixoto committed Jul 06, 2016 482 483 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 484 be accessed by inspecting the posterior odds ratio  Tiago Peixoto committed Jan 28, 2017 485 [peixoto-nonparametric-2017]_  Tiago Peixoto committed Jul 06, 2016 486 487 488  .. math::  Tiago Peixoto committed Oct 14, 2016 489 490  \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 491 492 493 494  &= \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 495 496 497 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 498 499 500  .. testcode:: model-selection  Tiago Peixoto committed Jan 02, 2017 501  print(u"ln \u039b: ", state_dc.entropy() - state_ndc.entropy())  Tiago Peixoto committed Jul 06, 2016 502 503 504 505  .. testoutput:: model-selection :options: +NORMALIZE_WHITESPACE  Tiago Peixoto committed Mar 11, 2018 506  ln Λ: -223.144303...  Tiago Peixoto committed Jul 06, 2016 507 508 509 510  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 Mar 11, 2018 511 particular degree-corrected fit is around :math:\mathrm{e}^{233} \approx 10^{96}  Tiago Peixoto committed Jul 06, 2016 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 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())  Tiago Peixoto committed Jan 02, 2017 528  print(u"ln \u039b:\t\t\t", state_ndc.entropy() - state_dc.entropy())  Tiago Peixoto committed Jul 06, 2016 529 530 531 532  .. testoutput:: model-selection :options: +NORMALIZE_WHITESPACE  Tiago Peixoto committed Mar 11, 2018 533 534 535  Non-degree-corrected DL: 1734.814739... Degree-corrected DL: 1780.576716... ln Λ: -45.761977...  Tiago Peixoto committed Jul 06, 2016 536   Tiago Peixoto committed Mar 11, 2018 537 538 Hence, with a posterior odds ratio of :math:\Lambda \approx \mathrm{e}^{-45} \approx 10^{-19} in favor of the non-degree-corrected model, it seems like the  Tiago Peixoto committed Jul 06, 2016 539 540 541 degree-corrected variant is an unnecessarily complex description for this network.  Tiago Peixoto committed Aug 15, 2017 542 543 544 545 .. _sampling: Sampling from the posterior distribution ----------------------------------------  Tiago Peixoto committed Jul 06, 2016 546 547 548  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  Tiago Peixoto committed Aug 15, 2017 549 550 551 552 553 probabilities. In such situations, one should instead sample partitions from the posterior distribution, instead of simply finding its maximum. One can then compute quantities that are averaged over the different model fits, weighted according to their posterior probabilities.  Tiago Peixoto committed Jul 06, 2016 554 555 556 557 558 559 560 561  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 _  Tiago Peixoto committed Aug 15, 2017 562 such moves so that, after a sufficiently long time, the partitions will  Tiago Peixoto committed Sep 20, 2017 563 564 565 566 567 be observed with the desired posterior probability. The algorithm is designed so that its run-time (i.e. each sweep of the MCMC) is linear on the number of edges in the network, and independent on the number of groups being used in the model, and hence is suitable for use on very large networks.  Tiago Peixoto committed Jul 06, 2016 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586  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.  Tiago Peixoto committed Aug 15, 2017 587 588 589  # Now we run 1,000 sweeps of the MCMC. Note that the number of groups # is allowed to change, so it will eventually move from the initial # value of B=20 to whatever is most appropriate for the data.  Tiago Peixoto committed Jul 06, 2016 590   Tiago Peixoto committed Mar 10, 2018 591  dS, nattempts, nmoves = state.mcmc_sweep(niter=1000)  Tiago Peixoto committed Jul 06, 2016 592 593 594 595 596 597  print("Change in description length:", dS) print("Number of accepted vertex moves:", nmoves) .. testoutput:: model-averaging  Tiago Peixoto committed Mar 11, 2018 598 599  Change in description length: -365.317522... Number of accepted vertex moves: 38213  Tiago Peixoto committed Jul 06, 2016 600 601 602 603  .. note:: Starting from a random partition is rarely the best option, since it  Tiago Peixoto committed Aug 15, 2017 604  may take a long time for it to equilibrate. It was done above simply  Tiago Peixoto committed Jul 06, 2016 605 606  as an illustration on how to initialize :class:~graph_tool.inference.BlockState by hand. Instead, a much  Tiago Peixoto committed Aug 15, 2017 607 608 609  better option in practice is to start from an approximation to the "ground state" obtained with :func:~graph_tool.inference.minimize_blockmodel_dl, e.g.  Tiago Peixoto committed Jul 06, 2016 610 611 612 613 614  .. testcode:: model-averaging state = gt.minimize_blockmodel_dl(g) state = state.copy(B=g.num_vertices())  Tiago Peixoto committed Mar 10, 2018 615  dS, nattempts, nmoves = state.mcmc_sweep(niter=1000)  Tiago Peixoto committed Jul 06, 2016 616 617 618 619 620 621  print("Change in description length:", dS) print("Number of accepted vertex moves:", nmoves) .. testoutput:: model-averaging  Tiago Peixoto committed Mar 11, 2018 622 623  Change in description length: 1.660677... Number of accepted vertex moves: 40461  Tiago Peixoto committed Jul 06, 2016 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643  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 Mar 11, 2018 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688  niter: 1 count: 0 breaks: 0 min_S: 706.26857 max_S: 708.14483 S: 708.14483 ΔS: 1.87626 moves: 418 niter: 2 count: 0 breaks: 0 min_S: 699.23453 max_S: 708.14483 S: 699.23453 ΔS: -8.91030 moves: 409 niter: 3 count: 0 breaks: 0 min_S: 699.23453 max_S: 715.33531 S: 715.33531 ΔS: 16.1008 moves: 414 niter: 4 count: 0 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 723.13301 ΔS: 7.79770 moves: 391 niter: 5 count: 1 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 702.93354 ΔS: -20.1995 moves: 411 niter: 6 count: 2 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 706.39029 ΔS: 3.45675 moves: 389 niter: 7 count: 3 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 706.80859 ΔS: 0.418293 moves: 404 niter: 8 count: 4 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 707.61960 ΔS: 0.811010 moves: 417 niter: 9 count: 5 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 706.46577 ΔS: -1.15383 moves: 392 niter: 10 count: 6 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 714.34671 ΔS: 7.88094 moves: 410 niter: 11 count: 7 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 706.43194 ΔS: -7.91477 moves: 383 niter: 12 count: 8 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 705.19434 ΔS: -1.23760 moves: 405 niter: 13 count: 9 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 702.21395 ΔS: -2.98039 moves: 423 niter: 14 count: 0 breaks: 1 min_S: 715.54878 max_S: 715.54878 S: 715.54878 ΔS: 13.3348 moves: 400 niter: 15 count: 0 breaks: 1 min_S: 715.54878 max_S: 716.65842 S: 716.65842 ΔS: 1.10964 moves: 413 niter: 16 count: 0 breaks: 1 min_S: 701.19994 max_S: 716.65842 S: 701.19994 ΔS: -15.4585 moves: 382 niter: 17 count: 1 breaks: 1 min_S: 701.19994 max_S: 716.65842 S: 715.56997 ΔS: 14.3700 moves: 394 niter: 18 count: 0 breaks: 1 min_S: 701.19994 max_S: 719.25577 S: 719.25577 ΔS: 3.68580 moves: 404 niter: 19 count: 0 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 723.78811 ΔS: 4.53233 moves: 413 niter: 20 count: 1 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 709.77340 ΔS: -14.0147 moves: 387 niter: 21 count: 2 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 714.14891 ΔS: 4.37551 moves: 419 niter: 22 count: 3 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 722.05875 ΔS: 7.90984 moves: 399 niter: 23 count: 4 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 714.32503 ΔS: -7.73371 moves: 422 niter: 24 count: 5 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 708.53927 ΔS: -5.78576 moves: 392 niter: 25 count: 6 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 714.05889 ΔS: 5.51962 moves: 404 niter: 26 count: 7 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 713.93196 ΔS: -0.126937 moves: 414 niter: 27 count: 8 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 709.49863 ΔS: -4.43333 moves: 410 niter: 28 count: 9 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 707.42167 ΔS: -2.07696 moves: 397 niter: 29 count: 0 breaks: 1 min_S: 699.89982 max_S: 723.78811 S: 699.89982 ΔS: -7.52185 moves: 388 niter: 30 count: 0 breaks: 1 min_S: 698.57305 max_S: 723.78811 S: 698.57305 ΔS: -1.32677 moves: 391 niter: 31 count: 1 breaks: 1 min_S: 698.57305 max_S: 723.78811 S: 706.02629 ΔS: 7.45324 moves: 412 niter: 32 count: 2 breaks: 1 min_S: 698.57305 max_S: 723.78811 S: 701.97778 ΔS: -4.04852 moves: 421 niter: 33 count: 3 breaks: 1 min_S: 698.57305 max_S: 723.78811 S: 707.50134 ΔS: 5.52356 moves: 410 niter: 34 count: 4 breaks: 1 min_S: 698.57305 max_S: 723.78811 S: 708.56686 ΔS: 1.06552 moves: 424 niter: 35 count: 0 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 724.07361 ΔS: 15.5067 moves: 399 niter: 36 count: 1 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 723.51969 ΔS: -0.553915 moves: 384 niter: 37 count: 2 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 702.36708 ΔS: -21.1526 moves: 406 niter: 38 count: 3 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 707.60129 ΔS: 5.23420 moves: 405 niter: 39 count: 4 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 709.67542 ΔS: 2.07413 moves: 400 niter: 40 count: 5 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 714.52753 ΔS: 4.85212 moves: 398 niter: 41 count: 6 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 707.86563 ΔS: -6.66190 moves: 409 niter: 42 count: 7 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 718.80926 ΔS: 10.9436 moves: 400 niter: 43 count: 8 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 716.37312 ΔS: -2.43615 moves: 378 niter: 44 count: 9 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 713.76944 ΔS: -2.60368 moves: 399 niter: 45 count: 10 breaks: 2 min_S: 698.57305 max_S: 724.07361 S: 715.29009 ΔS: 1.52066 moves: 421  Tiago Peixoto committed Jul 06, 2016 689   Tiago Peixoto committed Nov 01, 2016 690 Note that the value of wait above was made purposefully low so that  Tiago Peixoto committed Jul 06, 2016 691 the output would not be overly long. The most appropriate value requires  Tiago Peixoto committed Nov 01, 2016 692 experimentation, but a typically good value is wait=1000.  Tiago Peixoto committed Jul 06, 2016 693 694 695 696 697 698 699  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  Tiago Peixoto committed Aug 15, 2017 700 701 702 the example below to collect the posterior vertex marginals (via :class:~graph_tool.inference.BlockState.collect_vertex_marginals), i.e. the posterior probability that a node belongs to a given group:  Tiago Peixoto committed Jul 06, 2016 703 704 705 706 707 708 709 710 711 712 713 714  .. 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)  Tiago Peixoto committed Aug 15, 2017 715 716  # Now we collect the marginals for exactly 100,000 sweeps, at # intervals of 10 sweeps:  Tiago Peixoto committed Jul 06, 2016 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  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  Tiago Peixoto committed Aug 15, 2017 747 748  # Now we collect the marginals for exactly 100,000 sweeps, at # intervals of 10 sweeps:  Tiago Peixoto committed Jul 06, 2016 749 750 751 752 753 754 755 756 757  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  Tiago Peixoto committed Feb 27, 2017 758  bar(Bs[idx], h[idx] / h.sum(), width=1, color="#ccb974")  Tiago Peixoto committed Jul 06, 2016 759 760  gca().set_xticks([6,7,8,9]) xlabel("$B$")  Tiago Peixoto committed Oct 14, 2016 761  ylabel(r"$P(B|\boldsymbol G)$")  Tiago Peixoto committed Jul 06, 2016 762 763 764 765 766  savefig("lesmis-B-posterior.svg") .. figure:: lesmis-B-posterior.* :align: center  Tiago Peixoto committed Aug 15, 2017 767 768 769  Marginal posterior probability of the number of nonempty groups for the network of characters in the novel Les Misérables, according to the degree-corrected SBM.  Tiago Peixoto committed Jul 06, 2016 770 771 772 773 774 775 776 777 778 779 780 781 782 783  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  Tiago Peixoto committed Nov 01, 2016 784  option sampling=True.  Tiago Peixoto committed Jul 06, 2016 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811  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  Tiago Peixoto committed Mar 10, 2018 812  dS, nattempts, nmoves = state.mcmc_sweep(niter=1000)  Tiago Peixoto committed Jul 06, 2016 813 814 815 816 817 818  print("Change in description length:", dS) print("Number of accepted vertex moves:", nmoves) .. testoutput:: nested-model-averaging  Tiago Peixoto committed Mar 11, 2018 819 820  Change in description length: 2.371018... Number of accepted vertex moves: 56087  Tiago Peixoto committed Jul 06, 2016 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853  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 854 855 856  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 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877  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 878  f, ax = plt.subplots(1, 5, figsize=(10, 3))  Tiago Peixoto committed Sep 22, 2016 879  for i, h_ in enumerate(h[:5]):  Tiago Peixoto committed Jul 06, 2016 880 881  Bs = np.arange(len(h_)) idx = h_ > 0  Tiago Peixoto committed Feb 27, 2017 882  ax[i].bar(Bs[idx], h_[idx] / h_.sum(), width=1, color="#ccb974")  Tiago Peixoto committed Sep 22, 2016 883 884  ax[i].set_xticks(Bs[idx]) ax[i].set_xlabel("$B_{%d}$" % i)  Tiago Peixoto committed Oct 14, 2016 885  ax[i].set_ylabel(r"$P(B_{%d}|\boldsymbol G)$" % i)  Tiago Peixoto committed Jul 06, 2016 886  locator = MaxNLocator(prune='both', nbins=5)  Tiago Peixoto committed Sep 22, 2016 887  ax[i].yaxis.set_major_locator(locator)  Tiago Peixoto committed Jul 06, 2016 888 889 890 891 892 893  tight_layout() savefig("lesmis-nested-B-posterior.svg") .. figure:: lesmis-nested-B-posterior.* :align: center  Tiago Peixoto committed Aug 15, 2017 894  Marginal posterior probability of the number of nonempty groups  Tiago Peixoto committed Jul 06, 2016 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932  :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 933 934 **model class** provides a better fit of the data, considering all possible parameter choices. This is done by evaluating the model  Tiago Peixoto committed Aug 15, 2017 935 evidence summed over all possible partitions [peixoto-nonparametric-2017]_:  Tiago Peixoto committed Jul 06, 2016 936 937 938  .. math::  Tiago Peixoto committed Oct 14, 2016 939  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 940 941 942 943 944 945 946 947 948 949 950  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 951  \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 952 953 954 955 956 957  \underbrace{- \sum_{\boldsymbol b}q(\boldsymbol b)\ln q(\boldsymbol b)}_{\mathcal{S}} where .. math::  Tiago Peixoto committed Oct 14, 2016 958  q(\boldsymbol b) = \frac{P(\boldsymbol G,\boldsymbol b)}{\sum_{\boldsymbol b'}P(\boldsymbol G,\boldsymbol b')}  Tiago Peixoto committed Jul 06, 2016 959   Tiago Peixoto committed Aug 15, 2017 960 is the posterior probability of partition :math:\boldsymbol b. The  Tiago Peixoto committed Jul 06, 2016 961 962 963 964 965 966 967 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  Tiago Peixoto committed Aug 15, 2017 968 969 970 971 single partition with a very large probability, the entropy will tend to zero. However, if there are many partitions with similar probabilities --- meaning that there is no single partition that describes the network uniquely well --- it will take a large value instead.  Tiago Peixoto committed Jul 06, 2016 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996  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 997  q_i(r) = P(b_i = r | \boldsymbol G)  Tiago Peixoto committed Jul 06, 2016 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009  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 1010 A more accurate assumption is called the Bethe approximation  Tiago Peixoto committed Jul 06, 2016 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 [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 1024  q_{ij}(r, s) = P(b_i = r, b_j = s|\boldsymbol G)  Tiago Peixoto committed Jul 06, 2016 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 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071  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"]  Tiago Peixoto committed Feb 10, 2017 1113  nL = 10  Tiago Peixoto committed Jul 06, 2016 1114 1115 1116 1117 1118 1119  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.  Tiago Peixoto committed Feb 10, 2017 1120  bs += [np.zeros(1)] * (nL - len(bs)) # Augment it to L = 10 with  Tiago Peixoto committed Jul 06, 2016 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135  # 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 1136 1137  # 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 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149  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 Mar 11, 2018 1150 1151  Model evidence for deg_corr = True: -551.228195... (mean field), -740.460493... (Bethe) Model evidence for deg_corr = False: -544.660366... (mean field), -649.135026... (Bethe)  Tiago Peixoto committed Jul 06, 2016 1152   Tiago Peixoto committed Jul 20, 2016 1153 1154 1155 1156 1157 1158 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 1159   Tiago Peixoto committed Aug 15, 2017 1160 .. _weights:  Tiago Peixoto committed Jul 06, 2016 1161   Tiago Peixoto committed Aug 15, 2017 1162 1163 Edge weights and covariates ---------------------------  Tiago Peixoto committed Jul 06, 2016 1164   Tiago Peixoto committed Aug 15, 2017 1165 1166 1167 1168 1169 1170 1171 1172 1173 Very often networks cannot be completely represented by simple graphs, but instead have arbitrary "weights" :math:x_{ij} on the edges. Edge weights can be continuous or discrete numbers, and either strictly positive or positive or negative, depending on context. The SBM can be extended to cover these cases by treating edge weights as covariates that are sampled from some distribution conditioned on the node partition [aicher-learning-2015]_ [peixoto-weighted-2017]_, i.e. .. math::  Tiago Peixoto committed Jul 06, 2016 1174   Tiago Peixoto committed Aug 15, 2017 1175 1176  P(\boldsymbol x,\boldsymbol G|\boldsymbol b) = P(\boldsymbol x|\boldsymbol G,\boldsymbol b) P(\boldsymbol G|\boldsymbol b),  Tiago Peixoto committed Jul 06, 2016 1177   Tiago Peixoto committed Aug 15, 2017 1178 1179 1180 1181 1182 1183 1184 1185 where :math:P(\boldsymbol G|\boldsymbol b) is the likelihood of the unweighted SBM described previously, and :math:P(\boldsymbol x|\boldsymbol G,\boldsymbol b) is the integrated likelihood of the edge weights .. math:: P(\boldsymbol x|\boldsymbol G,\boldsymbol b) =  Tiago Peixoto committed Sep 20, 2017 1186  \prod_{r\le s}\int P({\boldsymbol x}_{rs}|\gamma)P(\gamma)\,\mathrm{d}\gamma,  Tiago Peixoto committed Aug 15, 2017 1187   Tiago Peixoto committed Sep 20, 2017 1188 1189 1190 1191 1192 1193 1194 where :math:P({\boldsymbol x}_{rs}|\gamma) is some model for the weights :math:{\boldsymbol x}_{rs} between groups :math:(r,s), conditioned on some parameter :math:\gamma, sampled from its prior :math:P(\gamma). A hierarchical version of the model can also be implemented by replacing this prior by a nested sequence of priors and hyperpriors, as described in [peixoto-weighted-2017]_. The posterior partition distribution is then simply  Tiago Peixoto committed Aug 15, 2017 1195 1196 1197 1198 1199 1200 1201 1202 1203 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 1239 1240 1241 1242  .. math:: P(\boldsymbol b | \boldsymbol G,\boldsymbol x) = \frac{P(\boldsymbol x|\boldsymbol G,\boldsymbol b) P(\boldsymbol G|\boldsymbol b) P(\boldsymbol b)}{P(\boldsymbol G,\boldsymbol x)}, which can be sampled from, or maximized, just like with the unweighted case, but will use the information on the weights to guide the partitions. A variety of weight models is supported, reflecting different kinds of edge covariates: .. csv-table:: :header: "Name", "Domain", "Bounds", "Shape" :widths: 10, 5, 5, 5 :delim: | :align: center "real-exponential" | Real :math:(\mathbb{R}) | :math:[0,\infty] | Exponential _ "real-normal" | Real :math:(\mathbb{R}) | :math:[-\infty,\infty] | Normal _ "discrete-geometric" | Natural :math:(\mathbb{N}) | :math:[0,\infty] | Geometric _ "discrete-binomial" | Natural :math:(\mathbb{N}) | :math:[0,M] | Binomial _ "discrete-poisson" | Natural :math:(\mathbb{N}) | :math:[0,\infty] | Poisson _ In fact, the actual model implements microcanonical _ versions of these distributions that are asymptotically equivalent, as described in [peixoto-weighted-2017]_. These can be combined with arbitrary weight transformations to achieve a large family of associated distributions. For example, to use a log-normal _ weight model for positive real weights :math:\boldsymbol x, we can use the transformation :math:y_{ij} = \ln x_{ij} together with the "real-normal" model for :math:\boldsymbol y. To model weights that are positive or negative integers in :math:\mathbb{Z}, we could either subtract the minimum value, :math:y_{ij} = x_{ij} - x^*, with :math:x^*=\operatorname{min}_{ij}x_{ij}, and use any of the above models for non-negative integers in :math:\mathbb{N}, or alternatively, consider the sign as an additional covariate, i.e. :math:s_{ij} = [\operatorname{sign}(x_{ij})+1]/2 \in \{0,1\}, using the Binomial distribution with :math:M=1 (a.k.a. the Bernoulli distribution _), and any of the other discrete distributions for the magnitude, :math:y_{ij} = \operatorname{abs}(x_{ij}). The support for weighted networks is activated by passing the parameters recs and rec_types to :class:~graph_tool.inference.BlockState  Tiago Peixoto committed Sep 20, 2017 1243 1244 1245 1246 (or :class:~graph_tool.inference.OverlapBlockState), that specify the edge covariates (an edge :class:~graph_tool.PropertyMap) and their types (a string from the table above), respectively. Note that these parameters expect *lists*, so that multiple edge weights can be used  Tiago Peixoto committed Aug 15, 2017 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 simultaneously. For example, let us consider a network of suspected terrorists involved in the train bombing of Madrid on March 11, 2004 [hayes-connecting-2006]_. An edge indicates that a connection between the two persons have been identified, and the weight of the edge (an integer in the range :math:[0,3]) indicates the "strength" of the connection. We can apply the weighted SBM, using a Binomial model for the weights, as follows: .. testsetup:: weighted-model  Tiago Peixoto committed Jul 06, 2016 1259 1260 1261 1262 1263 1264  import os try: os.chdir("demos/inference") except FileNotFoundError: pass  Tiago Peixoto committed Jun 09, 2017 1265 1266  gt.seed_rng(42)  Tiago Peixoto committed Aug 15, 2017 1267 .. testcode:: weighted-model  Tiago Peixoto committed Jul 06, 2016 1268   Tiago Peixoto committed Aug 15, 2017 1269  g = gt.collection.konect_data["moreno_train"]  Tiago Peixoto committed Jul 06, 2016 1270   Tiago Peixoto committed Aug 15, 2017 1271 1272 1273 1274 1275 1276  # This network contains an internal edge property map with name # "weight" that contains the strength of interactions. The values # integers in the range [0, 3]. state = gt.minimize_nested_blockmodel_dl(g, state_args=dict(recs=[g.ep.weight], rec_types=["discrete-binomial"]))  Tiago Peixoto committed Jul 06, 2016 1277   Tiago Peixoto committed Sep 20, 2017 1278  state.draw(edge_color=g.ep.weight, ecmap=(matplotlib.cm.inferno, .6),  Tiago Peixoto committed Aug 15, 2017 1279 1280  eorder=g.ep.weight, edge_pen_width=gt.prop_to_size(g.ep.weight, 1, 4, power=1), edge_gradient=[], output="moreno-train-wsbm.svg")  Tiago Peixoto committed Jul 06, 2016 1281   Tiago Peixoto committed Aug 15, 2017 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 .. figure:: moreno-train-wsbm.* :align: center :width: 350px Best fit of the Binomial-weighted degree-corrected SBM for a network of terror suspects, using the strength of connection as edge covariates. The edge colors and widths correspond to the strengths. Model selection +++++++++++++++ In order to select the best weighted model, we proceed in the same manner as described in Sec. :ref:model_selection. However, when using  Tiago Peixoto committed Sep 20, 2017 1295 1296 transformations on continuous weights, we must include the associated scaling of the probability density, as described in  Tiago Peixoto committed Aug 15, 2017 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 [peixoto-weighted-2017]_. For example, consider a food web _ between species in south Florida [ulanowicz-network-2005]_. A directed link exists from species :math:i to :math:j if a biomass flow exists between them, and a weight :math:x_{ij} on this edge indicates the magnitude of biomass flow (a positive real value, i.e. :math:x_{ij}\in [0,\infty]). One possibility, therefore, is to use the "real-exponential" model, as follows:  Tiago Peixoto committed Jul 06, 2016 1307   Tiago Peixoto committed Aug 15, 2017 1308 1309 1310 1311 1312 1313 1314 .. testsetup:: food-web import os try: os.chdir("demos/inference") except FileNotFoundError: pass  Tiago Peixoto committed Sep 20, 2017 1315  gt.seed_rng(44)  Tiago Peixoto committed Aug 15, 2017 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327  .. testcode:: food-web g = gt.collection.konect_data["foodweb-baywet"] # This network contains an internal edge property map with name # "weight" that contains the biomass flow between species. The values # are continuous in the range [0, infinity]. state = gt.minimize_nested_blockmodel_dl(g, state_args=dict(recs=[g.ep.weight], rec_types=["real-exponential"]))  Tiago Peixoto committed Sep 20, 2017 1328  state.draw(edge_color=gt.prop_to_size(g.ep.weight, power=1, log=True), ecmap=(matplotlib.cm.inferno, .6),  Tiago Peixoto committed Aug 15, 2017 1329 1330 1331 1332  eorder=g.ep.weight, edge_pen_width=gt.prop_to_size(g.ep.weight, 1, 4, power=1, log=True), edge_gradient=[], output="foodweb-wsbm.svg") .. figure:: foodweb-wsbm.*  Tiago Peixoto committed Jul 06, 2016 1333 1334 1335  :align: center :width: 350px  Tiago Peixoto committed Aug 15, 2017 1336 1337 1338 1339  Best fit of the exponential-weighted degree-corrected SBM for a food web, using the biomass flow as edge covariates (indicated by the edge colors and widths).  Tiago Peixoto committed Sep 20, 2017 1340 Alternatively, we may consider a transformation of the type  Tiago Peixoto committed Aug 15, 2017 1341 1342 1343  .. math:: :label: log_transform  Tiago Peixoto committed Jul 06, 2016 1344   Tiago Peixoto committed Aug 15, 2017 1345  y_{ij} = \ln x_{ij}  Tiago Peixoto committed Jul 06, 2016 1346   Tiago Peixoto committed Aug 15, 2017 1347 1348 1349 so that :math:y_{ij} \in [-\infty,\infty]. If we use a model "real-normal" for :math:\boldsymbol y, it amounts to a log-normal _ model for  Tiago Peixoto committed Sep 20, 2017 1350 1351 1352 :math:\boldsymbol x. This can be a better choice if the weights are distributed across many orders of magnitude, or show multi-modality. We can fit this alternative model simply by using the transformed weights:  Tiago Peixoto committed Aug 15, 2017 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362  .. testcode:: food-web # Apply the weight transformation y = g.ep.weight.copy() y.a = log(y.a) state_ln = gt.minimize_nested_blockmodel_dl(g, state_args=dict(recs=[y], rec_types=["real-normal"]))  Tiago Peixoto committed Sep 20, 2017 1363  state_ln.draw(edge_color=gt.prop_to_size(g.ep.weight, power=1, log=True), ecmap=(matplotlib.cm.inferno, .6),  Tiago Peixoto committed Aug 15, 2017 1364 1365 1366 1367 1368 1369  eorder=g.ep.weight, edge_pen_width=gt.prop_to_size(g.ep.weight, 1, 4, power=1, log=True), edge_gradient=[], output="foodweb-wsbm-lognormal.svg") .. figure:: foodweb-wsbm-lognormal.* :align: center :width: 350px  Tiago Peixoto committed Jul 06, 2016 1370   Tiago Peixoto committed Aug 15, 2017 1371 1372 1373 1374  Best fit of the log-normal-weighted degree-corrected SBM for a food web, using the biomass flow as edge covariates (indicated by the edge colors and widths).  Tiago Peixoto committed Sep 20, 2017 1375 1376 At this point, we ask ourselves which of the above models yields the best fit of the data. This is answered by performing model selection via  Tiago Peixoto committed Aug 15, 2017 1377 1378 1379 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 posterior odds ratios just like in Sec. :ref:model_selection. However, here we need to take into account the scaling of the probability density incurred by the variable transformation, i.e. .. math:: P(\boldsymbol x | \boldsymbol G, \boldsymbol b) = P(\boldsymbol y(\boldsymbol x) | \boldsymbol G, \boldsymbol b) \prod_{ij}\left[\frac{\mathrm{d}y_{ij}}{\mathrm{d}x_{ij}}(x_{ij})\right]^{A_{ij}}. In the particular case of Eq. :eq:log_transform, we have .. math:: \prod_{ij}\left[\frac{\mathrm{d}y_{ij}}{\mathrm{d}x_{ij}}(x_{ij})\right]^{A_{ij}} = \prod_{ij}\frac{1}{x_{ij}^{A_{ij}}}. Therefore, we can compute the posterior odds ratio between both models as: .. testcode:: food-web L1 = -state.entropy() L2 = -state_ln.entropy() - log(g.ep.weight.a).sum() print(u"ln \u039b: ", L2 - L1) .. testoutput:: food-web :options: +NORMALIZE_WHITESPACE  Tiago Peixoto committed Mar 11, 2018 1406  ln Λ: -70.145685...  Tiago Peixoto committed Aug 15, 2017 1407   Tiago Peixoto committed Mar 11, 2018 1408 A value of :math:\Lambda \approx \mathrm{e}^{-70} \approx 10^{-30} in  Tiago Peixoto committed Sep 20, 2017 1409 1410 1411 favor the exponential model indicates that the log-normal model does not provide a better fit for this particular data. Based on this, we conclude that the exponential model should be preferred in this case.  Tiago Peixoto committed Aug 15, 2017 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468  Posterior sampling ++++++++++++++++++ The procedure to sample from the posterior distribution is identical to what is described in Sec. :ref:sampling, but with the appropriate initialization, i.e. .. testcode:: weighted-model state = gt.BlockState(g, B=20, recs=[g.ep.weight], rec_types=["discrete