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 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 Jun 09, 2017 362  gt.seed_rng(43)  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 Jun 09, 2017 414 415 416 417  l: 0, N: 297, B: 14 l: 1, N: 14, B: 5 l: 2, N: 5, B: 2 l: 3, N: 2, 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 Jun 09, 2017 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 Jun 09, 2017 449  2  Tiago Peixoto committed May 23, 2017 450  1  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 Sep 23, 2017 479 480  Non-degree-corrected DL: 8511.005312... Degree-corrected DL: 8225.167736...  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 Sep 23, 2017 506  ln Λ: -285.837575...  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 Sep 20, 2017 511 particular degree-corrected fit is around :math:\mathrm{e}^{327} \approx 10^{142}  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 Sep 23, 2017 533 534 535  Non-degree-corrected DL: 1733.525685... Degree-corrected DL: 1791.750418... ln Λ: -58.224733...  Tiago Peixoto committed Jul 06, 2016 536   Tiago Peixoto committed Sep 23, 2017 537 538 Hence, with a posterior odds ratio of :math:\Lambda \approx \mathrm{e}^{-58} \approx 10^{-26} 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 590 591 592 593 594 595 596 597  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 Aug 15, 2017 598 599  Change in description length: -345.376523... Number of accepted vertex moves: 34222  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 610 611 612 613 614 615 616 617 618 619 620 621  .. 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 Aug 15, 2017 622 623  Change in description length: 16.124022... Number of accepted vertex moves: 41393  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 Aug 15, 2017 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705  niter: 1 count: 0 breaks: 0 min_S: 713.72081 max_S: 720.04438 S: 713.72081 ΔS: -6.32357 moves: 418 niter: 2 count: 0 breaks: 0 min_S: 713.72081 max_S: 728.31214 S: 728.31214 ΔS: 14.5913 moves: 384 niter: 3 count: 0 breaks: 0 min_S: 713.70119 max_S: 728.31214 S: 713.70119 ΔS: -14.6110 moves: 443 niter: 4 count: 1 breaks: 0 min_S: 713.70119 max_S: 728.31214 S: 722.48803 ΔS: 8.78684 moves: 391 niter: 5 count: 2 breaks: 0 min_S: 713.70119 max_S: 728.31214 S: 727.05935 ΔS: 4.57131 moves: 378 niter: 6 count: 3 breaks: 0 min_S: 713.70119 max_S: 728.31214 S: 727.29821 ΔS: 0.238862 moves: 344 niter: 7 count: 0 breaks: 0 min_S: 713.70119 max_S: 743.00358 S: 743.00358 ΔS: 15.7054 moves: 376 niter: 8 count: 0 breaks: 0 min_S: 711.80965 max_S: 743.00358 S: 711.80965 ΔS: -31.1939 moves: 382 niter: 9 count: 1 breaks: 0 min_S: 711.80965 max_S: 743.00358 S: 712.92615 ΔS: 1.11651 moves: 343 niter: 10 count: 2 breaks: 0 min_S: 711.80965 max_S: 743.00358 S: 721.94043 ΔS: 9.01428 moves: 388 niter: 11 count: 3 breaks: 0 min_S: 711.80965 max_S: 743.00358 S: 719.13006 ΔS: -2.81037 moves: 362 niter: 12 count: 4 breaks: 0 min_S: 711.80965 max_S: 743.00358 S: 729.78095 ΔS: 10.6509 moves: 383 niter: 13 count: 5 breaks: 0 min_S: 711.80965 max_S: 743.00358 S: 720.04992 ΔS: -9.73104 moves: 376 niter: 14 count: 6 breaks: 0 min_S: 711.80965 max_S: 743.00358 S: 732.90657 ΔS: 12.8567 moves: 387 niter: 15 count: 7 breaks: 0 min_S: 711.80965 max_S: 743.00358 S: 717.42580 ΔS: -15.4808 moves: 380 niter: 16 count: 8 breaks: 0 min_S: 711.80965 max_S: 743.00358 S: 716.75399 ΔS: -0.671812 moves: 359 niter: 17 count: 0 breaks: 0 min_S: 711.80965 max_S: 745.15972 S: 745.15972 ΔS: 28.4057 moves: 350 niter: 18 count: 1 breaks: 0 min_S: 711.80965 max_S: 745.15972 S: 728.99832 ΔS: -16.1614 moves: 389 niter: 19 count: 2 breaks: 0 min_S: 711.80965 max_S: 745.15972 S: 720.84596 ΔS: -8.15237 moves: 348 niter: 20 count: 0 breaks: 0 min_S: 709.75049 max_S: 745.15972 S: 709.75049 ΔS: -11.0955 moves: 392 niter: 21 count: 1 breaks: 0 min_S: 709.75049 max_S: 745.15972 S: 721.10373 ΔS: 11.3532 moves: 341 niter: 22 count: 2 breaks: 0 min_S: 709.75049 max_S: 745.15972 S: 718.50836 ΔS: -2.59537 moves: 354 niter: 23 count: 3 breaks: 0 min_S: 709.75049 max_S: 745.15972 S: 714.36017 ΔS: -4.14819 moves: 375 niter: 24 count: 0 breaks: 0 min_S: 707.10762 max_S: 745.15972 S: 707.10762 ΔS: -7.25255 moves: 367 niter: 25 count: 1 breaks: 0 min_S: 707.10762 max_S: 745.15972 S: 708.42197 ΔS: 1.31435 moves: 372 niter: 26 count: 0 breaks: 0 min_S: 704.56635 max_S: 745.15972 S: 704.56635 ΔS: -3.85562 moves: 346 niter: 27 count: 1 breaks: 0 min_S: 704.56635 max_S: 745.15972 S: 725.76740 ΔS: 21.2011 moves: 338 niter: 28 count: 2 breaks: 0 min_S: 704.56635 max_S: 745.15972 S: 708.78787 ΔS: -16.9795 moves: 378 niter: 29 count: 3 breaks: 0 min_S: 704.56635 max_S: 745.15972 S: 722.03356 ΔS: 13.2457 moves: 382 niter: 30 count: 0 breaks: 0 min_S: 704.10199 max_S: 745.15972 S: 704.10199 ΔS: -17.9316 moves: 375 niter: 31 count: 1 breaks: 0 min_S: 704.10199 max_S: 745.15972 S: 713.64366 ΔS: 9.54166 moves: 382 niter: 32 count: 2 breaks: 0 min_S: 704.10199 max_S: 745.15972 S: 727.65050 ΔS: 14.0068 moves: 383 niter: 33 count: 3 breaks: 0 min_S: 704.10199 max_S: 745.15972 S: 720.51443 ΔS: -7.13607 moves: 379 niter: 34 count: 4 breaks: 0 min_S: 704.10199 max_S: 745.15972 S: 726.77412 ΔS: 6.25969 moves: 366 niter: 35 count: 5 breaks: 0 min_S: 704.10199 max_S: 745.15972 S: 722.96778 ΔS: -3.80634 moves: 382 niter: 36 count: 6 breaks: 0 min_S: 704.10199 max_S: 745.15972 S: 717.65450 ΔS: -5.31328 moves: 394 niter: 37 count: 7 breaks: 0 min_S: 704.10199 max_S: 745.15972 S: 734.02750 ΔS: 16.3730 moves: 377 niter: 38 count: 8 breaks: 0 min_S: 704.10199 max_S: 745.15972 S: 727.36795 ΔS: -6.65954 moves: 393 niter: 39 count: 9 breaks: 0 min_S: 704.10199 max_S: 745.15972 S: 719.39773 ΔS: -7.97022 moves: 382 niter: 40 count: 0 breaks: 1 min_S: 718.06061 max_S: 718.06061 S: 718.06061 ΔS: -1.33712 moves: 411 niter: 41 count: 0 breaks: 1 min_S: 708.36787 max_S: 718.06061 S: 708.36787 ΔS: -9.69274 moves: 398 niter: 42 count: 0 breaks: 1 min_S: 708.36787 max_S: 729.98460 S: 729.98460 ΔS: 21.6167 moves: 406 niter: 43 count: 1 breaks: 1 min_S: 708.36787 max_S: 729.98460 S: 719.27340 ΔS: -10.7112 moves: 383 niter: 44 count: 2 breaks: 1 min_S: 708.36787 max_S: 729.98460 S: 709.89100 ΔS: -9.38239 moves: 409 niter: 45 count: 3 breaks: 1 min_S: 708.36787 max_S: 729.98460 S: 721.29921 ΔS: 11.4082 moves: 383 niter: 46 count: 0 breaks: 1 min_S: 706.67224 max_S: 729.98460 S: 706.67224 ΔS: -14.6270 moves: 405 niter: 47 count: 1 breaks: 1 min_S: 706.67224 max_S: 729.98460 S: 711.87311 ΔS: 5.20087 moves: 373 niter: 48 count: 2 breaks: 1 min_S: 706.67224 max_S: 729.98460 S: 708.20851 ΔS: -3.66460 moves: 367 niter: 49 count: 3 breaks: 1 min_S: 706.67224 max_S: 729.98460 S: 712.26954 ΔS: 4.06103 moves: 368 niter: 50 count: 4 breaks: 1 min_S: 706.67224 max_S: 729.98460 S: 717.69181 ΔS: 5.42227 moves: 396 niter: 51 count: 5 breaks: 1 min_S: 706.67224 max_S: 729.98460 S: 716.64174 ΔS: -1.05008 moves: 395 niter: 52 count: 0 breaks: 1 min_S: 706.67224 max_S: 731.96439 S: 731.96439 ΔS: 15.3226 moves: 387 niter: 53 count: 1 breaks: 1 min_S: 706.67224 max_S: 731.96439 S: 722.51613 ΔS: -9.44825 moves: 411 niter: 54 count: 2 breaks: 1 min_S: 706.67224 max_S: 731.96439 S: 719.18164 ΔS: -3.33449 moves: 414 niter: 55 count: 3 breaks: 1 min_S: 706.67224 max_S: 731.96439 S: 712.43942 ΔS: -6.74222 moves: 395 niter: 56 count: 4 breaks: 1 min_S: 706.67224 max_S: 731.96439 S: 720.71508 ΔS: 8.27565 moves: 395 niter: 57 count: 5 breaks: 1 min_S: 706.67224 max_S: 731.96439 S: 718.75450 ΔS: -1.96058 moves: 379 niter: 58 count: 6 breaks: 1 min_S: 706.67224 max_S: 731.96439 S: 710.43596 ΔS: -8.31854 moves: 428 niter: 59 count: 7 breaks: 1 min_S: 706.67224 max_S: 731.96439 S: 723.89819 ΔS: 13.4622 moves: 408 niter: 60 count: 8 breaks: 1 min_S: 706.67224 max_S: 731.96439 S: 718.87456 ΔS: -5.02363 moves: 435 niter: 61 count: 9 breaks: 1 min_S: 706.67224 max_S: 731.96439 S: 721.20227 ΔS: 2.32772 moves: 399 niter: 62 count: 10 breaks: 2 min_S: 706.67224 max_S: 731.96439 S: 726.81344 ΔS: 5.61116 moves: 383  Tiago Peixoto committed Jul 06, 2016 706   Tiago Peixoto committed Nov 01, 2016 707 Note that the value of wait above was made purposefully low so that  Tiago Peixoto committed Jul 06, 2016 708 the output would not be overly long. The most appropriate value requires  Tiago Peixoto committed Nov 01, 2016 709 experimentation, but a typically good value is wait=1000.  Tiago Peixoto committed Jul 06, 2016 710 711 712 713 714 715 716  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 717 718 719 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 720 721 722 723 724 725 726 727 728 729 730 731  .. 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 732 733  # Now we collect the marginals for exactly 100,000 sweeps, at # intervals of 10 sweeps:  Tiago Peixoto committed Jul 06, 2016 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763  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 764 765  # Now we collect the marginals for exactly 100,000 sweeps, at # intervals of 10 sweeps:  Tiago Peixoto committed Jul 06, 2016 766 767 768 769 770 771 772 773 774  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 775  bar(Bs[idx], h[idx] / h.sum(), width=1, color="#ccb974")  Tiago Peixoto committed Jul 06, 2016 776 777  gca().set_xticks([6,7,8,9]) xlabel("$B$")  Tiago Peixoto committed Oct 14, 2016 778  ylabel(r"$P(B|\boldsymbol G)$")  Tiago Peixoto committed Jul 06, 2016 779 780 781 782 783  savefig("lesmis-B-posterior.svg") .. figure:: lesmis-B-posterior.* :align: center  Tiago Peixoto committed Aug 15, 2017 784 785 786  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 787 788 789 790 791 792 793 794 795 796 797 798 799 800  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 801  option sampling=True.  Tiago Peixoto committed Jul 06, 2016 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835  Here we perform the sampling of hierarchical partitions using the same network as above. .. testcode:: nested-model-averaging g = gt.collection.data["lesmis"] state = gt.minimize_nested_blockmodel_dl(g) # Initialize he Markov # chain from the "ground # state" # Before doing model averaging, the need to create a NestedBlockState # by passing sampling = True. # We also want to increase the maximum hierarchy depth to L = 10 # We can do both of the above by copying. bs = state.get_bs() # Get hierarchical partition. bs += [np.zeros(1)] * (10 - len(bs)) # Augment it to L = 10 with # single-group levels. state = state.copy(bs=bs, sampling=True) # Now we run 1000 sweeps of the MCMC dS, nmoves = state.mcmc_sweep(niter=1000) print("Change in description length:", dS) print("Number of accepted vertex moves:", nmoves) .. testoutput:: nested-model-averaging  Tiago Peixoto committed Sep 20, 2017 836 837  Change in description length: 23.368680... Number of accepted vertex moves: 46167  Tiago Peixoto committed Jul 06, 2016 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870  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 871 872 873  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 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894  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 895  f, ax = plt.subplots(1, 5, figsize=(10, 3))  Tiago Peixoto committed Sep 22, 2016 896  for i, h_ in enumerate(h[:5]):  Tiago Peixoto committed Jul 06, 2016 897 898  Bs = np.arange(len(h_)) idx = h_ > 0  Tiago Peixoto committed Feb 27, 2017 899  ax[i].bar(Bs[idx], h_[idx] / h_.sum(), width=1, color="#ccb974")  Tiago Peixoto committed Sep 22, 2016 900 901  ax[i].set_xticks(Bs[idx]) ax[i].set_xlabel("$B_{%d}$" % i)  Tiago Peixoto committed Oct 14, 2016 902  ax[i].set_ylabel(r"$P(B_{%d}|\boldsymbol G)$" % i)  Tiago Peixoto committed Jul 06, 2016 903  locator = MaxNLocator(prune='both', nbins=5)  Tiago Peixoto committed Sep 22, 2016 904  ax[i].yaxis.set_major_locator(locator)  Tiago Peixoto committed Jul 06, 2016 905 906 907 908 909 910  tight_layout() savefig("lesmis-nested-B-posterior.svg") .. figure:: lesmis-nested-B-posterior.* :align: center  Tiago Peixoto committed Aug 15, 2017 911  Marginal posterior probability of the number of nonempty groups  Tiago Peixoto committed Jul 06, 2016 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949  :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 950 951 **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 952 evidence summed over all possible partitions [peixoto-nonparametric-2017]_:  Tiago Peixoto committed Jul 06, 2016 953 954 955  .. math::  Tiago Peixoto committed Oct 14, 2016 956  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 957 958 959 960 961 962 963 964 965 966 967  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 968  \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 969 970 971 972 973 974  \underbrace{- \sum_{\boldsymbol b}q(\boldsymbol b)\ln q(\boldsymbol b)}_{\mathcal{S}} where .. math::  Tiago Peixoto committed Oct 14, 2016 975  q(\boldsymbol b) = \frac{P(\boldsymbol G,\boldsymbol b)}{\sum_{\boldsymbol b'}P(\boldsymbol G,\boldsymbol b')}  Tiago Peixoto committed Jul 06, 2016 976   Tiago Peixoto committed Aug 15, 2017 977 is the posterior probability of partition :math:\boldsymbol b. The  Tiago Peixoto committed Jul 06, 2016 978 979 980 981 982 983 984 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 985 986 987 988 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 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013  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 1014  q_i(r) = P(b_i = r | \boldsymbol G)  Tiago Peixoto committed Jul 06, 2016 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026  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 1027 A more accurate assumption is called the Bethe approximation  Tiago Peixoto committed Jul 06, 2016 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 [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 1041  q_{ij}(r, s) = P(b_i = r, b_j = s|\boldsymbol G)  Tiago Peixoto committed Jul 06, 2016 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 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088  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 1130  nL = 10  Tiago Peixoto committed Jul 06, 2016 1131 1132 1133 1134 1135 1136  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 1137  bs += [np.zeros(1)] * (nL - len(bs)) # Augment it to L = 10 with  Tiago Peixoto committed Jul 06, 2016 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152  # 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 1153 1154  # 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 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166  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 Jun 09, 2017 1167 1168  Model evidence for deg_corr = True: -554.46079... (mean field), -699.6147... (Bethe) Model evidence for deg_corr = False: -548.6576... (mean field), -635.5347... (Bethe)  Tiago Peixoto committed Jul 06, 2016 1169   Tiago Peixoto committed Jul 20, 2016 1170 1171 1172 1173 1174 1175 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 1176   Tiago Peixoto committed Aug 15, 2017 1177 .. _weights:  Tiago Peixoto committed Jul 06, 2016 1178   Tiago Peixoto committed Aug 15, 2017 1179 1180 Edge weights and covariates ---------------------------  Tiago Peixoto committed Jul 06, 2016 1181   Tiago Peixoto committed Aug 15, 2017 1182 1183 1184 1185 1186 1187 1188 1189 1190 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 1191   Tiago Peixoto committed Aug 15, 2017 1192 1193  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 1194   Tiago Peixoto committed Aug 15, 2017 1195 1196 1197 1198 1199 1200 1201 1202 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 1203  \prod_{r\le s}\int P({\boldsymbol x}_{rs}|\gamma)P(\gamma)\,\mathrm{d}\gamma,  Tiago Peixoto committed Aug 15, 2017 1204   Tiago Peixoto committed Sep 20, 2017 1205 1206 1207 1208 1209 1210 1211 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 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 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259  .. 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 1260 1261 1262 1263 (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 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 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 1276 1277 1278 1279 1280 1281  import os try: os.chdir("demos/inference") except FileNotFoundError: pass  Tiago Peixoto committed Jun 09, 2017 1282 1283  gt.seed_rng(42)  Tiago Peixoto committed Aug 15, 2017 1284 .. testcode:: weighted-model  Tiago Peixoto committed Jul 06, 2016 1285   Tiago Peixoto committed Aug 15, 2017 1286  g = gt.collection.konect_data["moreno_train"]  Tiago Peixoto committed Jul 06, 2016 1287   Tiago Peixoto committed Aug 15, 2017 1288 1289 1290 1291 1292 1293  # 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 1294   Tiago Peixoto committed Sep 20, 2017 1295  state.draw(edge_color=g.ep.weight, ecmap=(matplotlib.cm.inferno, .6),  Tiago Peixoto committed Aug 15, 2017 1296 1297  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 1298   Tiago Peixoto committed Aug 15, 2017 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 .. 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 1312 1313 transformations on continuous weights, we must include the associated scaling of the probability density, as described in  Tiago Peixoto committed Aug 15, 2017 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 [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 1324   Tiago Peixoto committed Aug 15, 2017 1325 1326 1327 1328 1329 1330 1331 .. testsetup:: food-web import os try: os.chdir("demos/inference") except FileNotFoundError: pass  Tiago Peixoto committed Sep 20, 2017 1332  gt.seed_rng(44)  Tiago Peixoto committed Aug 15, 2017 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344  .. 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 1345  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 1346 1347 1348 1349  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 1350 1351 1352  :align: center :width: 350px  Tiago Peixoto committed Aug 15, 2017 1353 1354 1355 1356  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 1357 Alternatively, we may consider a transformation of the type  Tiago Peixoto committed Aug 15, 2017 1358 1359 1360  .. math:: :label: log_transform  Tiago Peixoto committed Jul 06, 2016 1361   Tiago Peixoto committed Aug 15, 2017 1362  y_{ij} = \ln x_{ij}  Tiago Peixoto committed Jul 06, 2016 1363   Tiago Peixoto committed Aug 15, 2017 1364 1365 1366 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 1367 1368 1369 :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 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379  .. 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 1380  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 1381 1382 1383 1384 1385 1386  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 1387   Tiago Peixoto committed Aug 15, 2017 1388 1389 1390 1391  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 1392 1393 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 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 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 Sep 20, 2017 1423  ln Λ: -43.189790...  Tiago Peixoto committed Aug 15, 2017 1424   Tiago Peixoto committed Sep 20, 2017 1425 1426 1427 1428 A value of :math:\Lambda \approx \mathrm{e}^{-43} \approx 10^{-19} in 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 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 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485  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-poisson"]) or for the nested model .. testcode:: weighted-model state = gt.NestedBlockState(g, bs=[np.random.randint(0, 20, g.num_vertices())] + [zeros(1)] * 10, state_args=dict(recs=[g.ep.weight], rec_types=["discrete-poisson"])) Layered networks ---------------- The edges of the network may be distributed in discrete "layers", representing distinct types if interactions [peixoto-inferring-2015]_. 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 categorical edge covariates (not to be confused with the weighted models in Sec. :ref:weights). 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. As an example, let us consider a social network of tribes, where two types of interactions were recorded, amounting to either friendship or enmity [read-cultures-1954]_. We may apply the layered model by separating these two types of interactions in two layers: .. testsetup:: layered-model import os try: os.chdir("demos/inference") except FileNotFoundError: pass gt.seed_rng(42)  Tiago Peixoto committed Jul 06, 2016 1486 1487 .. testcode:: layered-model  Tiago Peixoto committed Aug 15, 2017 1488 1489 1490 1491  g = gt.collection.konect_data["ucidata-gama"] # The edge types are stored in the edge property map "weights".  Tiago Peixoto committed Sep 20, 2017 1492  # Note the different meanings of the two 'layers' parameters below: The  Tiago Peixoto committed Aug 15, 2017 1493 1494  # first enables the use of LayeredBlockState, and the second selects # the 'edge layers' version (instead of 'edge covariates').  Tiago Peixoto committed Jul 06, 2016 1495   Tiago Peixoto committed Aug 15, 2017 1496 1497  state = gt.minimize_nested_blockmodel_dl(g, layers=True, state_args=dict(ec=g.ep.weight, layers=True))  Tiago Peixoto committed Jul 06, 2016 1498   Tiago Peixoto committed Aug 15, 2017 1499  state.draw(edge_color=g.ep.weight, edge_gradient=[],  Tiago Peixoto committed Sep 20, 2017 1500  ecmap=(matplotlib.cm.coolwarm_r, .6), edge_pen_width=5,  Tiago Peixoto committed Aug 15, 2017 1501 1502 1503  output="tribes-sbm-edge-layers.svg") .. figure:: tribes-sbm-edge-layers.*  Tiago Peixoto committed Jul 06, 2016 1504 1505 1506  :align: center :width: 350px  Tiago Peixoto committed Aug 15, 2017 1507 1508 1509  Best fit of the degree-corrected SBM with edge layers for a network of tribes, with edge layers shown as colors. The groups show two enemy tribes.  Tiago Peixoto committed Jul 06, 2016 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525  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 Feb 23, 2017 1526 1527 1528 1529 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 either to the missing or spurious edges. We may compute the posterior of :math:\delta \boldsymbol G as  Tiago Peixoto committed Aug 15, 2017 1530 [valles-catala-consistency-2017]_  Tiago Peixoto committed Jul 06, 2016 1531 1532 1533 1534  .. math:: :label: posterior-missing  Tiago Peixoto committed Aug 15, 2017 1535  P(\delta \boldsymbol G | \boldsymbol G) \propto  Tiago Peixoto committed Sep 20, 2017 1536  \sum_{\boldsymbol b}\frac{P(\boldsymbol G \cup \delta\boldsymbol G| \boldsymbol b)}{P(\boldsymbol G| \boldsymbol b)}P(\boldsymbol b | \boldsymbol G)  Tiago Peixoto committed Jul 06, 2016 1537   Tiago Peixoto committed Aug 15, 2017 1538 1539 1540 1541 1542 1543 1544 1545 up to a normalization constant. Although the normalization constant is difficult to obtain in general (since we need to perform a sum over all possible spurious/missing edges), the numerator of 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 alternative predictive hypotheses :math:\{\delta \boldsymbol G_i\} via their likelihood ratios  Tiago Peixoto committed Jul 06, 2016 1546 1547 1548  .. math::  Tiago Peixoto committed Oct 14, 2016 1549  \lambda_i = \frac{P(\delta \boldsymbol G_i | \boldsymbol G)}{\sum_j P(\delta \boldsymbol G_j | \boldsymbol G)}  Tiago Peixoto committed Jul 06, 2016 1550   Tiago Peixoto committed Aug 15, 2017 1551 which do not depend on the normalization constant.  Tiago Peixoto committed Jul 06, 2016 1552   Tiago Peixoto committed Feb 23, 2017 1553 The values :math:P(\delta \boldsymbol G | \boldsymbol G, \boldsymbol b)  Tiago Peixoto committed Oct 14, 2016 1554 can be computed with  Tiago Peixoto committed Jul 06, 2016 1555 1556 1557 1558 1559 1560 1561 1562 1563 1564 1565 1566 :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:  Tiago Peixoto committed Nov 05, 2016 1567 1568 1569 1570 1571 1572  import os try: os.chdir("demos/inference") except FileNotFoundError: pass  Tiago Peixoto committed Jul 06, 2016 1573 1574 1575  g = gt.collection.data["football"].copy() color = g.new_vp("string", val="#cccccc") ecolor = g.new_ep("string", val="#cccccc")  Tiago Peixoto committed Nov 05, 2016 1576  ewidth = g.new_ep("double", 1)  Tiago Peixoto committed Jul 06, 2016 1577 1578  e = g.add_edge(101, 102) ecolor[e] = "#a40000"  Tiago Peixoto committed Nov 05, 2016 1579  ewidth[e] = 5  Tiago Peixoto committed Jul 06, 2016 1580 1581  e = g.add_edge(17, 56) ecolor[e] = "#a40000"  Tiago Peixoto committed Nov 05, 2016 1582  ewidth[e] = 5  Tiago Peixoto committed Jul 06, 2016 1583 1584 1585 1586  eorder = g.edge_index.copy("int") gt.graph_draw(g, pos=g.vp.pos, vertex_color=color, vertex_fill_color=color, edge_color=ecolor,  Tiago Peixoto committed Nov 05, 2016 1587  eorder=eorder, edge_pen_width=ewidth,  Tiago Peixoto committed Feb 05, 2017 1588  output="football_missing.svg")  Tiago Peixoto committed Jul 06, 2016 1589 1590 1591 1592 1593 1594 1595 1596 1597  .. 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.  Tiago Peixoto committed Feb 05, 2017 1598 1599 .. testsetup:: missing-edges  Tiago Peixoto committed Sep 20, 2017 1600  gt.seed_rng(7)  Tiago Peixoto committed Feb 05, 2017 1601   Tiago Peixoto committed Jul 06, 2016 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 .. 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 1621 1622  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 1623 1624 1625  probs[0].append(p1) probs[1].append(p2)  Tiago Peixoto committed Feb 05, 2017 1626 1627  # Now we collect the probabilities for exactly 100,000 sweeps gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10),  Tiago Peixoto committed Jul 06, 2016 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650  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 09, 2017 1651 1652  likelihood-ratio for (101, 102): 0.37... likelihood-ratio for (17, 56): 0.62...  Tiago Peixoto committed Jul 06, 2016 1653   Tiago Peixoto committed Nov 27, 2016 1654 1655 From which we can conclude that edge :math:(17, 56) is more likely than :math:(101, 102) to be a missing edge.  Tiago Peixoto committed Jul 06, 2016 1656 1657 1658 1659 1660 1661 1662  The prediction using the non-nested model can be performed in an entirely analogous fashion. References ----------  Tiago Peixoto committed Jun 09, 2017 1663 1664 1665 .. [peixoto-bayesian-2017] Tiago P. Peixoto, "Bayesian stochastic blockmodeling", :arxiv:1705.10225  Tiago Peixoto committed Jul 06, 2016 1666 1667 .. [holland-stochastic-1983] Paul W. Holland, Kathryn Blackmond Laskey, Samuel Leinhardt, "Stochastic blockmodels: First steps", Social Networks  Tiago Peixoto committed Aug 15, 2017 1668  Volume 5, Issue 2, Pages 109-137 (1983). :doi:10.1016/0378-8733(83)90021-7  Tiago Peixoto committed Jul 06, 2016 1669 1670 1671  .. [karrer-stochastic-2011] Brian Karrer, M. E. J. Newman "Stochastic blockmodels and community structure in networks", Phys. Rev. E 83,  Tiago Peixoto committed Aug 15, 2017 1672  016107 (2011). :doi:10.1103/PhysRevE.83.016107, :arxiv:1008.3926  Tiago Peixoto committed Jul 06, 2016 1673   Tiago Peixoto committed Jan 28, 2017 1674 1675 .. [peixoto-nonparametric-2017] Tiago P. Peixoto, "Nonparametric Bayesian inference of the microcanonical stochastic block model",  Tiago Peixoto committed Aug 15, 2017 1676  Phys. Rev. E 95 012317 (2017). :doi:10.1103/PhysRevE.95.012317,  Tiago Peixoto committed Oct 14, 2016 1677  :arxiv:1610.02703  Tiago Peixoto committed Jul 06, 2016 1678 1679  .. [peixoto-parsimonious-2013] Tiago P. Peixoto, "Parsimonious module  Tiago Peixoto committed Aug 15, 2017 1680  inference in large networks", Phys. Rev. Lett. 110, 148701 (2013).  Tiago Peixoto committed Jul 06, 2016 1681 1682 1683 1684  :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",  Tiago Peixoto committed Aug 15, 2017 1685  Phys. Rev. X 4, 011047 (2014). :doi:10.1103/PhysRevX.4.011047,  Tiago Peixoto committed Jul 06, 2016 1686 1687 1688 1689  :arxiv:1310.4377. .. [peixoto-model-2016] Tiago P. Peixoto, "Model selection and hypothesis testing for large-scale network models with overlapping groups",  Tiago Peixoto committed Aug 15, 2017 1690  Phys. Rev. X 5, 011033 (2016). :doi:10.1103/PhysRevX.5.011033,  Tiago Peixoto committed Jul 06, 2016 1691 1692  :arxiv:1409.3059.  Tiago Peixoto committed Aug 15, 2017 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 .. [peixoto-inferring-2015] Tiago P. Peixoto, "Inferring the mesoscale structure of layered, edge-valued and time-varying networks", Phys. Rev. E 92, 042807 (2015). :doi:10.1103/PhysRevE.92.042807, :arxiv:1504.02381 .. [aicher-learning-2015] Christopher Aicher, Abigail Z. Jacobs, and Aaron Clauset, "Learning Latent Block Structure in Weighted Networks", Journal of Complex Networks 3(2). 221-248 (2015). :doi:10.1093/comnet/cnu026, :arxiv:1404.0431 .. [peixoto-weighted-2017] Tiago P. Peixoto, "Nonparametric weighted stochastic block models", :arxiv:1708.01432  Tiago Peixoto committed Jul 06, 2016 1706 1707 .. [peixoto-efficient-2014] Tiago P. Peixoto, "Efficient Monte Carlo and greedy heuristic for the inference of stochastic block models", Phys.  Tiago Peixoto committed Aug 15, 2017 1708  Rev. E 89, 012804 (2014). :doi:10.1103/PhysRevE.89.012804,  Tiago Peixoto committed Jul 06, 2016 1709 1710 1711 1712  :arxiv:1310.4378 .. [clauset-hierarchical-2008] Aaron Clauset, Cristopher Moore, M. E. J. Newman, "Hierarchical structure and the prediction of  Tiago Peixoto committed Aug 15, 2017 1713  missing links in networks", Nature 453, 98-101 (2008).  Tiago Peixoto committed Jul 06, 2016 1714 1715 1716 1717  :doi:10.1038/nature06830 .. [guimera-missing-2009] Roger Guimerà, Marta Sales-Pardo, "Missing and spurious interactions and the reconstruction of complex networks", PNAS  Tiago Peixoto committed Aug 15, 2017 1718  vol. 106 no. 52 (2009). :doi:10.1073/pnas.0908366106  Tiago Peixoto committed Jul 06, 2016 1719   Tiago Peixoto committed Aug 15, 2017 1720 1721 1722 1723 .. [valles-catala-consistency-2017] Toni Vallès-Català, Tiago P. Peixoto, Roger Guimerà, Marta Sales-Pardo, "On the consistency between model selection and link prediction in networks". :arxiv:1705.07967  Tiago Peixoto committed Jul 06, 2016 1724 .. [mezard-information-2009] Marc Mézard, Andrea Montanari, "Information,  Tiago Peixoto committed Aug 15, 2017 1725  Physics, and Computation", Oxford Univ Press (2009).  Tiago Peixoto committed Jul 06, 2016 1726 1727  :DOI:10.1093/acprof:oso/9780198570837.001.0001  Tiago Peixoto committed Sep 20, 2017 1728 1729 1730 1731 1732 ..