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 Tiago Peixoto committed Jun 28, 2018 1 2 3 4 5 6 7 Network reconstruction ---------------------- 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  Tiago Peixoto committed May 26, 2019 8 9 10 11 12 13 either missing or are outcomes of mistakes in measurement, or is not even observed at all. In this situation, we can use statistical inference to reconstruct the original network. Following [peixoto-reconstructing-2018]_, if :math:\boldsymbol{\mathcal{D}} is the observed data, the network can be reconstructed according to the posterior distribution,  Tiago Peixoto committed Jun 28, 2018 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58  .. math:: P(\boldsymbol A, \boldsymbol b | \boldsymbol{\mathcal{D}}) = \frac{P(\boldsymbol{\mathcal{D}} | \boldsymbol A)P(\boldsymbol A, \boldsymbol b)}{P(\boldsymbol{\mathcal{D}})} where the likelihood :math:P(\boldsymbol{\mathcal{D}}|\boldsymbol A) models the measurement process, and for the prior :math:P(\boldsymbol A, \boldsymbol b) we use the SBM as before. This means that when performing reconstruction, we sample both the community structure :math:\boldsymbol b and the network :math:\boldsymbol A itself from the posterior distribution. From it, we can obtain the marginal probability of each edge, .. math:: \pi_{ij} = \sum_{\boldsymbol A, \boldsymbol b}A_{ij}P(\boldsymbol A, \boldsymbol b | \boldsymbol{\mathcal{D}}). Based on the marginal posterior probabilities, the best estimate for the whole underlying network :math:\boldsymbol{\hat{A}} is given by the maximum of this distribution, .. math:: \hat A_{ij} = \begin{cases} 1 & \text{ if } \pi_{ij} > \frac{1}{2},\\ 0 & \text{ if } \pi_{ij} < \frac{1}{2}.\\ \end{cases} We can also make estimates :math:\hat y of arbitrary scalar network properties :math:y(\boldsymbol A) via posterior averages, .. math:: \begin{align} \hat y &= \sum_{\boldsymbol A, \boldsymbol b}y(\boldsymbol A)P(\boldsymbol A, \boldsymbol b | \boldsymbol{\mathcal{D}}),\\ \sigma^2_y &= \sum_{\boldsymbol A, \boldsymbol b}(y(\boldsymbol A)-\hat y)^2P(\boldsymbol A, \boldsymbol b | \boldsymbol{\mathcal{D}}) \end{align} with uncertainty given by :math:\sigma_y. This is gives us a complete probabilistic reconstruction framework that fully reflects both the information and the uncertainty in the measurement data. Furthermore, the use of the SBM means that the reconstruction can take advantage of the *correlations* observed in the data to further inform it, which generally can lead to substantial improvements  Tiago Peixoto committed May 26, 2019 59 [peixoto-reconstructing-2018]_ [peixoto-network-2019]_.  Tiago Peixoto committed Jun 28, 2018 60 61 62 63 64 65 66 67 68 69 70  In graph-tool there is support for reconstruction with the above framework for three measurement processes: 1. Repeated measurements with uniform errors (via :class:~graph_tool.inference.uncertain_blockmodel.MeasuredBlockState), 2. Repeated measurements with heterogeneous errors (via :class:~graph_tool.inference.uncertain_blockmodel.MixedMeasuredBlockState), and 3. Extraneously obtained edge probabilities (via :class:~graph_tool.inference.uncertain_blockmodel.UncertainBlockState), which we describe in the following.  Tiago Peixoto committed May 26, 2019 71 72 73 In addition, it is also possible to reconstruct networks from observed dynamical, as described in :ref:reconstruction_dynamics.  Tiago Peixoto committed Jun 28, 2018 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 Measured networks +++++++++++++++++ This model assumes that the node pairs :math:(i,j) were measured :math:n_{ij} times, and an edge has been recorded :math:x_{ij} times, where a missing edge occurs with probability :math:p and a spurious edge occurs with probability :math:q, uniformly for all node pairs, yielding a likelihood .. math:: P(\boldsymbol x | \boldsymbol n, \boldsymbol A, p, q) = \prod_{i__, which specify the amount of prior knowledge we have on the noise parameters. An important special case, which is the default unless otherwise specified, is when we are completely agnostic *a priori* about the noise magnitudes, and all hyperparameters are unity, .. math:: P(\boldsymbol x | \boldsymbol n, \boldsymbol A) \equiv P(\boldsymbol x | \boldsymbol n, \boldsymbol A, \alpha=1,\beta=1,\mu=1,\nu=1). In this situation the priors :math:P(p|\alpha=1,\beta=1) and :math:P(q|\mu=1,\nu=1) are uniform distribution in the interval :math:[0,1]. .. note:: It is important to emphasize that since this approach makes use of the *correlations* between edges to inform the reconstruction, as described by the inferred SBM, this means it can also be used when only single measurements have been performed, :math:n_{ij}=1, and the error magnitudes :math:p and :math:q are unknown. Since every arbitrary adjacency matrix can be cast in this setting, this method can be used to reconstruct networks for which no error assessments of any kind have been provided. Below, we illustrate how the reconstruction can be performed with a simple example, using :class:~graph_tool.inference.uncertain_blockmodel.MeasuredBlockState: .. testsetup:: measured import os try: os.chdir("demos/inference") except FileNotFoundError: pass np.random.seed(42) gt.seed_rng(44) .. testcode:: measured g = gt.collection.data["lesmis"].copy() # pretend we have measured and observed each edge twice n = g.new_ep("int", 2) # number of measurements x = g.new_ep("int", 2) # number of observations e = g.edge(11, 36) x[e] = 1 # pretend we have observed edge (11, 36) only once e = g.add_edge(15, 73) n[e] = 2 # pretend we have measured non-edge (15, 73) twice, x[e] = 1 # but observed it as an edge once. bs = [g.get_vertices()] + [zeros(1)] * 5 # initial hierarchical partition # We inititialize MeasuredBlockState, assuming that each non-edge has # been measured only once (as opposed to twice for the observed # edges), as specified by the 'n_default' and 'x_default' parameters. state = gt.MeasuredBlockState(g, n=n, n_default=1, x=x, x_default=0, state_args=dict(bs=bs)) # We will first equilibrate the Markov chain gt.mcmc_equilibrate(state, wait=1000, mcmc_args=dict(niter=10)) # Now we collect the marginals for exactly 100,000 sweeps, at # intervals of 10 sweeps: u = None # marginal posterior edge probabilities pv = None # marginal posterior group membership probabilities cs = [] # average local clustering coefficient def collect_marginals(s): global pv, u, cs u = s.collect_marginal(u) bstate = s.get_block_state() pv = bstate.levels[0].collect_vertex_marginals(pv) cs.append(gt.local_clustering(s.get_graph()).fa.mean()) gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10), callback=collect_marginals) eprob = u.ep.eprob print("Posterior probability of edge (11, 36):", eprob[u.edge(11, 36)]) print("Posterior probability of non-edge (15, 73):", eprob[u.edge(15, 73)]) print("Estimated average local clustering: %g ± %g" % (np.mean(cs), np.std(cs))) Which yields the following output: .. testoutput:: measured  Tiago Peixoto committed May 26, 2019 192 193 194  Posterior probability of edge (11, 36): 0.890889... Posterior probability of non-edge (15, 73): 0.056005... Estimated average local clustering: 0.572758 ± 0.003998...  Tiago Peixoto committed Jun 28, 2018 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251  We have a successful reconstruction, where both ambiguous adjacency matrix entries are correctly recovered. The value for the average clustering coefficient is also correctly estimated, and is compatible with the true value :math:0.57313675, within the estimated error. Below we visualize the maximum marginal posterior estimate of the reconstructed network: .. testcode:: measured # The maximum marginal posterior estimator can be obtained by # filtering the edges with probability larger than .5 u = gt.GraphView(u, efilt=u.ep.eprob.fa > .5) # Mark the recovered true edges as red, and the removed spurious edges as green ecolor = u.new_ep("vector", val=[0, 0, 0, .6]) for e in u.edges(): if g.edge(e.source(), e.target()) is None or (e.source(), e.target()) == (11, 36): ecolor[e] = [1, 0, 0, .6] for e in g.edges(): if u.edge(e.source(), e.target()) is None: ne = u.add_edge(e.source(), e.target()) ecolor[ne] = [0, 1, 0, .6] # Duplicate the internal block state with the reconstructed network # u, for visualization purposes. bstate = state.get_block_state() bstate = bstate.levels[0].copy(g=u) pv = u.own_property(pv) edash = u.new_ep("vector") edash[u.edge(15, 73)] = [.1, .1, 0] bstate.draw(pos=u.own_property(g.vp.pos), vertex_shape="pie", vertex_pie_fractions=pv, edge_color=ecolor, edge_dash_style=edash, edge_gradient=None, output="lesmis-reconstruction-marginals.svg") .. figure:: lesmis-reconstruction-marginals.* :align: center :width: 450px Reconstructed network of characters in the novel Les Misérables, assuming that each edge has been measured and recorded twice, and each non-edge has been measured only once, with the exception of edge (11, 36), shown in red, and non-edge (15, 73), shown in green, which have been measured twice and recorded as an edge once. Despite the ambiguity, both errors are successfully corrected by the reconstruction. The pie fractions on the nodes correspond to the probability of being in group associated with the respective color. Heterogeneous errors ^^^^^^^^^^^^^^^^^^^^ In a more general scenario the measurement errors can be different for each node pair, i.e. :math:p_{ij} and :math:q_{ij} are the missing  Tiago Peixoto committed Sep 12, 2018 252 and spurious edge probabilities for node pair :math:(i,j). The  Tiago Peixoto committed Jun 28, 2018 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 measurement likelihood then becomes .. math:: P(\boldsymbol x | \boldsymbol n, \boldsymbol A, \boldsymbol p, \boldsymbol q) = \prod_{i__, like before. Instead of pre-specifying the hyperparameters, we include them from the posterior distribution .. math:: P(\boldsymbol A, \boldsymbol b, \alpha,\beta,\mu,\nu | \boldsymbol x, \boldsymbol n) = \frac{P(\boldsymbol x | \boldsymbol n, \boldsymbol A, \alpha,\beta,\mu,\nu)P(\boldsymbol A, \boldsymbol b)P(\alpha,\beta,\mu,\nu)}{P(\boldsymbol x| \boldsymbol n)}, where :math:P(\alpha,\beta,\mu,\nu)\propto 1 is a uniform hyperprior. Operationally, the inference with this model works similarly to the one with uniform error rates, as we see with the same example: .. testcode:: measured state = gt.MixedMeasuredBlockState(g, n=n, n_default=1, x=x, x_default=0, state_args=dict(bs=bs)) # We will first equilibrate the Markov chain gt.mcmc_equilibrate(state, wait=1000, mcmc_args=dict(niter=10)) # Now we collect the marginals for exactly 100,000 sweeps, at # intervals of 10 sweeps: u = None # marginal posterior edge probabilities pv = None # marginal posterior group membership probabilities cs = [] # average local clustering coefficient gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10), callback=collect_marginals) eprob = u.ep.eprob print("Posterior probability of edge (11, 36):", eprob[u.edge(11, 36)]) print("Posterior probability of non-edge (15, 73):", eprob[u.edge(15, 73)]) print("Estimated average local clustering: %g ± %g" % (np.mean(cs), np.std(cs))) Which yields: .. testoutput:: measured  Tiago Peixoto committed May 26, 2019 313 314 315  Posterior probability of edge (11, 36): 0.515651... Posterior probability of non-edge (15, 73): 0.009000... Estimated average local clustering: 0.571673 ± 0.003228...  Tiago Peixoto committed Jun 28, 2018 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440  The results are very similar to the ones obtained with the uniform model in this case, but can be quite different in situations where a large number of measurements has been performed (see [peixoto-reconstructing-2018]_ for details). Extraneous error estimates ++++++++++++++++++++++++++ In some situations the edge uncertainties are estimated by means other than repeated measurements, using domain-specific models. Here we consider the general case where the error estimates are extraneously provided as independent edge probabilities :math:\boldsymbol Q, .. math:: P_Q(\boldsymbol A | \boldsymbol Q) = \prod_{i .5) # Mark the recovered true edges as red, and the removed spurious edges as green ecolor = u.new_ep("vector", val=[0, 0, 0, .6]) edash = u.new_ep("vector") for e in u.edges(): if g.edge(e.source(), e.target()) is None or (e.source(), e.target()) == (11, 36): ecolor[e] = [1, 0, 0, .6] for e in g.edges(): if u.edge(e.source(), e.target()) is None: ne = u.add_edge(e.source(), e.target()) ecolor[ne] = [0, 1, 0, .6] if (e.source(), e.target()) == (15, 73): edash[ne] = [.1, .1, 0] bstate = state.get_block_state() bstate = bstate.levels[0].copy(g=u) pv = u.own_property(pv) bstate.draw(pos=u.own_property(g.vp.pos), vertex_shape="pie", vertex_pie_fractions=pv, edge_color=ecolor, edge_dash_style=edash, edge_gradient=None, output="lesmis-uncertain-reconstruction-marginals.svg") .. figure:: lesmis-uncertain-reconstruction-marginals.* :align: center :width: 450px Reconstructed network of characters in the novel Les Misérables, assuming that each edge as a measurement probability of :math:.98. Edge (11, 36), shown in red, and non-edge (15, 73), shown in green, both have probability :math:0.5. Despite the ambiguity, both errors are successfully corrected by the reconstruction. The pie fractions on the nodes correspond to the probability of being in group associated with the respective color.  Tiago Peixoto committed May 26, 2019 487 488  .. include:: _reconstruction_dynamics.rst