.. _inference-howto: Inferring modular network structure =================================== graph-tool includes algorithms to identify the large-scale structure of networks in the :mod:~graph_tool.inference submodule. Here we explain the basic functionality with self-contained examples. For a more thorough theoretical introduction to the methods described here, the reader is referred to [peixoto-bayesian-2017]_. 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, we define a model that generates a network :math:\boldsymbol G with a probability .. math:: :label: model-likelihood P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b) 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 _ posterior .. math:: :label: model-posterior-sum P(\boldsymbol b | \boldsymbol G) = \frac{\sum_{\boldsymbol\theta}P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)P(\boldsymbol\theta, \boldsymbol b)}{P(\boldsymbol G)} where :math:P(\boldsymbol\theta, \boldsymbol b) is the prior probability _ of the model parameters, and .. math:: :label: model-evidence P(\boldsymbol G) = \sum_{\boldsymbol\theta,\boldsymbol b}P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)P(\boldsymbol\theta, \boldsymbol b) 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 .. math:: :label: model-posterior P(\boldsymbol b | \boldsymbol G) = \frac{P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b)P(\boldsymbol\theta, \boldsymbol b)}{P(\boldsymbol G)} with :math:\boldsymbol\theta above being the only choice compatible with :math:\boldsymbol G and :math:\boldsymbol b. The inference procedures considered here will consist in either finding a network partition that maximizes Eq. :eq:model-posterior, or sampling different partitions according its posterior probability. As we will show below, this approach also enables the comparison of different models according to statistical evidence (a.k.a. model selection). Minimum description length (MDL) ++++++++++++++++++++++++++++++++ We note that Eq. :eq:model-posterior can be written as .. math:: P(\boldsymbol b | \boldsymbol G) = \frac{\exp(-\Sigma)}{P(\boldsymbol G)} where .. math:: :label: model-dl \Sigma = -\ln P(\boldsymbol G|\boldsymbol\theta, \boldsymbol b) - \ln P(\boldsymbol\theta, \boldsymbol b) is called the **description length** of the network :math:\boldsymbol G. It measures the amount of information _ required to describe the data, if we encode _ it using the particular parametrization of the generative model given by :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 _ method. This approach corresponds to an implementation of Occam's razor _, where the simplest model is selected, among all possibilities with the same explanatory power. The selection is based on the statistical evidence available, and therefore will not overfit _, i.e. mistake stochastic fluctuations for actual structure. 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]_. 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 [peixoto-nonparametric-2017]_ of the basic or "traditional" version takes as parameters the partition of the nodes into groups :math:\boldsymbol b and a :math:B\times B matrix of edge counts :math:\boldsymbol e, where :math:e_{rs} is the number of edges between groups :math:r and :math:s. Given these constraints, the edges are then placed randomly. Hence, nodes that belong to the same group possess the same probability of being connected with other nodes of the network. An example of a possible parametrization is given in the following figure. .. testcode:: sbm-example :hide: import os try: os.chdir("demos/inference") except FileNotFoundError: pass g = gt.load_graph("blockmodel-example.gt.gz") gt.graph_draw(g, pos=g.vp.pos, vertex_size=10, vertex_fill_color=g.vp.bo, vertex_color="#333333", edge_gradient=g.new_ep("vector", val=[0]), output="sbm-example.svg") ers = g.gp.w from pylab import * figure() matshow(log(ers)) xlabel("Group $r$") ylabel("Group $s$") gca().xaxis.set_label_position("top") savefig("sbm-example-ers.svg") .. table:: :class: figure +----------------------------------+------------------------------+ |.. figure:: sbm-example-ers.svg |.. figure:: sbm-example.svg | | :width: 300px | :width: 300px | | :align: center | :align: center | | | | | Matrix of edge counts | Generated network. | | :math:\boldsymbol e between | | | groups. | | +----------------------------------+------------------------------+ .. note:: We emphasize that no constraints are imposed on what kind of modular structure is allowed, as the matrix of edge counts :math:e is unconstrained. Hence, we can detect the putatively typical pattern of "community structure" _, i.e. when nodes are connected mostly to other nodes of the same group, if it happens to be the most likely network description, but we can also detect a large multiplicity of other patterns, such as bipartiteness _, core-periphery, and many others, all under the same inference framework. Although quite general, the traditional model assumes that the edges are placed randomly inside each group, and 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]_). The nested stochastic block model +++++++++++++++++++++++++++++++++ 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 [peixoto-parsimonious-2013]_. In order to circumvent this, we need to replace the noninformative priors used by a hierarchy of priors and hyperpriors, which amounts to a nested SBM, where the groups themselves are clustered into groups, and the matrix :math:e of edge counts are generated from another SBM, and so on recursively [peixoto-hierarchical-2014]_, as illustrated below. .. figure:: nested-diagram.* :width: 400px :align: center Example of a nested SBM with three levels. 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 multilevel hierarchical description of the network. With such a description, we can uncover structural patterns at multiple scales, representing different levels of coarse-graining. 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 gt.seed_rng(7) .. testcode:: football g = gt.collection.data["football"] print(g) which yields .. testoutput:: football we then fit the degree-corrected model by calling .. testcode:: football state = gt.minimize_blockmodel_dl(g) This returns a :class:~graph_tool.inference.BlockState object that includes the inference results. .. note:: The inference algorithm used is stochastic by nature, and may return a 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 _, hence there is no efficient algorithm that is guaranteed to always find the best answer. Because of this, typically one would call the algorithm many times, and select the partition with the largest posterior probability of Eq. :eq:model-posterior, or equivalently, the minimum description length of Eq. :eq:model-dl. The description length of a fit can be obtained with the :meth:~graph_tool.inference.BlockState.entropy method. See also Sec. :ref:sec_model_selection below. We may perform a drawing of the partition obtained via the :mod:~graph_tool.inference.BlockState.draw method, that functions as a convenience wrapper to the :func:~graph_tool.draw.graph_draw function .. testcode:: football state.draw(pos=g.vp.pos, output="football-sbm-fit.svg") which yields the following image. .. figure:: football-sbm-fit.* :align: center :width: 400px Stochastic block model inference of a network of American college football teams. The colors correspond to inferred group membership of the nodes. We can obtain the group memberships as a :class:~graph_tool.PropertyMap on the vertices via the :mod:~graph_tool.inference.BlockState.get_blocks method: .. testcode:: football b = state.get_blocks() r = b[10] # group membership of vertex 10 print(r) which yields: .. testoutput:: football 3 We may also access the matrix of edge counts between groups via :mod:~graph_tool.inference.BlockState.get_matrix .. testcode:: football e = state.get_matrix() matshow(e.todense()) savefig("football-edge-counts.svg") .. figure:: football-edge-counts.* :align: center Matrix of edge counts between groups. We may obtain the same matrix of edge counts as a graph, which has internal edge and vertex property maps with the edge and vertex counts, respectively: .. testcode:: football bg = state.get_bg() ers = state.mrs # edge counts nr = state.wr # node counts .. _sec_model_selection: Hierarchical partitions +++++++++++++++++++++++ The inference of the nested family of SBMs is done in a similar manner, but we must use instead the :func:~graph_tool.inference.minimize_nested_blockmodel_dl function. We illustrate its use with the neural network of the C. elegans _ worm: .. testsetup:: celegans gt.seed_rng(47) .. testcode:: celegans g = gt.collection.data["celegansneural"] print(g) which has 297 vertices and 2359 edges. .. testoutput:: celegans A hierarchical fit of the degree-corrected model is performed as follows. .. testcode:: celegans state = gt.minimize_nested_blockmodel_dl(g) 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 l: 0, N: 297, B: 17 l: 1, N: 17, B: 9 l: 2, N: 9, B: 3 l: 3, N: 3, B: 1 The hierarchical levels themselves are represented by individual :meth:~graph_tool.inference.BlockState instances obtained via the :meth:~graph_tool.inference.NestedBlockState.get_levels() method: .. testcode:: celegans levels = state.get_levels() for s in levels: print(s) .. testoutput:: celegans , at 0x...> , at 0x...> , at 0x...> , at 0x...> This means that we can inspect the hierarchical partition just as before: .. testcode:: celegans r = levels[0].get_blocks()[46] # group membership of node 46 in level 0 print(r) r = levels[0].get_blocks()[r] # group membership of node 46 in level 1 print(r) r = levels[0].get_blocks()[r] # group membership of node 46 in level 2 print(r) .. testoutput:: celegans 7 0 0 .. _model_selection: 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 .. testsetup:: model-selection gt.seed_rng(43) .. 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 Non-degree-corrected DL: 8456.994339... Degree-corrected DL: 8233.850036... Since it yields the smallest description length, the degree-corrected fit should be preferred. The statistical significance of the choice can be accessed by inspecting the posterior odds ratio [peixoto-nonparametric-2017]_ .. math:: \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})} \\ &= \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 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 .. testcode:: model-selection print(u"ln \u039b: ", state_dc.entropy() - state_ndc.entropy()) .. testoutput:: model-selection :options: +NORMALIZE_WHITESPACE ln Λ: -223.144303... 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 particular degree-corrected fit is around :math:\mathrm{e}^{233} \approx 10^{96} times more likely than the non-degree corrected one, and hence it can be safely concluded that it provides a substantially better fit. Although it is often true that the degree-corrected model provides a better fit for many empirical networks, there are also exceptions. For example, for the American football network above, we have: .. testcode:: model-selection g = gt.collection.data["football"] state_ndc = gt.minimize_nested_blockmodel_dl(g, deg_corr=False) state_dc = gt.minimize_nested_blockmodel_dl(g, deg_corr=True) print("Non-degree-corrected DL:\t", state_ndc.entropy()) print("Degree-corrected DL:\t", state_dc.entropy()) print(u"ln \u039b:\t\t\t", state_ndc.entropy() - state_dc.entropy()) .. testoutput:: model-selection :options: +NORMALIZE_WHITESPACE Non-degree-corrected DL: 1734.814739... Degree-corrected DL: 1780.576716... ln Λ: -45.761977... Hence, with a posterior odds ratio of :math:\Lambda \approx \mathrm{e}^{-45} \approx 10^{-19} in favor of the non-degree-corrected model, it seems like the degree-corrected variant is an unnecessarily complex description for this network. .. _sampling: Sampling from the posterior distribution ---------------------------------------- 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 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. Full support for model averaging is implemented in graph-tool via an efficient Markov chain Monte Carlo (MCMC) _ algorithm [peixoto-efficient-2014]_. It works by attempting to move nodes into different groups with specific probabilities, and accepting or rejecting _ such moves so that, after a sufficiently long time, the partitions will 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. 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. # 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. dS, nattempts, nmoves = state.mcmc_sweep(niter=1000) print("Change in description length:", dS) print("Number of accepted vertex moves:", nmoves) .. testoutput:: model-averaging Change in description length: -365.317522... Number of accepted vertex moves: 38213 .. note:: Starting from a random partition is rarely the best option, since it may take a long time for it to equilibrate. It was done above simply as an illustration on how to initialize :class:~graph_tool.inference.BlockState by hand. Instead, a much better option in practice is to start from an approximation to the "ground state" obtained with :func:~graph_tool.inference.minimize_blockmodel_dl, e.g. .. testcode:: model-averaging state = gt.minimize_blockmodel_dl(g) state = state.copy(B=g.num_vertices()) dS, nattempts, nmoves = state.mcmc_sweep(niter=1000) print("Change in description length:", dS) print("Number of accepted vertex moves:", nmoves) .. testoutput:: model-averaging Change in description length: 1.660677... Number of accepted vertex moves: 40461 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 niter: 1 count: 0 breaks: 0 min_S: 706.26857 max_S: 708.14483 S: 708.14483 ΔS: 1.87626 moves: 418 niter: 2 count: 0 breaks: 0 min_S: 699.23453 max_S: 708.14483 S: 699.23453 ΔS: -8.91030 moves: 409 niter: 3 count: 0 breaks: 0 min_S: 699.23453 max_S: 715.33531 S: 715.33531 ΔS: 16.1008 moves: 414 niter: 4 count: 0 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 723.13301 ΔS: 7.79770 moves: 391 niter: 5 count: 1 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 702.93354 ΔS: -20.1995 moves: 411 niter: 6 count: 2 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 706.39029 ΔS: 3.45675 moves: 389 niter: 7 count: 3 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 706.80859 ΔS: 0.418293 moves: 404 niter: 8 count: 4 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 707.61960 ΔS: 0.811010 moves: 417 niter: 9 count: 5 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 706.46577 ΔS: -1.15383 moves: 392 niter: 10 count: 6 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 714.34671 ΔS: 7.88094 moves: 410 niter: 11 count: 7 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 706.43194 ΔS: -7.91477 moves: 383 niter: 12 count: 8 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 705.19434 ΔS: -1.23760 moves: 405 niter: 13 count: 9 breaks: 0 min_S: 699.23453 max_S: 723.13301 S: 702.21395 ΔS: -2.98039 moves: 423 niter: 14 count: 0 breaks: 1 min_S: 715.54878 max_S: 715.54878 S: 715.54878 ΔS: 13.3348 moves: 400 niter: 15 count: 0 breaks: 1 min_S: 715.54878 max_S: 716.65842 S: 716.65842 ΔS: 1.10964 moves: 413 niter: 16 count: 0 breaks: 1 min_S: 701.19994 max_S: 716.65842 S: 701.19994 ΔS: -15.4585 moves: 382 niter: 17 count: 1 breaks: 1 min_S: 701.19994 max_S: 716.65842 S: 715.56997 ΔS: 14.3700 moves: 394 niter: 18 count: 0 breaks: 1 min_S: 701.19994 max_S: 719.25577 S: 719.25577 ΔS: 3.68580 moves: 404 niter: 19 count: 0 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 723.78811 ΔS: 4.53233 moves: 413 niter: 20 count: 1 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 709.77340 ΔS: -14.0147 moves: 387 niter: 21 count: 2 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 714.14891 ΔS: 4.37551 moves: 419 niter: 22 count: 3 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 722.05875 ΔS: 7.90984 moves: 399 niter: 23 count: 4 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 714.32503 ΔS: -7.73371 moves: 422 niter: 24 count: 5 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 708.53927 ΔS: -5.78576 moves: 392 niter: 25 count: 6 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 714.05889 ΔS: 5.51962 moves: 404 niter: 26 count: 7 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 713.93196 ΔS: -0.126937 moves: 414 niter: 27 count: 8 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 709.49863 ΔS: -4.43333 moves: 410 niter: 28 count: 9 breaks: 1 min_S: 701.19994 max_S: 723.78811 S: 707.42167 ΔS: -2.07696 moves: 397 niter: 29 count: 0 breaks: 1 min_S: 699.89982 max_S: 723.78811 S: 699.89982 ΔS: -7.52185 moves: 388 niter: 30 count: 0 breaks: 1 min_S: 698.57305 max_S: 723.78811 S: 698.57305 ΔS: -1.32677 moves: 391 niter: 31 count: 1 breaks: 1 min_S: 698.57305 max_S: 723.78811 S: 706.02629 ΔS: 7.45324 moves: 412 niter: 32 count: 2 breaks: 1 min_S: 698.57305 max_S: 723.78811 S: 701.97778 ΔS: -4.04852 moves: 421 niter: 33 count: 3 breaks: 1 min_S: 698.57305 max_S: 723.78811 S: 707.50134 ΔS: 5.52356 moves: 410 niter: 34 count: 4 breaks: 1 min_S: 698.57305 max_S: 723.78811 S: 708.56686 ΔS: 1.06552 moves: 424 niter: 35 count: 0 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 724.07361 ΔS: 15.5067 moves: 399 niter: 36 count: 1 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 723.51969 ΔS: -0.553915 moves: 384 niter: 37 count: 2 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 702.36708 ΔS: -21.1526 moves: 406 niter: 38 count: 3 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 707.60129 ΔS: 5.23420 moves: 405 niter: 39 count: 4 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 709.67542 ΔS: 2.07413 moves: 400 niter: 40 count: 5 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 714.52753 ΔS: 4.85212 moves: 398 niter: 41 count: 6 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 707.86563 ΔS: -6.66190 moves: 409 niter: 42 count: 7 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 718.80926 ΔS: 10.9436 moves: 400 niter: 43 count: 8 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 716.37312 ΔS: -2.43615 moves: 378 niter: 44 count: 9 breaks: 1 min_S: 698.57305 max_S: 724.07361 S: 713.76944 ΔS: -2.60368 moves: 399 niter: 45 count: 10 breaks: 2 min_S: 698.57305 max_S: 724.07361 S: 715.29009 ΔS: 1.52066 moves: 421 Note that the value of wait above was made purposefully low so that the output would not be overly long. The most appropriate value requires experimentation, but a typically good value is wait=1000. The function :func:~graph_tool.inference.mcmc_equilibrate accepts a callback argument that takes an optional function to be invoked after each call to :meth:~graph_tool.inference.BlockState.mcmc_sweep. This function should accept a single parameter which will contain the actual :class:~graph_tool.inference.BlockState instance. We will use this in the example below to collect the posterior vertex marginals (via :class:~graph_tool.inference.BlockState.collect_vertex_marginals), i.e. the posterior probability that a node belongs to a given group: .. testcode:: model-averaging # We will first equilibrate the Markov chain gt.mcmc_equilibrate(state, wait=1000, mcmc_args=dict(niter=10)) pv = None def collect_marginals(s): global pv pv = s.collect_vertex_marginals(pv) # Now we collect the marginals for exactly 100,000 sweeps, at # intervals of 10 sweeps: gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10), callback=collect_marginals) # Now the node marginals are stored in property map pv. We can # visualize them as pie charts on the nodes: state.draw(pos=g.vp.pos, vertex_shape="pie", vertex_pie_fractions=pv, edge_gradient=None, output="lesmis-sbm-marginals.svg") .. figure:: lesmis-sbm-marginals.* :align: center :width: 450px Marginal probabilities of group memberships of the network of characters in the novel Les Misérables, according to the degree-corrected SBM. The pie fractions _ on the nodes correspond to the probability of being in group associated with the respective color. We can also obtain a marginal probability on the number of groups itself, as follows. .. testcode:: model-averaging h = np.zeros(g.num_vertices() + 1) def collect_num_groups(s): B = s.get_nonempty_B() h[B] += 1 # Now we collect the marginals for exactly 100,000 sweeps, at # intervals of 10 sweeps: gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10), callback=collect_num_groups) .. testcode:: model-averaging :hide: figure() Bs = np.arange(len(h)) idx = h > 0 bar(Bs[idx], h[idx] / h.sum(), width=1, color="#ccb974") gca().set_xticks([6,7,8,9]) xlabel("$B$") ylabel(r"$P(B|\boldsymbol G)$") savefig("lesmis-B-posterior.svg") .. figure:: lesmis-B-posterior.* :align: center 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. Hierarchical partitions +++++++++++++++++++++++ We can also perform model averaging using the nested SBM, which will give us a distribution over hierarchies. The whole procedure is fairly analogous, but now we make use of :class:~graph_tool.inference.NestedBlockState instances. .. note:: When using :class:~graph_tool.inference.NestedBlockState instances to perform model averaging, they need to be constructed with the option sampling=True. Here we perform the sampling of hierarchical partitions using the same network as above. .. testcode:: nested-model-averaging g = gt.collection.data["lesmis"] state = gt.minimize_nested_blockmodel_dl(g) # Initialize he Markov # chain from the "ground # state" # Before doing model averaging, the need to create a NestedBlockState # by passing sampling = True. # We also want to increase the maximum hierarchy depth to L = 10 # We can do both of the above by copying. bs = state.get_bs() # Get hierarchical partition. bs += [np.zeros(1)] * (10 - len(bs)) # Augment it to L = 10 with # single-group levels. state = state.copy(bs=bs, sampling=True) # Now we run 1000 sweeps of the MCMC dS, nattempts, nmoves = state.mcmc_sweep(niter=1000) print("Change in description length:", dS) print("Number of accepted vertex moves:", nmoves) .. testoutput:: nested-model-averaging Change in description length: 2.371018... Number of accepted vertex moves: 56087 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 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 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() f, ax = plt.subplots(1, 5, figsize=(10, 3)) for i, h_ in enumerate(h[:5]): Bs = np.arange(len(h_)) idx = h_ > 0 ax[i].bar(Bs[idx], h_[idx] / h_.sum(), width=1, color="#ccb974") ax[i].set_xticks(Bs[idx]) ax[i].set_xlabel("$B_{%d}$" % i) ax[i].set_ylabel(r"$P(B_{%d}|\boldsymbol G)$" % i) locator = MaxNLocator(prune='both', nbins=5) ax[i].yaxis.set_major_locator(locator) tight_layout() savefig("lesmis-nested-B-posterior.svg") .. figure:: lesmis-nested-B-posterior.* :align: center Marginal posterior probability of the number of nonempty groups :math:B_l at each hierarchy level :math:l for the network of characters in the novel Les Misérables, according to the nested degree-corrected SBM. Below we obtain some hierarchical partitions sampled from the posterior distribution. .. testcode:: nested-model-averaging for i in range(10): state.mcmc_sweep(niter=1000) state.draw(output="lesmis-partition-sample-%i.svg" % i, empty_branches=False) .. image:: lesmis-partition-sample-0.svg :width: 200px .. image:: lesmis-partition-sample-1.svg :width: 200px .. image:: lesmis-partition-sample-2.svg :width: 200px .. image:: lesmis-partition-sample-3.svg :width: 200px .. image:: lesmis-partition-sample-4.svg :width: 200px .. image:: lesmis-partition-sample-5.svg :width: 200px .. image:: lesmis-partition-sample-6.svg :width: 200px .. image:: lesmis-partition-sample-7.svg :width: 200px .. image:: lesmis-partition-sample-8.svg :width: 200px .. image:: lesmis-partition-sample-9.svg :width: 200px Model class selection +++++++++++++++++++++ When averaging over partitions, we may be interested in evaluating which **model class** provides a better fit of the data, considering all possible parameter choices. This is done by evaluating the model evidence summed over all possible partitions [peixoto-nonparametric-2017]_: .. math:: P(\boldsymbol G) = \sum_{\boldsymbol\theta,\boldsymbol b}P(\boldsymbol G,\boldsymbol\theta, \boldsymbol b) = \sum_{\boldsymbol b}P(\boldsymbol G,\boldsymbol b). 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 \ln P(\boldsymbol G) = \underbrace{\sum_{\boldsymbol b}q(\boldsymbol b)\ln P(\boldsymbol G,\boldsymbol b)}_{-\left<\Sigma\right>}\; \underbrace{- \sum_{\boldsymbol b}q(\boldsymbol b)\ln q(\boldsymbol b)}_{\mathcal{S}} where .. math:: q(\boldsymbol b) = \frac{P(\boldsymbol G,\boldsymbol b)}{\sum_{\boldsymbol b'}P(\boldsymbol G,\boldsymbol b')} is the posterior probability of partition :math:\boldsymbol b. The first term of Eq. :eq:free-energy (the "negative energy") is minus the average of description length :math:\left<\Sigma\right>, weighted according to the posterior distribution. The second term :math:\mathcal{S} is the entropy _ of the posterior distribution, and measures, in a sense, the "quality of fit" of the model: If the posterior is very "peaked", i.e. dominated by a single partition with a very large 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. 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:: q_i(r) = P(b_i = r | \boldsymbol G) 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. A more accurate assumption is called the Bethe approximation [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:: q_{ij}(r, s) = P(b_i = r, b_j = s|\boldsymbol G) 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"] nL = 10 for deg_corr in [True, False]: state = gt.minimize_nested_blockmodel_dl(g, deg_corr=deg_corr) # Initialize the Markov # chain from the "ground # state" bs = state.get_bs() # Get hierarchical partition. bs += [np.zeros(1)] * (nL - len(bs)) # Augment it to L = 10 with # single-group levels. state = state.copy(bs=bs, sampling=True) dls = [] # description length history vm = [None] * len(state.get_levels()) # vertex marginals em = None # edge marginals def collect_marginals(s): global vm, em levels = s.get_levels() vm = [sl.collect_vertex_marginals(vm[l]) for l, sl in enumerate(levels)] em = levels[0].collect_edge_marginals(em) dls.append(s.entropy()) # Now we collect the marginal distributions for exactly 200,000 sweeps gt.mcmc_equilibrate(state, force_niter=20000, mcmc_args=dict(niter=10), 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 Model evidence for deg_corr = True: -551.228195... (mean field), -740.460493... (Bethe) Model evidence for deg_corr = False: -544.660366... (mean field), -649.135026... (Bethe) 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. .. _weights: Edge weights and covariates --------------------------- 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:: P(\boldsymbol x,\boldsymbol G|\boldsymbol b) = P(\boldsymbol x|\boldsymbol G,\boldsymbol b) P(\boldsymbol G|\boldsymbol b), 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) = \prod_{r\le s}\int P({\boldsymbol x}_{rs}|\gamma)P(\gamma)\,\mathrm{d}\gamma, 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 .. 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 (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 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 import os try: os.chdir("demos/inference") except FileNotFoundError: pass gt.seed_rng(42) .. testcode:: weighted-model g = gt.collection.konect_data["moreno_train"] # 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"])) state.draw(edge_color=g.ep.weight, ecmap=(matplotlib.cm.inferno, .6), 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") .. 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 transformations on continuous weights, we must include the associated scaling of the probability density, as described in [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: .. testsetup:: food-web import os try: os.chdir("demos/inference") except FileNotFoundError: pass gt.seed_rng(44) .. 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"])) state.draw(edge_color=gt.prop_to_size(g.ep.weight, power=1, log=True), ecmap=(matplotlib.cm.inferno, .6), 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.* :align: center :width: 350px 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). Alternatively, we may consider a transformation of the type .. math:: :label: log_transform y_{ij} = \ln x_{ij} 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 :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: .. 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"])) state_ln.draw(edge_color=gt.prop_to_size(g.ep.weight, power=1, log=True), ecmap=(matplotlib.cm.inferno, .6), 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 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). 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 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 ln Λ: -70.145685... A value of :math:\Lambda \approx \mathrm{e}^{-70} \approx 10^{-30} 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. 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) .. testcode:: layered-model g = gt.collection.konect_data["ucidata-gama"] # The edge types are stored in the edge property map "weights". # Note the different meanings of the two 'layers' parameters below: The # first enables the use of LayeredBlockState, and the second selects # the 'edge layers' version (instead of 'edge covariates'). state = gt.minimize_nested_blockmodel_dl(g, layers=True, state_args=dict(ec=g.ep.weight, layers=True)) state.draw(edge_color=g.ep.weight, edge_gradient=[], ecmap=(matplotlib.cm.coolwarm_r, .6), edge_pen_width=5, output="tribes-sbm-edge-layers.svg") .. figure:: tribes-sbm-edge-layers.* :align: center :width: 350px 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. 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]_. 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 [valles-catala-consistency-2017]_ .. math:: :label: posterior-missing P(\delta \boldsymbol G | \boldsymbol G) \propto \sum_{\boldsymbol b}\frac{P(\boldsymbol G \cup \delta\boldsymbol G| \boldsymbol b)}{P(\boldsymbol G| \boldsymbol b)}P(\boldsymbol b | \boldsymbol G) 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 .. math:: \lambda_i = \frac{P(\delta \boldsymbol G_i | \boldsymbol G)}{\sum_j P(\delta \boldsymbol G_j | \boldsymbol G)} which do not depend on the normalization constant. The values :math:P(\delta \boldsymbol G | \boldsymbol G, \boldsymbol b) can be computed with :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: import os try: os.chdir("demos/inference") except FileNotFoundError: pass g = gt.collection.data["football"].copy() color = g.new_vp("string", val="#cccccc") ecolor = g.new_ep("string", val="#cccccc") ewidth = g.new_ep("double", 1) e = g.add_edge(101, 102) ecolor[e] = "#a40000" ewidth[e] = 5 e = g.add_edge(17, 56) ecolor[e] = "#a40000" ewidth[e] = 5 eorder = g.edge_index.copy("int") gt.graph_draw(g, pos=g.vp.pos, vertex_color=color, vertex_fill_color=color, edge_color=ecolor, eorder=eorder, edge_pen_width=ewidth, output="football_missing.svg") .. figure:: football_missing.* :align: center :width: 350px Two non-existing edges in the football network (in red): :math:(101,102) in the middle, and :math:(17,56) in the upper right region of the figure. .. testsetup:: missing-edges gt.seed_rng(7) .. 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): 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)) probs[0].append(p1) probs[1].append(p2) # Now we collect the probabilities for exactly 100,000 sweeps gt.mcmc_equilibrate(state, force_niter=10000, mcmc_args=dict(niter=10), callback=collect_edge_probs) def get_avg(p): p = np.array(p) pmax = p.max() p -= pmax return pmax + log(exp(p).mean()) p1 = get_avg(probs[0]) p2 = get_avg(probs[1]) p_sum = get_avg([p1, p2]) + log(2) l1 = p1 - p_sum l2 = p2 - p_sum print("likelihood-ratio for %s: %g" % (missing_edges[0], exp(l1))) print("likelihood-ratio for %s: %g" % (missing_edges[1], exp(l2))) .. testoutput:: missing-edges likelihood-ratio for (101, 102): 0.37... likelihood-ratio for (17, 56): 0.62... From which we can conclude that edge :math:(17, 56) is more likely than :math:(101, 102) to be a missing edge. The prediction using the non-nested model can be performed in an entirely analogous fashion. References ---------- .. [peixoto-bayesian-2017] Tiago P. Peixoto, "Bayesian stochastic blockmodeling", :arxiv:1705.10225 .. [holland-stochastic-1983] Paul W. Holland, Kathryn Blackmond Laskey, Samuel Leinhardt, "Stochastic blockmodels: First steps", Social Networks Volume 5, Issue 2, Pages 109-137 (1983). :doi:10.1016/0378-8733(83)90021-7 .. [karrer-stochastic-2011] Brian Karrer, M. E. J. Newman "Stochastic blockmodels and community structure in networks", Phys. Rev. E 83, 016107 (2011). :doi:10.1103/PhysRevE.83.016107, :arxiv:1008.3926 .. [peixoto-nonparametric-2017] Tiago P. Peixoto, "Nonparametric Bayesian inference of the microcanonical stochastic block model", Phys. Rev. E 95 012317 (2017). :doi:10.1103/PhysRevE.95.012317, :arxiv:1610.02703 .. 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