Bayesian Methods for Graph Clustering

A principled approach to characterize the hidden structure of networks is to formulate generative models and then infer their parameters from data. When the desired structure is composed of modules or "communities," a suitable choice for this task is the stochastic block model (SBM), where nodes are divided into groups, and the placement of edges is conditioned on the group memberships. Here, we present a nonparametric Bayesian method to infer the modular structure of empirical networks, including the number of modules and their hierarchical organization. We focus on a microcanonical variant of the SBM, where the structure is imposed via hard constraints, i.e., the generated networks are not allowed to violate the patterns imposed by the model. We show how this simple model variation allows simultaneously for two important improvements over more traditional inference approaches: (1) deeper Bayesian hierarchies, with noninformative priors replaced by sequences of priors and hyperpriors, which not only remove limitations that seriously degrade the inference on large networks but also reveal structures at multiple scales; (2) a very efficient inference algorithm that scales well not only for networks with a large number of nodes and edges but also with an unlimited number of modules. We show also how this approach can be used to sample modular hierarchies from the posterior distribution, as well as to perform model selection. We discuss and analyze the differences between sampling from the posterior and simply finding the single parameter estimate that maximizes it. Furthermore, we expose a direct equivalence between our microcanonical approach and alternative derivations based on the canonical SBM.

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“…This equivalence means that other Bayesian approaches such as Refs. [14][15][16][17][20][21][22], and those based on MDL, e.g. Refs.…”

Section: Nonparametric Bayesian Inferencementioning

confidence: 99%

Nonparametric Bayesian inference of the microcanonical stochastic block model

2017

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“…Other recent attempts to extend blockmodels take the flavor of mixture models that allow vertices to participate in overlapping groups [13] or to have mixed membership [14,15].…”

Section: Introductionmentioning

confidence: 99%

Stochastic blockmodels and community structure in networks

2011

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Stochastic blockmodels have been proposed as a tool for detecting community structure in networks as well as for generating synthetic networks for use as benchmarks. Most blockmodels, however, ignore variation in vertex degree, making them unsuitable for applications to real-world networks, which typically display broad degree distributions that can significantly distort the results. Here we demonstrate how the generalization of blockmodels to incorporate this missing element leads to an improved objective function for community detection in complex networks. We also propose a heuristic algorithm for community detection using this objective function or its non-degree-corrected counterpart and show that the degree-corrected version dramatically outperforms the uncorrected one in both real-world and synthetic networks.

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“…The models of the former case are mixtures of random graphs and are called Stochastic Block Models. Several statistical tools exist to infer their parameters Hofman and Wiggins, 2008;Latouche, Birmelé and Ambroise, 2008;Nowicki and Snijders, 2001). However, the stationarity of those models in terms of degree distribution is in our context a drawback as a high order theme involving vertices of small degree should be more significant than a high order theme involving hubs.…”

Section: The Random Graph Modelmentioning

confidence: 95%

“…It corresponds to Erdős-Rényi random graph model; Mixer the vertex classes and connectivity matrix of a Stochastic Block Model are inferred using the R-package mixer. It allows to choose between the classification method of Zanghi, Ambroise and Miele (2008), the variational frequentist method of Daudin, Picard and Robin (2008) and the bayesian procedure of (Latouche, Birmelé and Ambroise, 2008). Those three methods are of increasing precision for the parameter estimation of the model, but also of increasing running time; BLOCKS the vertex classes and connectivity matrix of a Stochastic Block Model are inferred using the procedure BLOCKS (Nowicki and Snijders, 2001) available in the STOCNET software (http://stat.gamma.rug.nl/ stocnet/).…”

Section: Stability With Respect To the Random Model Estimationmentioning

confidence: 99%

Detecting local network motifs

Birmelé¹

2012

Electron. J. Statist.

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Studying the structure of so-called real networks, that is networks obtained from sociological or biological data for instance, has become a major field of interest in the last decade. One way to deal with it is to consider that networks are partially built from small functional units called motifs, which can be found by looking for small subgraphs whose numbers of occurrences in the whole network are surprisingly high. In this article, we propose to define motifs through a local over-representation in the network and develop a statistic to detect them without relying on simulations. We then illustrate the performance of our procedure on simulated and real data, recovering already known biologically relevant motifs. Moreover, we explain how our method gives some information about the respective roles of the vertices in a motif. Detecting local network motifs 909 among all graphs having the same degree distribution as the network of interest. The stub-rewiring algorithm introduced by Milo et al. (2002) is used to sample under this model in several methods for motif detection, including those of Kashani et al. (2009); Kashtan et al. (2004); Wernicke and Rasche (2006), the method based on graph alignments of Berg and Lässig (2004) and the method devoted to labeled graphs of Banks et al. (2008). The earliest implementation of this method by Kashtan et al. (2004) was incorrect as the sampling was nonuniform (Wernicke and Rasche, 2006) but it was subsequently corrected. However, Artzy-Randrup et al. (2004) point out that it does not take into account the preferential links between some vertices and the high local density, which are two major features of biological networks. They show on a toy-example that the use of the stub-rewiring model and of a Z-score to detect motifs selects over-represented patterns in randomly chosen networks when the null model generates spatially clustered networks.Another way to define the null model is to consider a random graph model defined by a probability distribution. Literature about random graph models and their suitability to real networks is abundant (see e.g. Chung and Lu, 2006). Exponential family models have been developed to take the counting of small subgraphs into account (Holland and Leinhardt, 1970;Hunter and Handcock, 2006), but simulations are needed to compute normalization constants and the law of the motif counts seems to be analytically untractable. Another widely studied family of models are block models, which allow different link probabilities between classes of vertices and thus model the heterogeneity of connection patterns. Moreover, Picard et al. (2008) show that mean and variance calculation for the pattern counts are tractable under such models.Whatever the choice of the null model, the exact distributions of subgraph counts are unknown. Most of the detection algorithms assume a normal distribution and use the Z-score to compute p-values. However, Janson (1990) show that even asymptotically this assumption can be wrong. In the case of mixture models, Picard e...

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Bayesian Methods for Graph Clustering

Cited by 21 publications

References 16 publications

Nonparametric Bayesian inference of the microcanonical stochastic block model

Nonparametric Bayesian inference of the microcanonical stochastic block model

Stochastic blockmodels and community structure in networks

Detecting local network motifs

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