Model Selection for Degree-corrected Block Models

The proliferation of models for networks raises challenging problems of model selection: the data are sparse and globally dependent, and models are typically high-dimensional and have large numbers of latent variables. Together, these issues mean that the usual model-selection criteria do not work properly for networks. We illustrate these challenges, and show one way to resolve them, by considering the key network-analysis problem of dividing a graph into communities or blocks of nodes with homogeneous patterns of links to the rest of the network. The standard tool for doing this is the stochastic block model, under which the probability of a link between two nodes is a function solely of the blocks to which they belong. This imposes a homogeneous degree distribution within each block; this can be unrealistic, so degree-corrected block models add a parameter for each node, modulating its over-all degree. The choice between ordinary and degree-corrected block models matters because they make very different inferences about communities. We present the first principled and tractable approach to model selection between standard and degree-corrected block models, based on new large-graph asymptotics for the distribution of log-likelihood ratios under the stochastic block model, finding substantial departures from classical results for sparse graphs. We also develop linear-time approximations for log-likelihoods under both the stochastic block model and the degree-corrected model, using belief propagation. Applications to simulated and real networks show excellent agreement with our approximations. Our results thus both solve the practical problem of deciding on degree correction, and point to a general approach to model selection in network analysis

Bayesian stochastic blockmodels for community detection in networks and community-structured covariance selection

Networks have been widely used to describe interactions among objects in diverse fields. Given the interest in explaining a network by its structure, much attention has been drawn to finding clusters of nodes with dense connections within clusters but sparse connections between clusters. Such clusters are called communities, and identifying such clusters is known as community detection. Here, to perform community detection, I focus on stochastic blockmodels (SBM), a class of statistically-based generative models. I present a flexible SBM that represents different types of data as well as node attributes under a Bayesian framework. The proposed models explicitly capture community behavior by guaranteeing that connections are denser within communities than between communities. First, I present a degree-corrected SBM based on a logistic regression formulation to model binary networks. To fit the model, I obtain posterior samples via Gibbs sampling based on Polya-Gamma latent variables. I conduct inference based on a novel, canonically mapped centroid estimator that formally addresses label non-identifiability and captures representative community assignments. Next, to accommodate large-scale datasets, I further extend the degree-corrected SBM to a broader family of generalized linear models with group correction terms. To conduct exact inference efficiently, I develop an iteratively-reweighted least squares procedure that implicitly updates sufficient statistics on the network to obtain maximum a posteriori (MAP) estimators. I demonstrate the proposed model and estimation on simulated benchmark networks and various real-world datasets. Finally, I develop a Bayesian SBM for community-structured covariance selection. Here, I assume that the data at each node are Gaussian and a latent network where two nodes are not connected if their observations are conditionally independent given observations of other nodes. Under the context of biological and social applications, I expect that this latent network shows a block dependency structure that represents community behavior. Thus, to identify the latent network and detect communities, I propose a hierarchical prior in two levels: a spike-and-slab prior on off-diagonal entries of the concentration matrix for variable selection and a degree-corrected SBM to capture community behavior. I develop an efficient routine based on ridge regularization and MAP estimation to conduct inference

Model selection and hypothesis testing for large-scale network models with overlapping groups

The effort to understand network systems in increasing detail has resulted in a diversity of methods designed to extract their large-scale structure from data. Unfortunately, many of these methods yield diverging descriptions of the same network, making both the comparison and understanding of their results a difficult challenge. A possible solution to this outstanding issue is to shift the focus away from ad hoc methods and move towards more principled approaches based on statistical inference of generative models. As a result, we face instead the more well-defined task of selecting between competing generative processes, which can be done under a unified probabilistic framework. Here, we consider the comparison between a variety of generative models including features such as degree correction, where nodes with arbitrary degrees can belong to the same group, and community overlap, where nodes are allowed to belong to more than one group. Because such model variants possess an increasing number of parameters, they become prone to overfitting. In this work, we present a method of model selection based on the minimum description length criterion and posterior odds ratios that is capable of fully accounting for the increased degrees of freedom of the larger models, and selects the best one according to the statistical evidence available in the data. In applying this method to many empirical unweighted networks from different fields, we observe that community overlap is very often not supported by statistical evidence and is selected as a better model only for a minority of them. On the other hand, we find that degree correction tends to be almost universally favored by the available data, implying that intrinsic node proprieties (as opposed to group properties) are often an essential ingredient of network formation.Comment: 20 pages,7 figures, 1 tabl