Latent stochastic block models are flexible statistical models that are widely used in social network analysis. In recent years, efforts have been made to extend these models to temporal dynamic networks, whereby the connections between nodes are observed at a number of different times. In this paper we extend the original stochastic block model by using a Markovian property to describe the evolution of nodes' cluster memberships over time. We recast the problem of clustering the nodes of the network into a model-based context, and show that the integrated completed likelihood can be evaluated analytically for a number of likelihood models. Then, we propose a scalable greedy algorithm to maximise this quantity, thereby estimating both the optimal partition and the ideal number of groups in a single inferential framework. Finally we propose applications of our methodology to both real and artificial datasets.
We consider the hypothesis testing problem that two vertices $i$ and $j$ of a generalized random dot product graph have the same latent positions, possibly up to scaling. Special cases of this hypothesis test include testing whether two vertices in a stochastic block model or degree-corrected stochastic block model graph have the same block membership vectors, or testing whether two vertices in a popularity adjusted block model have the same community assignment. We propose several test statistics based on the empirical Mahalanobis distances between the $i$th and $j$th rows of either the adjac…
Modeling relations between individuals is a classical question in social sciences, ecology, etc. In order to uncover a latent structure in the data, a popular approach consists in clustering individuals according to the observed patterns of interactions. To do so, Stochastic block models (SBM) and Latent Block models (LBM) are standard tools for clustering the individuals with respect to their comportment in a unique network. However, when adopting an integrative point of view, individuals are not involved in a unique network but are part of several networks, resulting into a potentially compl…
The graphon (W-graph), including the stochastic block model as a special case, has been widely used in modeling and analyzing network data. This random graph model is well-characterized by its graphon function, and estimation of the graphon function has gained a lot of recent research interests. Most existing works focus on community detection in the latent space of the model, while adopting simple maximum likelihood or Bayesian estimates for the graphon or connectivity parameters given the identified communities. In this work, we propose a hierarchical Binomial model and develop a novel empir…
In cluster analysis interest lies in probabilistically capturing partitions of individuals, items or observations into groups, such that those belonging to the same group share similar attributes or relational profiles. Bayesian posterior samples for the latent allocation variables can be effectively obtained in a wide range of clustering models, including finite mixtures, infinite mixtures, hidden Markov models and block models for networks. However, due to the categorical nature of the clustering variables and the lack of scalable algorithms, summary tools that can interpret such samples are…
The latent stochastic block model is a flexible and widely used statistical model for the analysis of network data. Extensions of this model to a dynamic context often fail to capture the persistence of edges in contiguous network snapshots. The recently introduced stochastic block transition model addresses precisely this issue, by modelling the probabilities of creating a new edge and of maintaining an edge over time. Using a model-based clustering approach, this paper illustrates a methodology to fit stochastic block transition models under a Bayesian framework. The method relies on a greed…
Mixture models are probabilistic models aimed at uncovering and representing latent subgroups within a population. In the realm of network data analysis, the latent subgroups of nodes are typically identified by their connectivity behaviour, with nodes behaving similarly belonging to the same community. In this context, mixture modelling is pursued through stochastic blockmodelling. We consider stochastic blockmodels and some of their variants and extensions from a mixture modelling perspective. We also survey some of the main classes of estimation methods available, and propose an alternative…
We propose a mixture of latent trait models with common slope parameters (MCLT) for model-based clustering of high-dimensional binary data, a data type for which few established methods exist. Recent work on clustering of binary data, based on a $d$-dimensional Gaussian latent variable, is extended by incorporating common factor analyzers. Accordingly, our approach facilitates a low-dimensional visual representation of the clusters. We extend the model further by the incorporation of random block effects. The dependencies in each block are taken into account through block-specific parameters t…
In the framework of model-based clustering, a model allowing several latent class variables is proposed. This model assumes that the distribution of the observed data can be factorized into several independent blocks of variables. Each block is assumed to follow a latent class model ({\it i.e.,} mixture with conditional independence assumption). The proposed model includes variable selection, as a special case, and is able to cope with the mixed-data setting. The simplicity of the model allows to estimate the repartition of the variables into blocks and the mixture parameters simultaneously, t…
Predictive modelling of multivariate data where both the covariates and responses are high-dimensional is becoming an increasingly popular task in many data mining applications. Partial Least Squares (PLS) regression often turns out to be a useful model in these situations since it performs dimensionality reduction by assuming the existence of a small number of latent factors that may explain the linear dependence between input and output. In practice, the number of latent factors to be retained, which controls the complexity of the model and its predictive ability, has to be carefully selecte…
We present a selective review of statistical modeling of dynamic networks. We focus on models with latent variables, specifically, the latent space models and the latent class models (or stochastic blockmodels), which investigate both the observed features and the unobserved structure of networks. We begin with an overview of the static models, and then we introduce the dynamic extensions. For each dynamic model, we also discuss its applications that have been studied in the literature, with the data source listed in Appendix. Based on the review, we summarize a list of open problems and chall…
Observations consisting of measurements on relationships for pairs of objects arise in many settings, such as protein interaction and gene regulatory networks, collections of author-recipient email, and social networks. Analyzing such data with probabilisic models can be delicate because the simple exchangeability assumptions underlying many boilerplate models no longer hold. In this paper, we describe a latent variable model of such data called the mixed membership stochastic blockmodel. This model extends blockmodels for relational data to ones which capture mixed membership latent relationa…
A recent literature in econometrics models unobserved cross-sectional heterogeneity in panel data by assigning each cross-sectional unit a one-dimensional, discrete latent type. Such models have been shown to allow estimation and inference by regression clustering methods. This paper is motivated by the finding that the clustered heterogeneity models studied in this literature can be badly misspecified, even when the panel has significant discrete cross-sectional structure. To address this issue, we generalize previous approaches to discrete unobserved heterogeneity by allowing each unit to ha…
Fudan University（复旦大学）；Shanghai Center for Mathematical Sciences（上海数学中心）；University of Pennsylvania（宾夕法尼亚大学）；Shanghai University of Finance and Economics（上海财经大学）
We show that a simple community detection algorithm originated from stochastic blockmodel literature achieves consistency, and even optimality, for a broad and flexible class of sparse latent space models. The class of models includes latent eigenmodels (arXiv:0711.1146). The community detection algorithm is based on spectral clustering followed by local refinement via normalized edge counting.
Linköping University（林雪平大学）；Stockholm University（斯德哥尔摩大学）；University of Edinburgh（爱丁堡大学）
Dynamic transportation networks have been analyzed for years by means of static graph-based indicators in order to study the temporal evolution of relevant network components, and to reveal complex dependencies that would not be easily detected by a direct inspection of the data. This paper presents a state-of-the-art probabilistic latent network model to forecast multilayer dynamic graphs that are increasingly common in transportation and proposes a community-based extension to reduce the computational burden. Flexible time series analysis is obtained by modeling the probability of edges betw…
