Stochastic Block Model Based on Variational Inference and Its Extensions: An Application to Examine Global Migration Dynamics

Discrete latent variable models are widely used in statistics and related fields because they enable the formulation of flexible and interpretable models for analyzing data with complex dependence structures among variables. We focus specifically on stochastic block (SB) models based on discrete latent variables, which are widely used for modeling network data. We provide a review, first introducing SB model for simple graphs, namely undirected binary graphs without self-loops, which allows us to delve into advanced aspects of model formulation and estimation. In this context, the SB model assigns nodes to latent blocks, with the probability of an edge existing between nodes depending on block membership. We discuss key inferential aspects, including estimation of the parameters, conducted through a variational approximation of the expectation-maximization algorithm, prediction of the latent variable, and model selection. We also present several SB model extensions to realistically represent real-world networks, such as binary and weighted networks, dynamic networks, multiplex networks, bipartite and multipartite networks, and hypergraphs. Finally, to showcase the potential of the dynamic SB model, we illustrate an application aimed to identify groups of countries with similar migration flows, using total migrant stock data provided by the United Nations from 1990 to 2015.