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 challenges in dynamic network modeling with latent variables.

Graphical model has been widely used to investigate the complex dependence structure of high-dimensional data, and it is common to assume that observed data follow a homogeneous graphical model. However, observations usually come from different resources and have heterogeneous hidden commonality in real-world applications. Thus, it is of great importance to estimate heterogeneous dependencies and discover subpopulation with certain commonality across the whole population. In this work, we introduce a novel regularized estimation scheme for learning nonparametric mixture of Gaussian graphical models, which extends the methodology and applicability of Gaussian graphical models and mixture models. We propose a unified penalized likelihood approach to effectively estimate nonparametric functional parameters and heterogeneous graphical parameters. We further design an efficient generalized effective EM algorithm to address three significant challenges: high-dimensionality, non-convexity, and label switching. Theoretically, we study both the algorithmic convergence of our proposed algorithm and the asymptotic properties of our proposed estimators. Numerically, we demonstrate the performance of our method in simulation studies and a real application to estimate human brain functional connectivity from ADHD imaging data, where two heterogeneous conditional dependencies are explained through profiling demographic variables and supported by existing scientific findings.

Dynamic networks are a general language for describing time-evolving complex systems, and discrete time network models provide an emerging statistical technique for various applications. It is a fundamental research question to detect the community structure in time-evolving networks. However, due to significant computational challenges and difficulties in modeling communities of time-evolving networks, there is little progress in the current literature to effectively find communities in time-evolving networks. In this work, we propose a novel model-based clustering framework for time-evolving networks based on discrete time exponential-family random graph models. To choose the number of communities, we use conditional likelihood to construct an effective model selection criterion. Furthermore, we propose an efficient variational expectation-maximization (EM) algorithm to find approximate maximum likelihood estimates of network parameters and mixing proportions. By using variational methods and minorization-maximization (MM) techniques, our method has appealing scalability for large scale time-evolving networks. The power of our method is demonstrated in simulation studies and empirical applications to international trade networks and the collaboration networks of a large American research university.

Graphical models have received an increasing amount of attention in network psychometrics as a promising probabilistic approach to study the conditional relations among variables using graph theory. Despite recent advances, existing methods on graphical models usually assume a homogeneous population and focus on binary or continuous variables. However, ordinal variables are very popular in many areas of psychological science, and the population often consists of several different groups based on the heterogeneity in ordinal data. Driven by these needs, we introduce the finite mixture of ordinal graphical models to effectively study the heterogeneous conditional dependence relationships of ordinal data. We develop a penalized likelihood approach for model estimation, and design a generalized expectation-maximization (EM) algorithm to solve the significant computational challenges. We examine the performance of the proposed method and algorithm in simulation studies. Moreover, we demonstrate the potential usefulness of the proposed method in psychological science through a real application concerning the interests and attitudes related to fan avidity for students in a large public university in the United States.