Edge Estimation and Community Detection in Time-varying Networks

Abstract

In modern statistics and data science, there is a growing focus on network data that indicate interactions among a group of items in a complex system. Scientists are interested in these data as they can reveal important insights into the latent structure present among the nodes of a network. The emerging family of statistical methods effectively addresses these modeling demands in static networks. However, the evolving nature of network structures over time introduces unique challenges not present in static networks. Specifically, in dynamic networks, we want to characterize their smooth change which also controls the model complexity. To achieve this, we need to impose structural assumptions about the similarity of neighboring networks, and this usually will pose computational challenges.

This thesis studies three aspects of the statistical analysis in time-varying network problems. First, to identify the dynamic changes of associations among multivariate random variables, we propose a time-varying Gaussian graphical model with two different regularization methods imposed to characterize the smooth change of neighboring networks. These methods lead to non-trivial optimization problems that we solve by developing efficient computational methods based on the Alternating Direction Method of Multipliers algorithm. Second, given the observed time-varying financial relationships among nodes, such as their trading amounts in dollars, we propose new stochastic block models based on a restricted Tweedie distribution to accommodate non-negative continuous edge weights with a positive probability of zero counts. The model can capture dynamic nodal effects. We prove that the estimation of the dynamic covariate effects is asymptotically independent of assigned community labels, allowing for an efficient two-step algorithm. Third, when the timestamp of node interactions is accessible, we aim to enhance the modeling of the distribution of survival time of network interactions, especially in the presence of censoring. In addressing this, we employ Cox proportional hazard models to investigate the influence of community structures on the formation of networks. Overall, this thesis provides new methods for modeling and computing time-varying network problems.