Time‐Varying Degree‐Corrected Stochastic Block Models

Recent interest has emerged in community detection for dynamic networks, which are observed along a trajectory of points in time. In this paper, we present a time‐varying (t‐v) degree‐corrected stochastic block model to fit a dynamic network which allows evolving heterogeneity in the degrees of nodes within a community over time. In order to aggregate network information over time via a sliding window, we propose a smoothing‐based method that simultaneously allows to recover the: (i) t‐v node connection probabilities; (ii) t‐v community memberships; and (iii) t‐v node degree parameters. As a novelty compared to existing literature, we provide asymptotic theory for all of these three goals, that is, we derive rates of consistency of our smooth estimators for degree parameters and communities using a time‐localized profile‐likelihood approach as well as strong consistency of community membership estimation at every time point. Extensive simulation studies and applications to two different real data sets complete our work.