Dynamic clustering for heterophilic stochastic block models with time-varying node memberships

Summary We consider a time-ordered sequence of networks stemming from stochastic block models in which nodes gradually change their memberships over time, and no network at any single time point contains sufficient signal strength to recover its community structure. To estimate the time-varying community structure, we develop kernel-debiased sum of squares (KD-SoS), a method that performs spectral clustering after a debiased sum-of-squared aggregation of adjacency matrices. Our theory demonstrates, via a novel bias-variance decomposition, that KD-SoS achieves consistent community detection in each network, even for heterophilic networks, without requiring smoothness in the time-varying dynamics of between-community connectivities. We also prove the identifiability of aligning community structures across time based on how rapidly nodes change communities, and we develop a data-adaptive bandwidth-tuning procedure for KD-SoS. We demonstrate the utility and advantages of KD-SoS through simulations and a novel analysis of time-varying dynamics in gene coordination in the human developing brain.