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

We consider a time-ordered sequence of networks stemming from stochastic block models where nodes gradually change 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 KD-SoS (kernel debiased sum-of-square), a method performing 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 of each network even when heterophilic networks do not require 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 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 the time-varying dynamics in gene coordination in the human developing brain system.