  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.
