Online Learning for Autoregressive Multilayer Stochastic Block Models under Stationarity and Non-Stationarity

Abstract: Dynamic multilayer networks arise in many applications where multiple types of relations among a common set of nodes evolve over time. Existing approaches often assume temporal independence, focus on single-layer networks or impose stationarity, limiting their applicability in practice. In this paper, we introduce a first-order autoregressive multilayer stochastic block model (AR(1)-MSBM), in which edge formation and dissolution probabilities between consecutive time points are determined by latent community memberships and shared across layers. Under stationarity, we propose an online estimation procedure based on recursive updates and tensor-based spectral refinement. We establish non-asymptotic estimation rates, prove their minimax optimality and derive guarantees for community recovery. We further consider a non-stationary setting that allows both abrupt changes and gradual shifts, and develop an adaptive windowed online algorithm that automatically adjusts to unknown structural changes. Under a quasi-stationary segmentation framework, we derive estimation and community recovery guarantees that match the stationary results when applied segmentwise. Our theoretical findings are supported by extensive numerical experiments, with code available online.