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

TL;DR

This paper introduces an online learning framework for dynamic multilayer networks modeled by autoregressive stochastic block models, capable of handling both stationary and non-stationary changes with theoretical guarantees.

Contribution

It proposes a novel AR(1)-MSBM model with recursive estimation and spectral refinement, and develops adaptive algorithms for non-stationary environments with provable guarantees.

Findings

Establishes minimax optimal non-asymptotic estimation rates.

Provides guarantees for community recovery under stationarity.

Develops adaptive algorithms for structural changes in non-stationary settings.

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.