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

2026-04-27

AI Summary

Existing approaches to modeling dynamic multilayer networks often suffer from restrictive assumptions such as temporal independence, single-layer structure, or strong stationarity, limiting their ability to capture complex temporal dependencies and structural evolution in real-world scenarios. This work proposes a first-order autoregressive multilayer stochastic block model (AR(1)-MSBM), which, for the first time, integrates an autoregressive mechanism into the multilayer stochastic block model framework, thereby enabling a unified treatment of both stationary and non-stationary dynamic networks. To facilitate efficient inference, we develop an adaptive sliding-window online estimation algorithm that combines recursive updates, tensor spectral refinement, and a quasi-stationary segmentation strategy to automatically detect and respond to unknown structural shifts—both abrupt and gradual. Theoretical analysis establishes optimal non-asymptotic estimation rates and community recovery guarantees, while experiments demonstrate the method’s effectiveness and superiority across diverse dynamic network 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.

Problem

• dynamic multilayer networks
• temporal dependence
• non-stationarity
• community detection
• stochastic block models

Innovation

• autoregressive multilayer stochastic block model
• online learning
• non-stationarity
• tensor spectral refinement
• adaptive windowing