Uncertainty quantification for mixed membership in multilayer networks with degree heterogeneity using Gaussian variational inference

TL;DR

This paper introduces a Bayesian Gaussian variational inference method for quantifying uncertainty in multilayer network community detection, accommodating heterogeneity and providing confidence sets with theoretical guarantees.

Contribution

It develops a scalable inference algorithm for the ML-DCMM model and establishes a Bernstein--von Mises theorem for variational posteriors in this context.

Findings

Robust community membership estimates on U.S. airport network

Effective uncertainty quantification with confidence sets

Competitive performance against existing methods

Abstract

Analyzing multilayer networks is central to understanding complex relational measurements collected across multiple conditions or over time. A pivotal task in this setting is to quantify uncertainty in community structure while appropriately pooling information across layers and accommodating layer-specific heterogeneity. Building on the multilayer degree-corrected mixed-membership (ML-DCMM) model, which captures both stable community membership profiles and layer-specific vertex activity levels, we propose a Bayesian inference framework based on a spectral-assisted likelihood. We then develop a computationally efficient Gaussian variational inference algorithm implemented via stochastic gradient descent. Our theoretical analysis establishes a variational Bernstein--von Mises theorem, which provides a frequentist guarantee for using the variational posterior to construct confidence sets for mixed memberships. We demonstrate the utility of the method on a U.S. airport longitudinal network, where the procedure yields robust estimates, natural uncertainty quantification, and competitive performance relative to state-of-the-art methods.