Bias-Corrected Joint Spectral Embedding for Multilayer Networks with Invariant Subspace: Entrywise Eigenvector Perturbation and Inference

TL;DR

This paper introduces a bias-corrected joint spectral embedding method for multilayer networks, providing theoretical guarantees for subspace estimation and enabling accurate community detection and hypothesis testing.

Contribution

It develops a novel bias correction algorithm with rigorous entrywise eigenvector analysis, advancing inference in multilayer network models.

Findings

Sharp entrywise perturbation bounds established

Eigenvector central limit theorem proved

Effective for community detection and membership testing

Abstract

In this paper, we propose to estimate the invariant subspace across heterogeneous multiple networks using a novel bias-corrected joint spectral embedding algorithm. The proposed algorithm recursively calibrates the diagonal bias of the sum of squared network adjacency matrices by leveraging the closed-form bias formula and iteratively updates the subspace estimator using the most recent estimated bias. Correspondingly, we establish a complete recipe for the entrywise subspace estimation theory for the proposed algorithm, including a sharp entrywise subspace perturbation bound and the entrywise eigenvector central limit theorem. Leveraging these results, we settle two multiple network inference problems: the exact community detection in multilayer stochastic block models and the hypothesis testing of the equality of membership profiles in multilayer mixed membership models. Our proof relies on delicate leave-one-out and leave-two-out analyses that are specifically tailored to block-wise symmetric random matrices and a martingale argument that is of fundamental interest for the entrywise eigenvector central limit theorem.