Empirical Likelihood Test for Common Invariant Subspace of Multilayer Networks based on Monte Carlo Approximation

TL;DR

This paper introduces an empirical likelihood-based statistical test to identify a shared invariant subspace across multilayer networks, improving understanding of common structures in complex systems.

Contribution

It develops a novel EL-based hypothesis test for detecting shared subspaces in multilayer networks, with asymptotic analysis and simulation validation.

Findings

The test achieves satisfactory size and power in simulations.

Monte Carlo approximation effectively assesses the test's asymptotic behavior.

Application to real data demonstrates practical utility.

Abstract

Multilayer (or multiple) networks are widely used to represent diverse patterns of relationships among objects in increasingly complex real-world systems. Identifying a common invariant subspace across network layers has become an active area of research, as such a subspace can filter out layer-specific noise, facilitate cross-network comparisons, reduce dimensionality, and extract shared structural features of scientific interest. One statistical approach to detecting a common subspace is hypothesis testing, which evaluates whether the observed networks share a common latent structure. In this paper, we propose an empirical likelihood (EL) based test for this purpose. The null hypothesis states that all network layers share the same invariant subspace, whereas under the alternative hypothesis at least two layers differ in their subspaces. We study the asymptotic behavior of the proposed test via Monte Carlo approximation and assess its finite-sample performance through extensive simulations. The simulation results demonstrate that the proposed method achieves satisfactory size and power, and its practical utility is further illustrated with a real-data application.