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

Qianqian Yao

TL;DR

Addresses testing whether multiple network layers share a common invariant subspace. Proposes an Empirical Likelihood (EL) test using weighted degree differences and calibrates its null distribution via Monte Carlo approximation. Demonstrates through simulations that the EL test achieves correct size and superior power relative to a baseline WDDT method and shows robustness under rank-2 settings, with a real-data CS-Aarhus application supporting practical utility. The approach provides a nonparametric tool for cross-layer structural inference in multilayer networks and motivates extensions to higher-rank subspaces.

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.

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

TL;DR

Addresses testing whether multiple network layers share a common invariant subspace. Proposes an Empirical Likelihood (EL) test using weighted degree differences and calibrates its null distribution via Monte Carlo approximation. Demonstrates through simulations that the EL test achieves correct size and superior power relative to a baseline WDDT method and shows robustness under rank-2 settings, with a real-data CS-Aarhus application supporting practical utility. The approach provides a nonparametric tool for cross-layer structural inference in multilayer networks and motivates extensions to higher-rank subspaces.

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.
Paper Structure (10 sections, 1 theorem, 44 equations, 11 figures, 32 tables)

This paper contains 10 sections, 1 theorem, 44 equations, 11 figures, 32 tables.

Table of Contents

  1. Introduction
  2. Model and Hypothesis
  3. Empirical Likelihood Test
  4. Monte Carlo Approximation of the Null Distribution
  5. Simulation Studies
  6. Simulation
  7. Factors that Impact Performance of EL Test
  8. Robustness of EL test under Rank-2 Multilayer Networks
  9. Real Data Application
  10. Discussion

Key Result

Theorem 3.2

Let $X_1, X_2, ..., X_n \in \mathbb{R}^d$, where $d \geq 1$, be independent random variables with a common distribution $F_0$ with mean $\mu_0$ and finite variance covariance matrix $V_0$ of rank $q>0$. Then $-2 \text{log} R_n$ converges in distribution to a $\chi_{(q)}^2$ random variable as $n \rig

Figures (11)

  • Figure 1: Network or Graph and its Adjacency Matrix: node size $n=50$
  • Figure 2: Five Layers Real-World CS-Aarhus Networks
  • Figure 3: 3 Layers Multilayer Networks: share common subspace
  • Figure 4: 3 Layers Multilayer Networks: do not share common subspace
  • Figure 5: Monte Carlo Approximation Under Null Hypothesis
  • ...and 6 more figures

Theorems & Definitions (8)

  • Definition 2.1: Random Heterogeneous Graph (Erdős--Rényi 1960; Bollobás et al. 2007)
  • Definition 2.2: COmmon Subspace Independent Edge graphs (Arroyo et al. 2021)
  • Definition 2.3: DIverse MultiPLEx Generalized Dot Product Graph (Pensky and Wang 2024)
  • Definition 2.4: Rank-1 Random Multilayer Heterogeneous Graphs (Yuan and Yao 2025)
  • Definition 3.1: Owen 1988; Owen 2001; Lazar 2021
  • Theorem 3.2: Owen 1988; Owen 2001; Lazar 2021
  • Example 3.1
  • Example 3.2