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Learning cross-layer dependence structure in multilayer networks

Jiaheng Li, Jonathan R. Stewart

TL;DR

The paper addresses learning cross-layer dependence in multilayer networks by introducing a separable framework that decouples network formation from layer formation, enabling seamless extension of single-layer models via Markov random fields without latent variables. It derives non-asymptotic, minimax-optimal error bounds for maximum likelihood estimators and provides non-asymptotic normal approximation results that underpin FDR-controlled model selection. The authors validate their theory through simulation studies and apply the method to Lazega's lawyers network, illustrating interpretable cross-layer effects and accurate recovery of the basis network. Overall, the work delivers scalable, theoretically grounded tools for inferring cross-layer dependence in multilayer networks, with practical impact on analyzing complex relational data.

Abstract

We propose a novel class of separable multilayer network models to capture cross-layer dependencies in multilayer networks, enabling the analysis of how interactions in one or more layers may influence interactions in other layers. Our approach separates the network formation process from the layer formation process, and is able to extend existing single-layer network models to multilayer network models that accommodate cross-layer dependence. We establish non-asymptotic and minimax-optimal error bounds for maximum likelihood estimators and demonstrate the convergence rate in scenarios of increasing parameter dimension. Additionally, we establish non-asymptotic error bounds for multivariate normal approximations and propose a model selection method that controls the false discovery rate. Simulation studies and an application to the Lazega lawyers network show that our framework and method perform well in realistic settings.

Learning cross-layer dependence structure in multilayer networks

TL;DR

The paper addresses learning cross-layer dependence in multilayer networks by introducing a separable framework that decouples network formation from layer formation, enabling seamless extension of single-layer models via Markov random fields without latent variables. It derives non-asymptotic, minimax-optimal error bounds for maximum likelihood estimators and provides non-asymptotic normal approximation results that underpin FDR-controlled model selection. The authors validate their theory through simulation studies and apply the method to Lazega's lawyers network, illustrating interpretable cross-layer effects and accurate recovery of the basis network. Overall, the work delivers scalable, theoretically grounded tools for inferring cross-layer dependence in multilayer networks, with practical impact on analyzing complex relational data.

Abstract

We propose a novel class of separable multilayer network models to capture cross-layer dependencies in multilayer networks, enabling the analysis of how interactions in one or more layers may influence interactions in other layers. Our approach separates the network formation process from the layer formation process, and is able to extend existing single-layer network models to multilayer network models that accommodate cross-layer dependence. We establish non-asymptotic and minimax-optimal error bounds for maximum likelihood estimators and demonstrate the convergence rate in scenarios of increasing parameter dimension. Additionally, we establish non-asymptotic error bounds for multivariate normal approximations and propose a model selection method that controls the false discovery rate. Simulation studies and an application to the Lazega lawyers network show that our framework and method perform well in realistic settings.
Paper Structure (25 sections, 10 theorems, 218 equations, 10 figures, 7 tables)

This paper contains 25 sections, 10 theorems, 218 equations, 10 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Modeling cross-layer dependence in multilayer networks
  3. Separable multilayer network models
  4. Example of a multilayer network with pairwise interactions
  5. Estimation of cross-layer dependence structure
  6. Error of the normal approximation and model selection
  7. Model selection via univariate testing with FDR control
  8. Simulation studies
  9. Consistency
  10. Multivariate normality and model selection
  11. Application
  12. Discussion
  13. Proof of Proposition \ref{['prop:inference']}
  14. Proof of Lemma \ref{['lem:min-eig']}
  15. Concentration inequalities for multilayer networks
  16. ...and 10 more sections

Key Result

lemma 1

Consider a family $\{\mathbb{P}_{\boldsymbol{\theta}} : \boldsymbol{\theta} \in \mathbb{R}^p\}$ of separable multilayer network models satisfying general_model and an observation $\bm{x} \in \mathbb{X}$ of $\bm{X}$. Let $(\bm{x}, \bm{y})$ be the concordant pair where $\bm{y}$ is given by Proposition Then there exists a $p \times p$ matrix $I(\boldsymbol{\theta})$ such that for all $\{i,j\} \subse

Figures (10)

  • Figure 1: Multilayer networks specified by three different basis network structures: the latent space model (LSM), the exponential random graph model (ERGM), and the stochastic block model (SBM).
  • Figure 2: The relative $\ell_2$-errors between $\widetilde{\boldsymbol{\theta}}$ and $\boldsymbol{\theta}^\star$ decrease as the number of activated dyads increases. Each box is created by 250 replicates of multilayer networks.
  • Figure 3: Q-Q plots and $p$-values of six components of $\widetilde{\boldsymbol{\theta}}$ estimated from 250 multilayer network samples at size 1000 on the dense Bernoulli basis network.
  • Figure 4: The coworker layer, the advice layer and the friendship layer of Lazega's corporate law partnership network.
  • Figure 5: Box-plot of reproduced statistics from 10 simulated samples using the MPLE obtained from the Lazega's lawyer network. Red dots are values of the observed sufficient statistics of Lazega's lawyer network.
  • ...and 5 more figures

Theorems & Definitions (10)

  • lemma 1
  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Corollary 2
  • Theorem 3
  • lemma 2
  • lemma 3
  • lemma 4
  • lemma 5