Estimation and Statistical Inference for Generalized Multilayer Latent Space Model

TL;DR

A novel unfolding and fusion method is developed to facilitate estimation and establish both consistency and asymptotic normality for the estimated latent positions and connection matrices, which paves the way for statistical inference tasks in multilayer network applications.

Abstract

Multilayer networks have become increasingly ubiquitous across diverse scientific fields, ranging from social sciences and biology to economics and international relations. Despite their broad applications, the inferential theory for multilayer networks remains underdeveloped. In this paper, we propose a flexible latent space model for multilayer directed networks with various edge types, where each node is assigned with two latent positions capturing sending and receiving behaviors, and each layer has a connection matrix governing the layer-specific structure. Through nonlinear link functions, the proposed model represents the structure of a multilayer network as a tensor, which admits a Tucker low-rank decomposition. This formulation poses significant challenges on the estimation and statistical inference for the latent positions and connection matrices, where existing techniques are inapplicable. To tackle this issue, a novel unfolding and fusion method is developed to facilitate estimation. We establish both consistency and asymptotic normality for the estimated latent positions and connection matrices, which paves the way for statistical inference tasks in multilayer network applications, such as constructing confidence regions for the latent positions and testing whether two network layers share the same structure. We validate the proposed method through extensive simulation studies and demonstrate its practical utility on real-world data.

Figures (4)

• Figure 1: Boxplots for the estimation errors, $\Delta\boldsymbol \Theta,~\Delta\boldsymbol \Phi,~\Delta\boldsymbol \Lambda$, over 200 independent experiments under different settings. Columns correspond to $\boldsymbol{\Theta}$, $\boldsymbol{\Phi}$, and $\{\boldsymbol \Lambda_t\}_{t=1}^T$, while rows correspond to Gaussian, Poisson, and Binary settings.
• Figure 2: Empirical distributions of $[\widehat{\boldsymbol \Theta} - \boldsymbol \Theta^*\mathbf R_1]_{1,1},[\widehat{\boldsymbol \Phi} - \boldsymbol \Phi^*\mathbf R_2]_{1,1}$ and $[\widehat{\boldsymbol \Lambda}_1 - \mathbf R_1\boldsymbol \Lambda_1\mathbf R_2]_{1,1}$ after standardization ($n=1600$, $T=100$). Gray curves represent the standard normal distribution.
• Figure 3: Visualizations of the $\{\widehat{\boldsymbol \theta}_i\}_{i=1}^n$. Panel (a) shows the first two dimensions and panel (b) shows the first and third dimension. Countries are colored according to region.
• Figure 4: Panels (a)--(b) show the sequence and first-order difference for $[\widehat{\boldsymbol \Lambda}_t]_{1,1}$, while panels (c)--(d) display those for $[\widehat{\boldsymbol \Lambda}_t]_{3,3}$, with two $95\%$-confidence intervals at each time point in the difference plots: the black intervals are the original $95\%$-confidence intervals while the gray intervals are Bonferroni-corrected ones. The red dots indicate statistically significant change points identified using the Bonferroni-corrected confidence intervals.

Theorems & Definitions (18)

• Remark 1
• Remark 2
• Remark 3
• Lemma 1
• Proposition 1
• Remark 4
• Remark 5
• Remark 6
• Theorem 1
• Remark 7