Estimation and Statistical Inference for Generalized Multilayer Latent Space Model

TL;DR

This paper introduces a flexible latent space model for multilayer directed networks, enabling estimation and inference of latent positions and layer-specific structures, with proven consistency and asymptotic normality.

Contribution

It develops a novel unfolding and fusion estimation method for multilayer networks, addressing challenges in inference and establishing theoretical properties.

Findings

The method achieves consistent estimation of latent positions.

Asymptotic normality of estimators is established.

Practical utility demonstrated on real-world multilayer network data.

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.