Structure-Preserving Embedding of Multi-layer Networks

Yaoming Zhen, Shirong Xu, Junhui Wang

Keywords: Community detection, latent space model, multi-layer network, network embedding, tensor decomposition

Abstract: This paper investigates structure-preserving embedding for multi-layer networks with community structure. We propose a novel generative tensor-based latent space model (TLSM) that allows heterogeneity among vertices. It embeds vertices into a low-dimensional latent space so that vertices within the same community are close to each other in the ambient space, and captures layer heterogeneity through a layer-effect factor matrix. With a general and flexible tensor decomposition on the expected network adjacency tensor, TLSM is dedicated to preserving the original vertex relations and layer-specific effects in the network embedding. An efficient alternative updating scheme is developed to estimate the model parameters and conduct community detection simultaneously. Theoretically, we establish the asymptotic consistencies of TLSM in terms of both multi-layer network estimation and community detection. The theoretical results are supported by extensive numerical experiments on both synthetic and real-life multi-layer networks.