A General Latent Embedding Approach for Modeling Non-uniform High-dimensional Sparse Hypergraphs with Multiplicity

TL;DR

This paper introduces a versatile latent embedding method for modeling complex high-dimensional hypergraphs that can handle non-uniformity and repeated hyperlinks, with theoretical guarantees and practical validation.

Contribution

It presents a novel, general approach integrating latent embeddings and degree heterogeneity for hypergraphs, overcoming uniformity and repetition limitations of prior models.

Findings

Theoretical identifiability conditions established.

Convergence rates and asymptotic distributions derived.

Algorithm validated through simulations and real data.

Abstract

Recent research has shown growing interest in modeling hypergraphs, which capture polyadic interactions among entities beyond traditional dyadic relations. However, most existing methodologies for hypergraphs face significant limitations, including their heavy reliance on uniformity restrictions for hyperlink orders and their inability to account for repeated observations of identical hyperlinks. In this work, we introduce a novel and general latent embedding approach that addresses these challenges through the integration of latent embeddings, vertex degree heterogeneity parameters, and an order-adjusting parameter. Theoretically, we investigate the identifiability conditions for the latent embeddings and associated parameters, and we establish the convergence rates of their estimators along with asymptotic distributions. Computationally, we employ a projected gradient ascent algorithm for parameter estimation. Comprehensive simulation studies demonstrate the effectiveness of the algorithm and validate the theoretical findings. Moreover, an application to a co-citation hypergraph illustrates the advantages of the proposed method.