Modeling Hypergraphs with Diversity and Heterogeneous Popularity

TL;DR

This paper introduces a novel hypergraph modeling approach based on determinantal point processes that emphasizes diversity and node popularity, extending beyond similarity-driven models and accommodating complex hypergraph structures.

Contribution

It proposes a new latent space hypergraph model driven by diversity and popularity, with theoretical guarantees and an efficient estimation algorithm, applicable to broad hypergraph types.

Findings

Model captures hyperedge diversity effectively

Algorithm demonstrates strong performance in simulations

Application reveals meaningful ingredient embeddings

Abstract

While relations among individuals make an important part of data with scientific and business interests, existing statistical modeling of relational data has mainly been focusing on dyadic relations, i.e., those between two individuals. This article addresses the less studied, though commonly encountered, polyadic relations that can involve more than two individuals. In particular, we propose a new latent space model for hypergraphs using determinantal point processes, which is driven by the diversity within hyperedges and each node's popularity. This model mechanism is in contrast to existing hypergraph models, which are predominantly driven by similarity rather than diversity. Additionally, the proposed model accommodates broad types of hypergraphs, with no restriction on the cardinality and multiplicity of hyperedges, which previous models often have. Consistency and asymptotic normality of the maximum likelihood estimates of the model parameters have been established. The proof is challenging, owing to the special configuration of the parameter space. Further, we apply the projected accelerated gradient descent algorithm to obtain the parameter estimates, and we show its effectiveness in simulation studies. We also demonstrate an application of the proposed model on the What's Cooking data and present the embedding of food ingredients learned from cooking recipes using the model.