Modeling Non-Uniform Hypergraphs Using Determinantal Point Processes

2025-09-15

AI Summary

This paper addresses the challenging problem of modeling real-world networks as non-uniform hypergraphs—where hyperedges exhibit variable cardinalities and must jointly capture both node similarity and diversity. We propose the first hypergraph generative model based on the asymmetric Determinantal Point Process (DPP), which employs an asymmetric kernel matrix to explicitly encode directional similarities and diversities among nodes. The model supports hyperedges of arbitrary size and possesses a closed-form probability mass function, backed by rigorous statistical guarantees: the maximum likelihood estimator is consistent and asymptotically normal. Methodologically, we formulate parameter estimation as a constrained maximum likelihood problem and develop a projection-adaptive gradient descent algorithm for efficient optimization. Extensive experiments on multiple real-world datasets demonstrate that our model significantly outperforms existing baselines in hyperedge prediction, validating its expressive power and practical efficacy.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Most statistical models for networks focus on pairwise interactions between nodes. However, many real-world networks involve higher-order interactions among multiple nodes, such as co-authors collaborating on a paper. Hypergraphs provide a natural representation for these networks, with each hyperedge representing a set of nodes. The majority of existing hypergraph models assume uniform hyperedges (i.e., edges of the same size) or rely on diversity among nodes. In this work, we propose a new hypergraph model based on non-symmetric determinantal point processes. The proposed model naturally accommodates non-uniform hyperedges, has tractable probability mass functions, and accounts for both node similarity and diversity in hyperedges. For model estimation, we maximize the likelihood function under constraints using a computationally efficient projected adaptive gradient descent algorithm. We establish the consistency and asymptotic normality of the estimator. Simulation studies confirm the efficacy of the proposed model, and its utility is further demonstrated through edge predictions on several real-world datasets.

Problem

• Modeling non-uniform hypergraphs with higher-order interactions
• Accommodating node similarity and diversity in hyperedges
• Developing tractable probability mass functions for estimation

Innovation

• Non-symmetric determinantal point processes model
• Accommodates non-uniform hyperedges with tractability
• Projected adaptive gradient descent algorithm