Heterophily-Agnostic Hypergraph Neural Networks with Riemannian Local Exchanger

TL;DR

A novel Heat-Exchanger with Adaptive Locality for Hypergraph Neural Network (HealHGNN), designed as a node-hyperedge bidirectional systems with linear complexity in the number of nodes and hyperedges, enabling heterophily-agnostic message passing with theoretical guarantees.

Abstract

Hypergraphs are the natural description of higher-order interactions among objects, widely applied in social network analysis, cross-modal retrieval, etc. Hypergraph Neural Networks (HGNNs) have become the dominant solution for learning on hypergraphs. Traditional HGNNs are extended from message passing graph neural networks, following the homophily assumption, and thus struggle with the prevalent heterophilic hypergraphs that call for long-range dependence modeling. In this paper, we achieve heterophily-agnostic message passing through the lens of Riemannian geometry. The key insight lies in the connection between oversquashing and hypergraph bottleneck within the framework of Riemannian manifold heat flow. Building on this, we propose the novel idea of locally adapting the bottlenecks of different subhypergraphs. The core innovation of the proposed mechanism is the design of an adaptive local (heat) exchanger. Specifically, it captures the rich long-range dependencies via the Robin condition, and preserves the representation distinguishability via source terms, thereby enabling heterophily-agnostic message passing with theoretical guarantees. Based on this theoretical foundation, we present a novel Heat-Exchanger with Adaptive Locality for Hypergraph Neural Network (HealHGNN), designed as a node-hyperedge bidirectional systems with linear complexity in the number of nodes and hyperedges. Extensive experiments on both homophilic and heterophilic cases show that HealHGNN achieves the state-of-the-art performance.

Figures (6)

• Figure 1: (A). Traditional dynamics in HGNN. (B). Illustration of our adaptive heat exchanger; bd denotes boundary nodes.
• Figure 2: Overall architecture of the proposed heat kernel based hypergraph neural network.
• Figure 3: Illustration of the Topological Structures
• Figure 4: Performance comparison on LRGB tasks.
• Figure 5: Information Transfer Performance on the Four Hypergraph Types with Hyperedge Size 4.

Theorems & Definitions (4)

• theorem 1: Cheeger Inequality 24-hgtheory
• theorem 2: Spectral Gap and Dirichlet Energy, \ref{['prf:Spectral Gap and Dirichlet Energy']}
• theorem 3: Flexibility of the Robin Condition, \ref{['prf:flexibility of rc']}
• theorem 4: Mode Preservation under Source Energy Injection, \ref{['prf:source']}