How Particle System Theory Enhances Hypergraph Message Passing

TL;DR

This paper introduces a hypergraph message passing framework inspired by particle systems, incorporating attraction, repulsion, and stochastic elements to improve deep message passing and handle complex higher-order relationships.

Contribution

It presents a novel particle system-inspired hypergraph message passing method that mitigates over-smoothing and captures complex interactions, with theoretical guarantees and empirical success.

Findings

Mitigates over-smoothing by maintaining hypergraph Dirichlet energy.

Achieves competitive performance on diverse hypergraph node classification tasks.

Enables deeper message passing through second-order system stability.

Abstract

Hypergraphs effectively model higher-order relationships in natural phenomena, capturing complex interactions beyond pairwise connections. We introduce a novel hypergraph message passing framework inspired by interacting particle systems, where hyperedges act as fields inducing shared node dynamics. By incorporating attraction, repulsion, and Allen-Cahn forcing terms, particles of varying classes and features achieve class-dependent equilibrium, enabling separability through the particle-driven message passing. We investigate both first-order and second-order particle system equations for modeling these dynamics, which mitigate over-smoothing and heterophily thus can capture complete interactions. The more stable second-order system permits deeper message passing. Furthermore, we enhance deterministic message passing with stochastic element to account for interaction uncertainties. We prove theoretically that our approach mitigates over-smoothing by maintaining a positive lower bound on the hypergraph Dirichlet energy during propagation and thus to enable hypergraph message passing to go deep. Empirically, our models demonstrate competitive performance on diverse real-world hypergraph node classification tasks, excelling on both homophilic and heterophilic datasets.