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How Particle System Theory Enhances Hypergraph Message Passing

Yixuan Ma, Kai Yi, Pietro Lio, Shi Jin, Yu Guang Wang

TL;DR

This paper addresses over-smoothing and heterophily in hypergraph neural networks by recasting hypergraph message passing as interacting-particle dynamics. It introduces Hypergraph Atomic Message Passing (HAMP) with two instantiations, HAMP-I (first-order) and HAMP-II (second-order), incorporating attractive and repulsive forces, Allen-Cahn damping, and optional Brownian noise to enable deep, stable propagation. The authors prove a positive lower bound on the hypergraph Dirichlet energy and establish L2-type separability results, ensuring anti-over-smoothing behavior, while empirical results on nine real-world hypergraph benchmarks demonstrate competitive performance, particularly on heterophilic data. The framework unifies diffusion perspectives with particle dynamics, offering a physically interpretable, scalable approach to modeling high-order interactions and uncertainty in complex systems, with potential for broader applications in biology and beyond.

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.

How Particle System Theory Enhances Hypergraph Message Passing

TL;DR

This paper addresses over-smoothing and heterophily in hypergraph neural networks by recasting hypergraph message passing as interacting-particle dynamics. It introduces Hypergraph Atomic Message Passing (HAMP) with two instantiations, HAMP-I (first-order) and HAMP-II (second-order), incorporating attractive and repulsive forces, Allen-Cahn damping, and optional Brownian noise to enable deep, stable propagation. The authors prove a positive lower bound on the hypergraph Dirichlet energy and establish L2-type separability results, ensuring anti-over-smoothing behavior, while empirical results on nine real-world hypergraph benchmarks demonstrate competitive performance, particularly on heterophilic data. The framework unifies diffusion perspectives with particle dynamics, offering a physically interpretable, scalable approach to modeling high-order interactions and uncertainty in complex systems, with potential for broader applications in biology and beyond.

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.

Paper Structure

This paper contains 45 sections, 9 theorems, 58 equations, 5 figures, 9 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hypergraphs.
  4. Message Passing in Hypergraphs.
  5. Neural ODEs.
  6. Hypergraph Message Passing based on Particle Dynamics
  7. Fields and Hypergraphs.
  8. Interaction Force.
  9. Damping Force.
  10. Hypergraph Dynamics.
  11. First-order ODEs.
  12. Second-order ODEs.
  13. From ODEs to SDEs.
  14. Hypergraph Message Passing.
  15. Scale Translation of Hypergraph: Diffusion and Particle Dynamics
  16. ...and 30 more sections

Key Result

Proposition 5.2

For Eq. eq:1st order, suppose the above assumptions are satisfied. Define the mean value $\mathbf{\Bar{x}} := \frac{1}{N}\sum\limits_{i=1}^N \mathbf{x}_i$, and the second moments $M_2(\mathbf{x}):= \sum \limits_{i=1}^N \mathbf{x}_i^2.$ Then for sufficiently large $N_1,N_2$, there exist constants $\l with a positive constant $\mu$, where $\widehat{M}_2(t):=M_2(\mathbf{x}^{(1)}(t))+M_2(\mathbf{x}^{(

Figures (5)

  • Figure 1: An illustration for HAMP framework. The property $\mathbf{p}$ can account for feature or velocity.
  • Figure 2: An empirical analysis of the depth-accuracy correlation in deep neural networks. The shaded area represents the standard deviation, helping to show the range of accuracy fluctuations.
  • Figure 3: Significance plot for noise on Senate dataset.
  • Figure 4: The t-SNE visualization of vertex representation evolution of HAMP-I (the first row) and HAMP-II (the second row) on Congress dataset. The colors represent the class labels.
  • Figure 5: Time-Memory tradeoff analysis of different methods on Walmart dataset. SD denotes the standard deviation of time consumption.

Theorems & Definitions (13)

  • Definition 5.1
  • Proposition 5.2: $L_2$ separation of HAMP-I
  • Proposition 5.3: $L_2$ separation of HAMP-II, fang2019emergent
  • Proposition 5.4: Lower bound of the Dirichlet energy
  • Lemma B.1: $L_2$ estimate for $M_2$
  • Proposition B.2
  • Lemma B.3
  • Lemma B.4
  • Proposition B.5: $L_2$ separation of HAMP-I
  • Remark B.6
  • ...and 3 more