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A Theoretical Formulation of Many-body Message Passing Neural Networks

Jiatong Han

TL;DR

The paper introduces Many-Body MPNN, a theoretical framework that extends traditional message passing to higher-order interactions by using motif-based, localized spectral filters weighted by edge Ricci curvatures. It establishes permutation invariance, derives sensitivity and energy bounds, and proves scalability to deeper and wider networks. Empirical results on synthetic energy regression and heterophilic graph classification show that the approach achieves high Dirichlet energy growth and favorable performance, while acknowledging computational trade-offs compared to two-body baselines. The work provides a principled, energy-aware perspective on graph representation learning and suggests directions toward learnable embeddings and broader applicability in complex graph topologies.

Abstract

We present many-body Message Passing Neural Network (MPNN) framework that models higher-order node interactions ($\ge 2$ nodes). We model higher-order terms as tree-shaped motifs, comprising a central node with its neighborhood, and apply localized spectral filters on motif Laplacian, weighted by global edge Ricci curvatures. We prove our formulation is invariant to neighbor node permutation, derive its sensitivity bound, and bound the range of learned graph potential. We run regression on graph energies to demonstrate that it scales well with deeper and wider network topology, and run classification on synthetic graph datasets with heterophily and show its consistently high Dirichlet energy growth. We open-source our code at https://github.com/JThh/Many-Body-MPNN.

A Theoretical Formulation of Many-body Message Passing Neural Networks

TL;DR

The paper introduces Many-Body MPNN, a theoretical framework that extends traditional message passing to higher-order interactions by using motif-based, localized spectral filters weighted by edge Ricci curvatures. It establishes permutation invariance, derives sensitivity and energy bounds, and proves scalability to deeper and wider networks. Empirical results on synthetic energy regression and heterophilic graph classification show that the approach achieves high Dirichlet energy growth and favorable performance, while acknowledging computational trade-offs compared to two-body baselines. The work provides a principled, energy-aware perspective on graph representation learning and suggests directions toward learnable embeddings and broader applicability in complex graph topologies.

Abstract

We present many-body Message Passing Neural Network (MPNN) framework that models higher-order node interactions ( nodes). We model higher-order terms as tree-shaped motifs, comprising a central node with its neighborhood, and apply localized spectral filters on motif Laplacian, weighted by global edge Ricci curvatures. We prove our formulation is invariant to neighbor node permutation, derive its sensitivity bound, and bound the range of learned graph potential. We run regression on graph energies to demonstrate that it scales well with deeper and wider network topology, and run classification on synthetic graph datasets with heterophily and show its consistently high Dirichlet energy growth. We open-source our code at https://github.com/JThh/Many-Body-MPNN.
Paper Structure (37 sections, 6 theorems, 21 equations, 6 figures)

This paper contains 37 sections, 6 theorems, 21 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Potential function.
  4. Formulation of Many-body MPNN
  5. Many-Body Mixing Bound: Sensitivity Bound
  6. Properties of Many-Body MPNN
  7. Invariance Property
  8. Energy Bounds
  9. Complexity Bounds
  10. Experiments
  11. Experimental Settings.
  12. Regressing Synthetic Random Graph Energies
  13. Classifying Synthetic Heterophilic Graph Nodes
  14. Efficiency Benchmarks
  15. Discussion
  16. ...and 22 more sections

Key Result

Theorem 4.1

The sensitivity bound of a many-body MPNN with update function as defined in fm:expanded is given by where $\nu \ge 2$ is the correlation order, $r$ is the shortest distance between node $v$ and node $u$.

Figures (6)

  • Figure 1: Test MSE Losses on 100 synthetic random graphs. We show the impact of varying number of layers, hidden dimensions, and max correlation orders on the energy regression. The upper row is regressing graph energies emphasizing clustering, with the lower emphasizing node distances.
  • Figure 2: Test accuracy and Dirichlet energy growth of models over 300 epochs on a synthetic heterophilic graph.
  • Figure 3: Runtime for different models, averaged over 30 runs. The models have varying numbers of layers, 16 hidden dimensions, and 4 correlation orders. The support for the Chebyshev expansion is 4. The batch size is 4, and the number of epochs is 50. Benchmarked on one NVIDIA RTX 2080 Ti GPU.
  • Figure 4: Log-energy growth from $k$-body term component ($k \le 5$) of Many-body MPNN.
  • Figure 5: Test MSE on regressing the synthetic energies of regular-shaped graphs when varying the number of layers from 1 to 32 for different models. The graph energy functions are different between emphases on node distances and clustering and hence the two rows of plots only share the x-axis.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Theorem 4.1
  • Theorem 5.1
  • Theorem 5.2
  • Proposition 5.3
  • proof
  • proof
  • Lemma 1.1
  • proof
  • Corollary 1.2
  • proof