A Theoretical Formulation of Many-body Message Passing Neural Networks

TL;DR

This paper introduces a novel many-body message passing neural network framework that models complex higher-order node interactions using spectral filters on motif Laplacians, ensuring permutation invariance and demonstrating scalability and effectiveness on graph regression and classification tasks.

Contribution

The paper presents a new theoretical formulation for many-body MPNNs that captures higher-order interactions with spectral filtering, invariance properties, and bounds, advancing graph neural network modeling.

Findings

Scales well with deeper and wider networks.

Achieves high Dirichlet energy growth on synthetic datasets.

Demonstrates effectiveness in graph regression and classification.

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.