# A Theoretical Formulation of Many-body Message Passing Neural Networks

arXiv: https://arxiv.org/abs/2407.11756 · HTML: https://mlsanity.com/paper/2407.11756/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 10 references that are in the map.
- First posted on arXiv 16 Jul 2024.

## Builds on (most cited first, 9 of 9)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [FastGCN: Fast Learning with Graph Convolutional Networks via Importance Sampling](https://mlsanity.com/paper/1801.10247/) (2018)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [A Survey on Oversmoothing in Graph Neural Networks](https://mlsanity.com/paper/2303.10993/) (2023)
- [On Over-Squashing in Message Passing Neural Networks: The Impact of Width, Depth, and Topology](https://mlsanity.com/paper/2302.02941/) (2023)
- [Understanding Oversquashing in GNNs through the Lens of Effective Resistance](https://mlsanity.com/paper/2302.06835/) (2023)
- [MotifNet: a motif-based Graph Convolutional Network for directed graphs](https://mlsanity.com/paper/1802.01572/) (2018)

## Abstract (arXiv)

We present many-body Message Passing Neural Network (MPNN) framework that models higher-order node interactions (≥ 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.