# Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?

arXiv: https://arxiv.org/abs/2410.11443 · HTML: https://mlsanity.com/paper/2410.11443/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 82 references that are in the map.
- First posted on arXiv 15 Oct 2024.

## Builds on (most cited first, 30 of 51)

- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Learning from Protein Structure with Geometric Vector Perceptrons](https://mlsanity.com/paper/2009.01411/) (2020)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations](https://mlsanity.com/paper/2306.12059/) (2023)
- [Geometric Latent Diffusion Models for 3D Molecule Generation](https://mlsanity.com/paper/2305.01140/) (2023)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Crystal Structure Prediction by Joint Equivariant Diffusion](https://mlsanity.com/paper/2309.04475/) (2023)
- [On the Expressive Power of Geometric Graph Neural Networks](https://mlsanity.com/paper/2301.09308/) (2023)
- [On the Universality of Rotation Equivariant Point Cloud Networks](https://mlsanity.com/paper/2010.02449/) (2020)
- [SE(3) Equivariant Graph Neural Networks with Complete Local Frames](https://mlsanity.com/paper/2110.14811/) (2021)
- [Conditional Antibody Design as 3D Equivariant Graph Translation](https://mlsanity.com/paper/2208.06073/) (2022)
- [FAENet: Frame Averaging Equivariant GNN for Materials Modeling](https://mlsanity.com/paper/2305.05577/) (2023)
- [ComENet: Towards Complete and Efficient Message Passing for 3D Molecular Graphs](https://mlsanity.com/paper/2206.08515/) (2022)
- [Equivariant Graph Mechanics Networks with Constraints](https://mlsanity.com/paper/2203.06442/) (2022)
- [Equivariant Flows: sampling configurations for multi-body systems with symmetric energies](https://mlsanity.com/paper/1910.00753/) (2019)
- [Molecule Generation by Principal Subgraph Mining and Assembling](https://mlsanity.com/paper/2106.15098/) (2021)

## Built on by (most cited first, 18 of 18)

- [The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products](https://mlsanity.com/paper/2506.13523/) (2025)
- [Relaxed Equivariance via Multitask Learning](https://mlsanity.com/paper/2410.17878/) (2024)
- [Siamese Foundation Models for Crystal Structure Prediction](https://mlsanity.com/paper/2503.10471/) (2025)
- [Fast and Distributed Equivariant Graph Neural Networks by Virtual Node Learning](https://mlsanity.com/paper/2506.19482/) (2025)
- [DualEquiNet: A Dual-Space Hierarchical Equivariant Network for Large Biomolecules](https://mlsanity.com/paper/2506.19862/) (2025)
- [Learning Inter-Atomic Potentials without Explicit Equivariance](https://mlsanity.com/paper/2510.00027/) (2025)
- [Geometric Hyena Networks for Large-scale Equivariant Learning](https://mlsanity.com/paper/2505.22560/) (2025)
- [Tokenizing Electron Cloud in Protein-Ligand Interaction Learning](https://mlsanity.com/paper/2505.19014/) (2025)
- [Equivariant Spherical Transformer for Efficient Molecular Modeling](https://mlsanity.com/paper/2505.23086/) (2025)
- [Universally Invariant Learning in Equivariant GNNs](https://mlsanity.com/paper/2510.13169/) (2025)
- [PAINET: A Principled Efficient Transformer for 3D Dynamics Modeling](https://mlsanity.com/paper/2510.04233/) (2025)
- [Geometric Mixture Models for Electrolyte Conductivity Prediction](https://mlsanity.com/paper/2510.15403/) (2025)
- [A Conformation-Centric Generative Foundation Model for Linear Polymer Modeling and Design](https://mlsanity.com/paper/2510.16023/) (2025)
- [GDEGAN: Gaussian Dynamic Equivariant Graph Attention Network for Ligand Binding Site Prediction](https://mlsanity.com/paper/2603.19817/) (2026)
- [ElemeNet: Multiscale Molecular Machine Learning with Uncertainty Quantification Across the Periodic Table](https://mlsanity.com/paper/2606.30961/) (2026)
- [Reducing Symmetry Increase in Equivariant Neural Networks](https://mlsanity.com/paper/2608.12010/) (2026)
- [Equivariant Sheaf Neural Networks: Learning Geometric Transport on Graphs](https://mlsanity.com/paper/2608.28853/) (2026)
- [Why Multi-Layer Message Passing Works: Completeness Theory for Graph Neural Network Interatomic Potentials](https://mlsanity.com/paper/2609.00528/) (2026)

## Abstract (arXiv)

Equivariant Graph Neural Networks (GNNs) that incorporate E(3) symmetry have achieved significant success in various scientific applications. As one of the most successful models, EGNN leverages a simple scalarization technique to perform equivariant message passing over only Cartesian vectors (i.e., 1st-degree steerable vectors), enjoying greater efficiency and efficacy compared to equivariant GNNs using higher-degree steerable vectors. This success suggests that higher-degree representations might be unnecessary. In this paper, we disprove this hypothesis by exploring the expressivity of equivariant GNNs on symmetric structures, including k-fold rotations and regular polyhedra. We theoretically demonstrate that equivariant GNNs will always degenerate to a zero function if the degree of the output representations is fixed to 1 or other specific values. Based on this theoretical insight, we propose HEGNN, a high-degree version of EGNN to increase the expressivity by incorporating high-degree steerable vectors while maintaining EGNN's efficiency through the scalarization trick. Our extensive experiments demonstrate that HEGNN not only aligns with our theoretical analyses on toy datasets consisting of symmetric structures, but also shows substantial improvements on more complicated datasets such as N-body and MD17. Our theoretical findings and empirical results potentially open up new possibilities for the research of equivariant GNNs.