mlsanityData to 7 Oct 2026

On the Expressive Power of Geometric Graph Neural Networks

arXiv 2301.09308 ↗First posted 23 Jan 2023cs.LG math.GR stat.MLcited by 89

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Abstract (arXiv)

The expressive power of Graph Neural Networks (GNNs) has been studied extensively through the Weisfeiler-Leman (WL) graph isomorphism test. However, standard GNNs and the WL framework are inapplicable for geometric graphs embedded in Euclidean space, such as biomolecules, materials, and other physical systems. In this work, we propose a geometric version of the WL test (GWL) for discriminating geometric graphs while respecting the underlying physical symmetries: permutations, rotation, reflection, and translation. We use GWL to characterise the expressive power of geometric GNNs that are invariant or equivariant to physical symmetries in terms of distinguishing geometric graphs. GWL unpacks how key design choices influence geometric GNN expressivity: (1) Invariant layers have limited expressivity as they cannot distinguish one-hop identical geometric graphs; (2) Equivariant layers distinguish a larger class of graphs by propagating geometric information beyond local neighbourhoods; (3) Higher order tensors and scalarisation enable maximally powerful geometric GNNs; and (4) GWL's discrimination-based perspective is equivalent to universal approximation. Synthetic experiments supplementing our results are available at urlhttps://github.com/chaitjo/geometric-gnn-dojo

Key facts. Cited by 89 papers in the map (data through 7 Oct 2026). Builds on 43 of its 83 references that are in the map. First posted on arXiv 23 Jan 2023.

Builds on

43 of 83 references.

References · 43

  1. 2018How Powerful are Graph Neural Networks?cited by 4,355
  2. 2017Neural Message Passing for Quantum Chemistrycited by 3,460
  3. 2019Fast Graph Representation Learning with PyTorch Geometriccited by 2,414
  4. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  5. 2021E(n) Equivariant Graph Neural Networkscited by 951
  6. 2018Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networkscited by 878
  7. 2020Learning to Simulate Complex Physics with Graph Networkscited by 867
  8. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
  9. 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 701
  10. 2020Benchmarking Graph Neural Networkscited by 672
  11. 2020On the Bottleneck of Graph Neural Networks and its Practical Implicationscited by 624
  12. 2020SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networkscited by 602
31 more
  1. 2020Directional Message Passing for Molecular Graphscited by 556
  2. 2020Principal Neighbourhood Aggregation for Graph Netscited by 477
  3. 2021Understanding over-squashing and bottlenecks on graphs via curvaturecited by 471
  4. 2022MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fieldscited by 467
  5. 2021Equivariant message passing for the prediction of tensorial properties and molecular spectracited by 438
  6. 2019Provably Powerful Graph Networkscited by 385
  7. 20183D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Datacited by 348
  8. 2021GemNet: Universal Directional Graph Neural Networks for Moleculescited by 342
  9. 2020Learning from Protein Structure with Geometric Vector Perceptronscited by 308
  10. 2019Cormorant: Covariant Molecular Neural Networkscited by 262
  11. 2019Revisiting Graph Neural Networks: All We Have is Low-Pass Filterscited by 260
  12. 2020The Open Catalyst 2020 (OC20) Dataset and Community Challengescited by 246
  13. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 225
  14. 2021Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networkscited by 219
  15. 2021Weisfeiler and Lehman Go Cellular: CW Networkscited by 218
  16. 2020Generalization and Representational Limits of Graph Neural Networkscited by 217
  17. 2019On the equivalence between graph isomorphism testing and function approximation with GNNscited by 172
  18. 2022e3nn: Euclidean Neural Networkscited by 171
  19. 2021Spherical Message Passing for 3D Graph Networkscited by 166
  20. 2022Protein Representation Learning by Geometric Structure Pretrainingcited by 161
  21. 2021Scalars are universal: Equivariant machine learning, structured like classical physicscited by 127
  22. 2021Weisfeiler and Leman go Machine Learning: The Story so farcited by 121
  23. 2022The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentialscited by 100
  24. 2019Graph Neural Networks for Decentralized Multi-Robot Path Planningcited by 84
  25. 2020On the Universality of Rotation Equivariant Point Cloud Networkscited by 80
  26. 2021SE(3) Equivariant Graph Neural Networks with Complete Local Framescited by 75
  27. 2022ComENet: Towards Complete and Efficient Message Passing for 3D Molecular Graphscited by 69
  28. 2022Benchmarking Graphormer on Large-Scale Molecular Modeling Datasetscited by 63
  29. 2022Theory of Graph Neural Networks: Representation and Learningcited by 60
  30. 2022Learning Hierarchical Protein Representations via Complete 3D Graph Networkscited by 54
  31. 2022Symmetry Group Equivariant Architectures for Physicscited by 23

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 22; 2024: 34; 2025: 25; 2026: 8.

89 papers.

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  1. 2023Attending to Graph Transformerscited by 81
  2. 2025All-atom Diffusion Transformers: Unified generative modelling of molecules and materialscited by 75
  3. 2023FAENet: Frame Averaging Equivariant GNN for Materials Modelingcited by 72
  4. 2023A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systemscited by 65
  5. 2023How does over-squashing affect the power of GNNs?cited by 53
  6. 2023A new perspective on building efficient and expressive 3D equivariant graph neural networkscited by 46
  7. 2023Geometry-Complete Diffusion for 3D Molecule Generation and Optimizationcited by 36
  8. 2024Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Productscited by 35
  9. 2023Neural Injective Functions for Multisets, Measures and Graphs via a Finite Witness Theoremcited by 33
  10. 2024A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applicationscited by 28
  11. 2023Learning Joint 2D & 3D Diffusion Models for Complete Molecule Generationcited by 27
  12. 2023Learning Probabilistic Symmetrization for Architecture Agnostic Equivariancecited by 25
38 more
  1. 2023Is Distance Matrix Enough for Geometric Deep Learning?cited by 22
  2. 2024E(n) Equivariant Topological Neural Networkscited by 22
  3. 2023On the Expressivity of Persistent Homology in Graph Learningcited by 18
  4. 2023gRNAde: Geometric Deep Learning for 3D RNA inverse designcited by 18
  5. 2023Generalist Equivariant Transformer Towards 3D Molecular Interaction Learningcited by 18
  6. 2024Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?cited by 18
  7. 2023Diffusion probabilistic models enhance variational autoencoder for crystal structure generative modelingcited by 17
  8. 2024Evaluating representation learning on the protein structure universecited by 16
  9. 2024TorchMD-Net 2.0: Fast Neural Network Potentials for Molecular Simulationscited by 15
  10. 2024Topological Neural Networks go Persistent, Equivariant, and Continuouscited by 15
  11. 2024VN-EGNN: E(3)-Equivariant Graph Neural Networks with Virtual Nodes Enhance Protein Binding Site Identificationcited by 14
  12. 2024Higher-Rank Irreducible Cartesian Tensors for Equivariant Message Passingcited by 13
  13. 2024Relaxed Equivariance via Multitask Learningcited by 13
  14. 2024Triplet Interaction Improves Graph Transformers: Accurate Molecular Graph Learning with Triplet Graph Transformerscited by 12
  15. 2024Euclidean Fast Attention -- Machine Learning Global Atomic Representations at Linear Costcited by 12
  16. 2024On the Completeness of Invariant Geometric Deep Learning Modelscited by 11
  17. 2023Complete Neural Networks for Complete Euclidean Graphscited by 10
  18. 2023Subequivariant Graph Reinforcement Learning in 3D Environmentscited by 10
  19. 2023Universal Representation of Permutation-Invariant Functions on Vectors and Tensorscited by 10
  20. 2024Weisfeiler Leman for Euclidean Equivariant Machine Learningcited by 10
  21. 2024AI-driven inverse design of materials: Past, present and futurecited by 10
  22. 2025Learning (Approximately) Equivariant Networks via Constrained Optimizationcited by 10
  23. 2023Three iterations of (d-1)-WL test distinguish non isometric clouds of d-dimensional pointscited by 9
  24. 2024A Characterization Theorem for Equivariant Networks with Point-wise Activationscited by 9
  25. 2025Kinetic Langevin Diffusion for Crystalline Materials Generationcited by 9
  26. 2024Beyond Sequence: Impact of Geometric Context for RNA Property Predictioncited by 8
  27. 2024FlowDock: Geometric Flow Matching for Generative Protein-Ligand Docking and Affinity Predictioncited by 8
  28. 2023Mixture of Weak & Strong Experts on Graphscited by 7
  29. 2024On the Expressive Power of Sparse Geometric MPNNscited by 7
  30. 2024Learning local equivariant representations for quantum operatorscited by 7
  31. 2026EquiformerV3: Scaling Efficient, Expressive, and General SE(3)-Equivariant Graph Attention Transformerscited by 7
  32. 2024On dimensionality of feature vectors in MPNNscited by 6
  33. 2024GeoMFormer: A General Architecture for Geometric Molecular Representation Learningcited by 6
  34. 2024Diss-l-ECT: Dissecting Graph Data with Local Euler Characteristic Transformscited by 6
  35. 2025Enhancing Graph Representation Learning with Localized Topological Featurescited by 6
  36. 2024Future Directions in the Theory of Graph Machine Learningcited by 5
  37. 2025DualEquiNet: A Dual-Space Hierarchical Equivariant Network for Large Biomoleculescited by 5
  38. 2025Learning Inter-Atomic Potentials without Explicit Equivariancecited by 5

50 most cited of 89 All 89 (JSON)