mlsanityData to 7 Oct 2026

Invariant and Equivariant Graph Networks

arXiv 1812.09902 ↗First posted 24 Dec 2018cs.LG stat.MLcited by 339

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Abstract (arXiv)

Invariant and equivariant networks have been successfully used for learning images, sets, point clouds, and graphs. A basic challenge in developing such networks is finding the maximal collection of invariant and equivariant linear layers. Although this question is answered for the first three examples (for popular transformations, at-least), a full characterization of invariant and equivariant linear layers for graphs is not known. In this paper we provide a characterization of all permutation invariant and equivariant linear layers for (hyper-)graph data, and show that their dimension, in case of edge-value graph data, is 2 and 15, respectively. More generally, for graph data defined on k-tuples of nodes, the dimension is the k-th and 2k-th Bell numbers. Orthogonal bases for the layers are computed, including generalization to multi-graph data. The constant number of basis elements and their characteristics allow successfully applying the networks to different size graphs. From the theoretical point of view, our results generalize and unify recent advancement in equivariant deep learning. In particular, we show that our model is capable of approximating any message passing neural network Applying these new linear layers in a simple deep neural network framework is shown to achieve comparable results to state-of-the-art and to have better expressivity than previous invariant and equivariant bases.

Key facts. Cited by 339 papers in the map (data through 7 Oct 2026). First posted on arXiv 24 Dec 2018.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 57; 2022: 66; 2023: 73; 2024: 63; 2025: 48; 2026: 32.

339 papers.

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  1. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  2. 2021Graph Neural Networks with Learnable Structural and Positional Representationscited by 326
  3. 2021Combinatorial optimization and reasoning with graph neural networkscited by 267
  4. 2021Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networkscited by 219
  5. 2021Weisfeiler and Lehman Go Cellular: CW Networkscited by 218
  6. 2021Identity-aware Graph Neural Networkscited by 197
  7. 2022Pure Transformers are Powerful Graph Learnerscited by 188
  8. 2021A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groupscited by 185
  9. 2022Sign and Basis Invariant Networks for Spectral Graph Representation Learningcited by 176
  10. 2021Frame Averaging for Invariant and Equivariant Network Designcited by 166
  11. 2021Equivariant Subgraph Aggregation Networkscited by 154
  12. 2021From Stars to Subgraphs: Uplifting Any GNN with Local Structure Awarenesscited by 154
38 more
  1. 2021Nested Graph Neural Networkscited by 136
  2. 2021Scalars are universal: Equivariant machine learning, structured like classical physicscited by 127
  3. 2021Weisfeiler and Leman go Machine Learning: The Story so farcited by 121
  4. 2022Understanding and Extending Subgraph GNNs by Rethinking Their Symmetriescited by 112
  5. 2023Equivariant Architectures for Learning in Deep Weight Spacescited by 106
  6. 2022Approximately Equivariant Networks for Imperfectly Symmetric Dynamicscited by 95
  7. 2022Equivariant and Stable Positional Encoding for More Powerful Graph Neural Networkscited by 94
  8. 2022A Generalization of ViT/MLP-Mixer to Graphscited by 93
  9. 2022How Powerful are K-hop Message Passing Graph Neural Networkscited by 90
  10. 2021GraphEBM: Molecular Graph Generation with Energy-Based Modelscited by 85
  11. 2021The Logic of Graph Neural Networkscited by 84
  12. 2021Size-Invariant Graph Representations for Graph Classification Extrapolationscited by 82
  13. 2022Neo-GNNs: Neighborhood Overlap-aware Graph Neural Networks for Link Predictioncited by 78
  14. 2021SE(3) Equivariant Graph Neural Networks with Complete Local Framescited by 75
  15. 2021Residual Pathway Priors for Soft Equivariance Constraintscited by 74
  16. 2024PirateNets: Physics-informed Deep Learning with Residual Adaptive Networkscited by 70
  17. 2023Permutation Equivariant Neural Functionalscited by 69
  18. 2024Polynormer: Polynomial-Expressive Graph Transformer in Linear Timecited by 64
  19. 2021Universal Approximation of Functions on Setscited by 62
  20. 2022Theory of Graph Neural Networks: Representation and Learningcited by 60
  21. 2023On the Connection Between MPNN and Graph Transformercited by 60
  22. 2022Interpretable Graph Neural Networks for Connectome-Based Brain Disorder Analysiscited by 58
  23. 2022Boosting the Cycle Counting Power of Graph Neural Networks with I2-GNNscited by 55
  24. 2023Graph Metanetworks for Processing Diverse Neural Architecturescited by 55
  25. 2022A Short Tutorial on The Weisfeiler-Lehman Test And Its Variantscited by 52
  26. 2022Theoretical Guarantees for Permutation-Equivariant Quantum Neural Networkscited by 52
  27. 2021Graph Neural Networks with Local Graph Parameterscited by 51
  28. 2022Message passing all the way upcited by 47
  29. 2021Walking Out of the Weisfeiler Leman Hierarchy: Graph Learning Beyond Message Passingcited by 46
  30. 2022Equivariant Hypergraph Diffusion Neural Operatorscited by 45
  31. 2023Approximation-Generalization Trade-offs under (Approximate) Group Equivariancecited by 45
  32. 2023Towards Understanding the Generalization of Graph Neural Networkscited by 41
  33. 2022Representation Theory for Geometric Quantum Machine Learningcited by 40
  34. 2021Autobahn: Automorphism-based Graph Neural Netscited by 39
  35. 2021Lipschitz Normalization for Self-Attention Layers with Application to Graph Neural Networkscited by 38
  36. 2023Equivariant Polynomials for Graph Neural Networkscited by 38
  37. 2021Category Theory in Machine Learningcited by 36
  38. 2021Top-N: Equivariant set and graph generation without exchangeabilitycited by 36

50 most cited of 339 All 339 (JSON)