# Covariant Compositional Networks For Learning Graphs

arXiv: https://arxiv.org/abs/1801.02144 · HTML: https://mlsanity.com/paper/1801.02144/

## Key facts

- Cited by 54 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jan 2018.

## Built on by (most cited first, 30 of 54)

- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [GemNet: Universal Directional Graph Neural Networks for Molecules](https://mlsanity.com/paper/2106.08903/) (2021)
- [Identity-aware Graph Neural Networks](https://mlsanity.com/paper/2101.10320/) (2021)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [Equivariant Subgraph Aggregation Networks](https://mlsanity.com/paper/2110.02910/) (2021)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Pre-training via Denoising for Molecular Property Prediction](https://mlsanity.com/paper/2206.00133/) (2022)
- [Weisfeiler and Leman go Machine Learning: The Story so far](https://mlsanity.com/paper/2112.09992/) (2021)
- [Understanding and Extending Subgraph GNNs by Rethinking Their Symmetries](https://mlsanity.com/paper/2206.11140/) (2022)
- [SE(3)-equivariant prediction of molecular wavefunctions and electronic densities](https://mlsanity.com/paper/2106.02347/) (2021)
- [Theory of Graph Neural Networks: Representation and Learning](https://mlsanity.com/paper/2204.07697/) (2022)
- [Simple GNN Regularisation for 3D Molecular Property Prediction & Beyond](https://mlsanity.com/paper/2106.07971/) (2021)
- [Equivariant Networks for Crystal Structures](https://mlsanity.com/paper/2211.15420/) (2022)
- [Multiresolution Equivariant Graph Variational Autoencoder](https://mlsanity.com/paper/2106.00967/) (2021)
- [Bispectral Neural Networks](https://mlsanity.com/paper/2209.03416/) (2022)
- [Equivariant Networks for Pixelized Spheres](https://mlsanity.com/paper/2106.06662/) (2021)
- [Going Deeper into Permutation-Sensitive Graph Neural Networks](https://mlsanity.com/paper/2205.14368/) (2022)
- [WGCN: Graph Convolutional Networks with Weighted Structural Features](https://mlsanity.com/paper/2104.14060/) (2021)
- [A Simple and Universal Rotation Equivariant Point-cloud Network](https://mlsanity.com/paper/2203.01216/) (2022)
- [Temporal Multiresolution Graph Neural Networks For Epidemic Prediction](https://mlsanity.com/paper/2205.14831/) (2022)
- [A Characterization Theorem for Equivariant Networks with Point-wise Activations](https://mlsanity.com/paper/2401.09235/) (2024)
- [E(3)-Equivariant Mesh Neural Networks](https://mlsanity.com/paper/2402.04821/) (2024)
- [Improving the Expressiveness of K-hop Message-Passing GNNs by Injecting Contextualized Substructure Information](https://mlsanity.com/paper/2406.19244/) (2024)
- [DE-HNN: An effective neural model for Circuit Netlist representation](https://mlsanity.com/paper/2404.00477/) (2024)
- [Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networks](https://mlsanity.com/paper/2409.11772/) (2024)
- [Predicting COVID-19 pandemic by spatio-temporal graph neural networks: A New Zealand's study](https://mlsanity.com/paper/2305.07731/) (2023)
- [MuGSI: Distilling GNNs with Multi-Granularity Structural Information for Graph Classification](https://mlsanity.com/paper/2406.19832/) (2024)
- [Equivariant and Invariant Reynolds Networks](https://mlsanity.com/paper/2110.08092/) (2021)
- [Learning Continuous Rotation Canonicalization with Radial Beam Sampling](https://mlsanity.com/paper/2206.10690/) (2022)
- [Multiresolution Graph Transformers and Wavelet Positional Encoding for Learning Hierarchical Structures](https://mlsanity.com/paper/2302.08647/) (2023)

## Abstract (arXiv)

Most existing neural networks for learning graphs address permutation invariance by conceiving of the network as a message passing scheme, where each node sums the feature vectors coming from its neighbors. We argue that this imposes a limitation on their representation power, and instead propose a new general architecture for representing objects consisting of a hierarchy of parts, which we call Covariant Compositional Networks (CCNs). Here, covariance means that the activation of each neuron must transform in a specific way under permutations, similarly to steerability in CNNs. We achieve covariance by making each activation transform according to a tensor representation of the permutation group, and derive the corresponding tensor aggregation rules that each neuron must implement. Experiments show that CCNs can outperform competing methods on standard graph learning benchmarks.