# Equivariant Subgraph Aggregation Networks

arXiv: https://arxiv.org/abs/2110.02910 · HTML: https://mlsanity.com/paper/2110.02910/

## Key facts

- Cited by 154 papers in the map (data through 14 Oct 2026).
- Builds on 63 of its 105 references that are in the map.
- First posted on arXiv 6 Oct 2021.

## Builds on (most cited first, 30 of 63)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Representation Learning on Graphs with Jumping Knowledge Networks](https://mlsanity.com/paper/1806.03536/) (2018)
- [Automatic chemical design using a data-driven continuous representation of molecules](https://mlsanity.com/paper/1610.02415/) (2016)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [Graph Neural Networks for Social Recommendation](https://mlsanity.com/paper/1902.07243/) (2019)
- [DropEdge: Towards Deep Graph Convolutional Networks on Node Classification](https://mlsanity.com/paper/1907.10903/) (2019)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (2020)
- [Junction Tree Variational Autoencoder for Molecular Graph Generation](https://mlsanity.com/paper/1802.04364/) (2018)
- [GraphSAINT: Graph Sampling Based Inductive Learning Method](https://mlsanity.com/paper/1907.04931/) (2019)
- [Cluster-GCN: An Efficient Algorithm for Training Deep and Large Graph Convolutional Networks](https://mlsanity.com/paper/1905.07953/) (2019)
- [Principal Neighbourhood Aggregation for Graph Nets](https://mlsanity.com/paper/2004.05718/) (2020)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Image Generation from Scene Graphs](https://mlsanity.com/paper/1804.01622/) (2018)
- [Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting](https://mlsanity.com/paper/2006.09252/) (2020)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Diffusion-Convolutional Neural Networks](https://mlsanity.com/paper/1511.02136/) (2015)

## Built on by (most cited first, 30 of 154)

- [Sign and Basis Invariant Networks for Spectral Graph Representation Learning](https://mlsanity.com/paper/2202.13013/) (2022)
- [From Stars to Subgraphs: Uplifting Any GNN with Local Structure Awareness](https://mlsanity.com/paper/2110.03753/) (2021)
- [Graph Inductive Biases in Transformers without Message Passing](https://mlsanity.com/paper/2305.17589/) (2023)
- [Weisfeiler and Leman go Machine Learning: The Story so far](https://mlsanity.com/paper/2112.09992/) (2021)
- [Rethinking the Expressive Power of GNNs via Graph Biconnectivity](https://mlsanity.com/paper/2301.09505/) (2023)
- [Understanding and Extending Subgraph GNNs by Rethinking Their Symmetries](https://mlsanity.com/paper/2206.11140/) (2022)
- [Universal Prompt Tuning for Graph Neural Networks](https://mlsanity.com/paper/2209.15240/) (2022)
- [SPECTRE: Spectral Conditioning Helps to Overcome the Expressivity Limits of One-shot Graph Generators](https://mlsanity.com/paper/2204.01613/) (2022)
- [How Powerful are K-hop Message Passing Graph Neural Networks](https://mlsanity.com/paper/2205.13328/) (2022)
- [Edge Directionality Improves Learning on Heterophilic Graphs](https://mlsanity.com/paper/2305.10498/) (2023)
- [Graph Mamba: Towards Learning on Graphs with State Space Models](https://mlsanity.com/paper/2402.08678/) (2024)
- [A Complete Expressiveness Hierarchy for Subgraph GNNs via Subgraph Weisfeiler-Lehman Tests](https://mlsanity.com/paper/2302.07090/) (2023)
- [Theory of Graph Neural Networks: Representation and Learning](https://mlsanity.com/paper/2204.07697/) (2022)
- [Ordered Subgraph Aggregation Networks](https://mlsanity.com/paper/2206.11168/) (2022)
- [Boosting the Cycle Counting Power of Graph Neural Networks with I2-GNNs](https://mlsanity.com/paper/2210.13978/) (2022)
- [Rethinking Tokenizer and Decoder in Masked Graph Modeling for Molecules](https://mlsanity.com/paper/2310.14753/) (2023)
- [A Theoretical Comparison of Graph Neural Network Extensions](https://mlsanity.com/paper/2201.12884/) (2022)
- [Message passing all the way up](https://mlsanity.com/paper/2202.11097/) (2022)
- [Linkless Link Prediction via Relational Distillation](https://mlsanity.com/paper/2210.05801/) (2022)
- [Beyond Weisfeiler-Lehman: A Quantitative Framework for GNN Expressiveness](https://mlsanity.com/paper/2401.08514/) (2024)
- [Equivariant Polynomials for Graph Neural Networks](https://mlsanity.com/paper/2302.11556/) (2023)
- [SpeqNets: Sparsity-aware Permutation-equivariant Graph Networks](https://mlsanity.com/paper/2203.13913/) (2022)
- [Cooperative Graph Neural Networks](https://mlsanity.com/paper/2310.01267/) (2023)
- [The expressive power of pooling in Graph Neural Networks](https://mlsanity.com/paper/2304.01575/) (2023)
- [An Empirical Study of Realized GNN Expressiveness](https://mlsanity.com/paper/2304.07702/) (2023)
- [On the Stability of Expressive Positional Encodings for Graphs](https://mlsanity.com/paper/2310.02579/) (2023)
- [Path Neural Networks: Expressive and Accurate Graph Neural Networks](https://mlsanity.com/paper/2306.05955/) (2023)
- [WL meet VC](https://mlsanity.com/paper/2301.11039/) (2023)
- [Graph Positional Encoding via Random Feature Propagation](https://mlsanity.com/paper/2303.02918/) (2023)
- [Towards Better Out-of-Distribution Generalization of Neural Algorithmic Reasoning Tasks](https://mlsanity.com/paper/2211.00692/) (2022)

## Abstract (arXiv)

Message-passing neural networks (MPNNs) are the leading architecture for deep learning on graph-structured data, in large part due to their simplicity and scalability. Unfortunately, it was shown that these architectures are limited in their expressive power. This paper proposes a novel framework called Equivariant Subgraph Aggregation Networks (ESAN) to address this issue. Our main observation is that while two graphs may not be distinguishable by an MPNN, they often contain distinguishable subgraphs. Thus, we propose to represent each graph as a set of subgraphs derived by some predefined policy, and to process it using a suitable equivariant architecture. We develop novel variants of the 1-dimensional Weisfeiler-Leman (1-WL) test for graph isomorphism, and prove lower bounds on the expressiveness of ESAN in terms of these new WL variants. We further prove that our approach increases the expressive power of both MPNNs and more expressive architectures. Moreover, we provide theoretical results that describe how design choices such as the subgraph selection policy and equivariant neural architecture affect our architecture's expressive power. To deal with the increased computational cost, we propose a subgraph sampling scheme, which can be viewed as a stochastic version of our framework. A comprehensive set of experiments on real and synthetic datasets demonstrates that our framework improves the expressive power and overall performance of popular GNN architectures.