# A Flexible, Equivariant Framework for Subgraph GNNs via Graph Products and Graph Coarsening

arXiv: https://arxiv.org/abs/2406.09291 · HTML: https://mlsanity.com/paper/2406.09291/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 41 references that are in the map.
- First posted on arXiv 13 Jun 2024.

## Builds on (most cited first, 30 of 36)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [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)
- [Strategies for Pre-training Graph Neural Networks](https://mlsanity.com/paper/1905.12265/) (2019)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Automatic chemical design using a data-driven continuous representation of molecules](https://mlsanity.com/paper/1610.02415/) (2016)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (2020)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [Residual Gated Graph ConvNets](https://mlsanity.com/paper/1711.07553/) (2017)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [ASAM: Adaptive Sharpness-Aware Minimization for Scale-Invariant Learning of Deep Neural Networks](https://mlsanity.com/paper/2102.11600/) (2021)
- [Long Range Graph Benchmark](https://mlsanity.com/paper/2206.08164/) (2022)
- [Universal Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/1905.04943/) (2019)
- [From Stars to Subgraphs: Uplifting Any GNN with Local Structure Awareness](https://mlsanity.com/paper/2110.03753/) (2021)
- [Equivariant Subgraph Aggregation Networks](https://mlsanity.com/paper/2110.02910/) (2021)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Nested Graph Neural Networks](https://mlsanity.com/paper/2110.13197/) (2021)
- [Weisfeiler and Leman go Machine Learning: The Story so far](https://mlsanity.com/paper/2112.09992/) (2021)
- [DropGNN: Random Dropouts Increase the Expressiveness of Graph Neural Networks](https://mlsanity.com/paper/2111.06283/) (2021)
- [Understanding and Extending Subgraph GNNs by Rethinking Their Symmetries](https://mlsanity.com/paper/2206.11140/) (2022)
- [Expressive Power of Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/2006.15646/) (2020)
- [Small ReLU networks are powerful memorizers: a tight analysis of memorization capacity](https://mlsanity.com/paper/1810.07770/) (2018)
- [Reconstruction for Powerful Graph Representations](https://mlsanity.com/paper/2110.00577/) (2021)
- [A Complete Expressiveness Hierarchy for Subgraph GNNs via Subgraph Weisfeiler-Lehman Tests](https://mlsanity.com/paper/2302.07090/) (2023)
- [Ordered Subgraph Aggregation Networks](https://mlsanity.com/paper/2206.11168/) (2022)

## Built on by (most cited first, 8 of 8)

- [Balancing Efficiency and Expressiveness: Subgraph GNNs with Walk-Based Centrality](https://mlsanity.com/paper/2501.03113/) (2025)
- [Demystifying Higher-Order Graph Neural Networks](https://mlsanity.com/paper/2406.12841/) (2024)
- [Spectral Graph Neural Networks are Incomplete on Graphs with a Simple Spectrum](https://mlsanity.com/paper/2506.05530/) (2025)
- [On topological descriptors for graph products](https://mlsanity.com/paper/2511.08846/) (2025)
- [Heat Kernel Goes Topological](https://mlsanity.com/paper/2507.12380/) (2025)
- [On the Expressive Power of GNNs to Solve Linear SDPs](https://mlsanity.com/paper/2604.27786/) (2026)
- [On The Expressive Power of GNN Derivatives](https://mlsanity.com/paper/2510.02565/) (2025)
- [Explaining Synergistic Effects in Social Recommendations](https://mlsanity.com/paper/2601.18151/) (2026)

## Abstract (arXiv)

Subgraph GNNs enhance message-passing GNNs expressivity by representing graphs as sets of subgraphs, demonstrating impressive performance across various tasks. However, their scalability is hindered by the need to process large numbers of subgraphs. While previous approaches attempted to generate smaller subsets of subgraphs through random or learnable sampling, these methods often yielded suboptimal selections or were limited to small subset sizes, ultimately compromising their effectiveness. This paper introduces a new Subgraph GNN framework to address these issues. Our approach diverges from most previous methods by associating subgraphs with node clusters rather than with individual nodes. We show that the resulting collection of subgraphs can be viewed as the product of coarsened and original graphs, unveiling a new connectivity structure on which we perform generalized message passing. Crucially, controlling the coarsening function enables meaningful selection of any number of subgraphs. In addition, we reveal novel permutation symmetries in the resulting node feature tensor, characterize associated linear equivariant layers, and integrate them into our Subgraph GNN. We also introduce novel node marking strategies and provide a theoretical analysis of their expressive power and other key aspects of our approach. Extensive experiments on multiple graph learning benchmarks demonstrate that our method is significantly more flexible than previous approaches, as it can seamlessly handle any number of subgraphs, while consistently outperforming baseline approaches. Our code is available at https://github.com/BarSGuy/Efficient-Subgraph-GNNs.