# Geometry-Aware Edge Pooling for Graph Neural Networks

arXiv: https://arxiv.org/abs/2506.11700 · HTML: https://mlsanity.com/paper/2506.11700/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 62 references that are in the map.
- First posted on arXiv 13 Jun 2025.

## Builds on (most cited first, 30 of 32)

- [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)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [Self-Attention Graph Pooling](https://mlsanity.com/paper/1904.08082/) (2019)
- [Diffusion Improves Graph Learning](https://mlsanity.com/paper/1911.05485/) (2019)
- [A Fair Comparison of Graph Neural Networks for Graph Classification](https://mlsanity.com/paper/1912.09893/) (2019)
- [Design Space for Graph Neural Networks](https://mlsanity.com/paper/2011.08843/) (2020)
- [Understanding Attention and Generalization in Graph Neural Networks](https://mlsanity.com/paper/1905.02850/) (2019)
- [ASAP: Adaptive Structure Aware Pooling for Learning Hierarchical Graph Representations](https://mlsanity.com/paper/1911.07979/) (2019)
- [Towards Sparse Hierarchical Graph Classifiers](https://mlsanity.com/paper/1811.01287/) (2018)
- [Rethinking pooling in graph neural networks](https://mlsanity.com/paper/2010.11418/) (2020)
- [Understanding Pooling in Graph Neural Networks](https://mlsanity.com/paper/2110.05292/) (2021)
- [Graph Neural Networks in TensorFlow and Keras with Spektral](https://mlsanity.com/paper/2006.12138/) (2020)
- [Edge Contraction Pooling for Graph Neural Networks](https://mlsanity.com/paper/1905.10990/) (2019)
- [Graph Pooling for Graph Neural Networks: Progress, Challenges, and Opportunities](https://mlsanity.com/paper/2204.07321/) (2022)
- [Metrics for Graph Comparison: A Practitioner's Guide](https://mlsanity.com/paper/1904.07414/) (2019)
- [The expressive power of pooling in Graph Neural Networks](https://mlsanity.com/paper/2304.01575/) (2023)
- [Hierarchical Representation Learning in Graph Neural Networks with Node Decimation Pooling](https://mlsanity.com/paper/1910.11436/) (2019)
- [Graph Neural Networks Use Graphs When They Shouldn't](https://mlsanity.com/paper/2309.04332/) (2023)
- [Distribution Knowledge Embedding for Graph Pooling](https://mlsanity.com/paper/2109.14333/) (2021)
- [Metric Space Magnitude for Evaluating the Diversity of Latent Representations](https://mlsanity.com/paper/2311.16054/) (2023)
- [No Metric to Rule Them All: Toward Principled Evaluations of Graph-Learning Datasets](https://mlsanity.com/paper/2502.02379/) (2025)
- [Metric Space Magnitude and Generalisation in Neural Networks](https://mlsanity.com/paper/2305.05611/) (2023)
- [A Non-Negative Factorization approach to node pooling in Graph Convolutional Neural Networks](https://mlsanity.com/paper/1909.03287/) (2019)
- [Topological Generalization Bounds for Discrete-Time Stochastic Optimization Algorithms](https://mlsanity.com/paper/2407.08723/) (2024)
- [Landmark Diffusion Maps (L-dMaps): Accelerated manifold learning out-of-sample extension](https://mlsanity.com/paper/1706.09396/) (2017)
- [Graph Pooling via Ricci Flow](https://mlsanity.com/paper/2407.04236/) (2024)
- [Boosting Graph Pooling with Persistent Homology](https://mlsanity.com/paper/2402.16346/) (2024)

## Built on by (most cited first, 5 of 5)

- [Learnable quantum spectral filters for hybrid graph neural networks](https://mlsanity.com/paper/2507.05640/) (2025)
- [Contraction and Hourglass Persistence for Learning on Graphs, Simplices, and Cells](https://mlsanity.com/paper/2604.17548/) (2026)
- [Geometric Flow enhanced Graph Coarsening](https://mlsanity.com/paper/2609.14962/) (2026)
- [Diversity Curves for Graph Representation Learning](https://mlsanity.com/paper/2605.06466/) (2026)
- [Train Small, Deploy Large: Zero-Shot GNN Transfer Through Geometric Renormalization](https://mlsanity.com/paper/2607.27767/) (2026)

## Abstract (arXiv)

Graph Neural Networks (GNNs) have shown significant success for graph-based tasks. Motivated by the prevalence of large datasets in real-world applications, pooling layers are crucial components of GNNs. By reducing the size of input graphs, pooling enables faster training and potentially better generalisation. However, existing pooling operations often optimise for the learning task at the expense of discarding fundamental graph structures, thus reducing interpretability. This leads to unreliable performance across dataset types, downstream tasks and pooling ratios. Addressing these concerns, we propose novel graph pooling layers for structure-aware pooling via edge collapses. Our methods leverage diffusion geometry and iteratively reduce a graph's size while preserving both its metric structure and its structural diversity. We guide pooling using magnitude, an isometry-invariant diversity measure, which permits us to control the fidelity of the pooling process. Further, we use the spread of a metric space as a faster and more stable alternative ensuring computational efficiency. Empirical results demonstrate that our methods (i) achieve top performance compared to alternative pooling layers across a range of diverse graph classification tasks, (ii) preserve key spectral properties of the input graphs, and (iii) retain high accuracy across varying pooling ratios.