# Curvature-based Pooling within Graph Neural Networks

arXiv: https://arxiv.org/abs/2308.16516 · HTML: https://mlsanity.com/paper/2308.16516/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 47 references that are in the map.
- First posted on arXiv 31 Aug 2023.

## Builds on (most cited first, 24 of 24)

- [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)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [MoleculeNet: A Benchmark for Molecular Machine Learning](https://mlsanity.com/paper/1703.00564/) (2017)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [Measuring and Relieving the Over-smoothing Problem for Graph Neural Networks from the Topological View](https://mlsanity.com/paper/1909.03211/) (2019)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Self-Attention Graph Pooling](https://mlsanity.com/paper/1904.08082/) (2019)
- [Revisiting Graph Neural Networks: All We Have is Low-Pass Filters](https://mlsanity.com/paper/1905.09550/) (2019)
- [A Note on Over-Smoothing for Graph Neural Networks](https://mlsanity.com/paper/2006.13318/) (2020)
- [Two Sides of the Same Coin: Heterophily and Oversmoothing in Graph Convolutional Neural Networks](https://mlsanity.com/paper/2102.06462/) (2021)
- [ASAP: Adaptive Structure Aware Pooling for Learning Hierarchical Graph Representations](https://mlsanity.com/paper/1911.07979/) (2019)
- [Hierarchical Inter-Message Passing for Learning on Molecular Graphs](https://mlsanity.com/paper/2006.12179/) (2020)
- [Memory-Based Graph Networks](https://mlsanity.com/paper/2002.09518/) (2020)
- [The expressive power of pooling in Graph Neural Networks](https://mlsanity.com/paper/2304.01575/) (2023)
- [Towards Interpretable Sparse Graph Representation Learning with Laplacian Pooling](https://mlsanity.com/paper/1905.11577/) (2019)
- [Hierarchical Adaptive Pooling by Capturing High-order Dependency for Graph Representation Learning](https://mlsanity.com/paper/2104.05960/) (2021)
- [Learning Graph Pooling and Hybrid Convolutional Operations for Text Representations](https://mlsanity.com/paper/1901.06965/) (2019)
- [Clique pooling for graph classification](https://mlsanity.com/paper/1904.00374/) (2019)
- [Transforming PageRank into an Infinite-Depth Graph Neural Network](https://mlsanity.com/paper/2207.00684/) (2022)
- [Forecasting Unobserved Node States with spatio-temporal Graph Neural Networks](https://mlsanity.com/paper/2211.11596/) (2022)

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

- [Exposition on over-squashing problem on GNNs: Current Methods, Benchmarks and Challenges](https://mlsanity.com/paper/2311.07073/) (2023)
- [Graph Pooling via Ricci Flow](https://mlsanity.com/paper/2407.04236/) (2024)
- [Foundations and Frontiers of Graph Learning Theory](https://mlsanity.com/paper/2407.03125/) (2024)
- [Conditional Shift-Robust Conformal Prediction for Graph Neural Network](https://mlsanity.com/paper/2405.11968/) (2024)
- [A roadmap for curvature-based geometric data analysis and learning](https://mlsanity.com/paper/2510.22599/) (2025)

## Abstract (arXiv)

Over-squashing and over-smoothing are two critical issues, that limit the capabilities of graph neural networks (GNNs). While over-smoothing eliminates the differences between nodes making them indistinguishable, over-squashing refers to the inability of GNNs to propagate information over long distances, as exponentially many node states are squashed into fixed-size representations. Both phenomena share similar causes, as both are largely induced by the graph topology. To mitigate these problems in graph classification tasks, we propose CurvPool, a novel pooling method. CurvPool exploits the notion of curvature of a graph to adaptively identify structures responsible for both over-smoothing and over-squashing. By clustering nodes based on the Balanced Forman curvature, CurvPool constructs a graph with a more suitable structure, allowing deeper models and the combination of distant information. We compare it to other state-of-the-art pooling approaches and establish its competitiveness in terms of classification accuracy, computational complexity, and flexibility. CurvPool outperforms several comparable methods across all considered tasks. The most consistent results are achieved by pooling densely connected clusters using the sum aggregation, as this allows additional information about the size of each pool.