# Understanding Pooling in Graph Neural Networks

arXiv: https://arxiv.org/abs/2110.05292 · HTML: https://mlsanity.com/paper/2110.05292/

## Key facts

- Cited by 69 papers in the map (data through 14 Oct 2026).
- Builds on 41 of its 70 references that are in the map.
- First posted on arXiv 11 Oct 2021.

## Builds on (most cited first, 30 of 41)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space](https://mlsanity.com/paper/1706.02413/) (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)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [Gated Graph Sequence Neural Networks](https://mlsanity.com/paper/1511.05493/) (2015)
- [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)
- [TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems](https://mlsanity.com/paper/1603.04467/) (2016)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Self-Attention Graph Pooling](https://mlsanity.com/paper/1904.08082/) (2019)
- [OctNet: Learning Deep 3D Representations at High Resolutions](https://mlsanity.com/paper/1611.05009/) (2016)
- [Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphs](https://mlsanity.com/paper/1704.02901/) (2017)
- [Diffusion-Convolutional Neural Networks](https://mlsanity.com/paper/1511.02136/) (2015)
- [Order Matters: Sequence to sequence for sets](https://mlsanity.com/paper/1511.06391/) (2015)
- [CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filters](https://mlsanity.com/paper/1705.07664/) (2017)
- [Universal Invariant and Equivariant Graph Neural Networks](https://mlsanity.com/paper/1905.04943/) (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)
- [SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernels](https://mlsanity.com/paper/1711.08920/) (2017)
- [ASAP: Adaptive Structure Aware Pooling for Learning Hierarchical Graph Representations](https://mlsanity.com/paper/1911.07979/) (2019)
- [On the Universality of Invariant Networks](https://mlsanity.com/paper/1901.09342/) (2019)
- [A Survey on The Expressive Power of Graph Neural Networks](https://mlsanity.com/paper/2003.04078/) (2020)
- [Towards Sparse Hierarchical Graph Classifiers](https://mlsanity.com/paper/1811.01287/) (2018)
- [Graph Convolutional Networks with EigenPooling](https://mlsanity.com/paper/1904.13107/) (2019)
- [Rethinking pooling in graph neural networks](https://mlsanity.com/paper/2010.11418/) (2020)
- [Graph Neural Networks in TensorFlow and Keras with Spektral](https://mlsanity.com/paper/2006.12138/) (2020)

## Built on by (most cited first, 30 of 69)

- [BrainGB: A Benchmark for Brain Network Analysis with Graph Neural Networks](https://mlsanity.com/paper/2204.07054/) (2022)
- [Ordered Subgraph Aggregation Networks](https://mlsanity.com/paper/2206.11168/) (2022)
- [Graph Pooling for Graph Neural Networks: Progress, Challenges, and Opportunities](https://mlsanity.com/paper/2204.07321/) (2022)
- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [The expressive power of pooling in Graph Neural Networks](https://mlsanity.com/paper/2304.01575/) (2023)
- [Graph Deep Learning for Time Series Forecasting](https://mlsanity.com/paper/2310.15978/) (2023)
- [LLM as GNN: Graph Vocabulary Learning for Text-Attributed Graph Foundation Models](https://mlsanity.com/paper/2503.03313/) (2025)
- [Graph-based Time Series Clustering for End-to-End Hierarchical Forecasting](https://mlsanity.com/paper/2305.19183/) (2023)
- [Graph-based Forecasting with Missing Data through Spatiotemporal Downsampling](https://mlsanity.com/paper/2402.10634/) (2024)
- [A Gromov--Wasserstein Geometric View of Spectrum-Preserving Graph Coarsening](https://mlsanity.com/paper/2306.08854/) (2023)
- [Understanding Virtual Nodes: Oversquashing and Node Heterogeneity](https://mlsanity.com/paper/2405.13526/) (2024)
- [Search to Fine-tune Pre-trained Graph Neural Networks for Graph-level Tasks](https://mlsanity.com/paper/2308.06960/) (2023)
- [Neural Bayes Estimators for Irregular Spatial Data using Graph Neural Networks](https://mlsanity.com/paper/2310.02600/) (2023)
- [Exact Computation of Any-Order Shapley Interactions for Graph Neural Networks](https://mlsanity.com/paper/2501.16944/) (2025)
- [Graph Pooling via Ricci Flow](https://mlsanity.com/paper/2407.04236/) (2024)
- [Heterogeneous Sheaf Neural Networks](https://mlsanity.com/paper/2409.08036/) (2024)
- [A Survey on Graph Condensation](https://mlsanity.com/paper/2402.02000/) (2024)
- [Boosting Graph Pooling with Persistent Homology](https://mlsanity.com/paper/2402.16346/) (2024)
- [Graph Coarsening with Message-Passing Guarantees](https://mlsanity.com/paper/2405.18127/) (2024)
- [Simplifying Clustering with Graph Neural Networks](https://mlsanity.com/paper/2207.08779/) (2022)
- [PatchGT: Transformer over Non-trainable Clusters for Learning Graph Representations](https://mlsanity.com/paper/2211.14425/) (2022)
- [Cluster-wise Graph Transformer with Dual-granularity Kernelized Attention](https://mlsanity.com/paper/2410.06746/) (2024)
- [SS-GNN: A Simple-Structured Graph Neural Network for Affinity Prediction](https://mlsanity.com/paper/2206.07015/) (2022)
- [Total Variation Graph Neural Networks](https://mlsanity.com/paper/2211.06218/) (2022)
- [A Comprehensive Graph Pooling Benchmark: Effectiveness, Robustness and Generalizability](https://mlsanity.com/paper/2406.09031/) (2024)
- [Pooling Strategies for Simplicial Convolutional Networks](https://mlsanity.com/paper/2210.05490/) (2022)
- [Adaptive Hierarchical SpatioTemporal Network for Traffic Forecasting](https://mlsanity.com/paper/2306.09386/) (2023)
- [MaxCutPool: differentiable feature-aware Maxcut for pooling in graph neural networks](https://mlsanity.com/paper/2409.05100/) (2024)
- [Geometry-Aware Edge Pooling for Graph Neural Networks](https://mlsanity.com/paper/2506.11700/) (2025)
- [Graph State-Space Models and Latent Relational Inference](https://mlsanity.com/paper/2301.01741/) (2023)

## Abstract (arXiv)

Inspired by the conventional pooling layers in convolutional neural networks, many recent works in the field of graph machine learning have introduced pooling operators to reduce the size of graphs. The great variety in the literature stems from the many possible strategies for coarsening a graph, which may depend on different assumptions on the graph structure or the specific downstream task. In this paper we propose a formal characterization of graph pooling based on three main operations, called selection, reduction, and connection, with the goal of unifying the literature under a common framework. Following this formalization, we introduce a taxonomy of pooling operators and categorize more than thirty pooling methods proposed in recent literature. We propose criteria to evaluate the performance of a pooling operator and use them to investigate and contrast the behavior of different classes of the taxonomy on a variety of tasks.