Structural Entropy Guided Graph Hierarchical Pooling
Abstract
Following the success of convolution on non-Euclidean space, the corresponding pooling approaches have also been validated on various tasks regarding graphs. However, because of the fixed compression quota and stepwise pooling design, these hierarchical pooling methods still suffer from local structure damage and suboptimal problem. In this work, inspired by structural entropy, we propose a hierarchical pooling approach, SEP, to tackle the two issues. Specifically, without assigning the layer-specific compression quota, a global optimization algorithm is designed to generate the cluster assignment matrices for pooling at once. Then, we present an illustration of the local structure damage from previous methods in the reconstruction of ring and grid synthetic graphs. In addition to SEP, we further design two classification models, SEP-G and SEP-N for graph classification and node classification, respectively. The results show that SEP outperforms state-of-the-art graph pooling methods on graph classification benchmarks and obtains superior performance on node classifications.
Benchmark Results
Tested on 10 datasets 42 baseline values reported too low
Tables show presented results in global context (best 3 papers + GCN + MLP).
Classic
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | CNA | 0.9575
±0.0058 | |
| 2 | IS-GIB | 0.9390
±0.0187 | |
| 3 | EERM | 0.9112
±0.0145 | |
| 20 | APPNP | 0.8142
±0.0026 best reported | best at pub: Tackling Over-Smoothing for General Graph Convolutional Networks (2020) best today: Feature Expansion for Graph Neural Networks (ICML 2023) |
| 35 | GAT | 0.8080
±0.0026 best reported | best at pub: Gophormer: Ego-Graph Transformer for Node Classification (2021) best today: Feature Expansion for Graph Neural Networks (ICML 2023) |
| 44 | GCN | 0.8033
±0.0088 best reported | best at pub: Gophormer: Ego-Graph Transformer for Node Classification (2021) |
| 61 | GraphSAGE | 0.7923
±0.0053 | |
| 66 | DGI | 0.7890
±0.0020 best reported | best at pub: Node Representation Learning in Graph via Node-to-Neighbourhood Mutual Information Maximization (2022) best today: Oversmoothing: A Nightmare for Graph Contrastive Learning? (2023) |
| 97 | FastGCN | 0.7765
±0.0005 best reported | |
| 96 | SGC | 0.7765
±0.0012 best reported | |
| 126 | MixHop | 0.7140
±0.0080 | prev best: Contrastive Laplacian Eigenmaps (NeurIPS 2022) |
| 264 | S^2GC | 0.7360
±0.0009 | prev best: PaSca: a Graph Neural Architecture Search System under the Scalable Paradigm (WWW 2022) |
| 306 | g-U-Nets | 0.7320
±0.0050 | This paper |
| 346 | SEP-N | 0.7290
±0.0070 | This paper |
| 350 | MLP | 0.7289
±0.0010 | |
| 389 | GIN | 0.7250
±0.0080 best reported | best at pub: Towards Self-Explainable Graph Neural Network (2021) |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | LGTL | 0.9518
±0.0011 | |
| 2 | MixHop (LO) | 0.9513
±0.0038 | |
| 3 | GraphSAGE (LO) | 0.9511
±0.0036 | |
| 18 | GCN | 0.9296
±0.0015 best reported | best at pub: Tackling Over-Smoothing for General Graph Convolutional Networks (2020) |
| 70 | MixHop | 0.9004
±0.0141 best reported | best at pub: Power up! Robust Graph Convolutional Network via Graph Powering (AAAI 2019) best today: Language is All a Graph Needs (2023) |
| 116 | SAGE | 0.8881
±0.0015 | |
| 141 | SGC | 0.8832
±0.0055 best reported | |
| 157 | GIN | 0.8810 best reported | best at pub: Learning Conjoint Attentions for Graph Neural Nets (NeurIPS 2021) best today: UGSL: A Unified Framework for Benchmarking Graph Structure Learning (2023) |
| 159 | FastGCN | 0.8800 best reported | best at pub: Tackling Over-Smoothing for General Graph Convolutional Networks (2020) best today: Training Robust Graph Neural Networks with Topology Adaptive Edge Dropping (2021) |
| 204 | DGI | 0.8708
±0.0017 best reported | |
| 210 | MLP | 0.8690
±0.0074 | |
| 433 | S^2GC | 0.8020
±0.0002 | prev best: PaSca: a Graph Neural Architecture Search System under the Scalable Paradigm (WWW 2022) |
| 434 | SEP-N | 0.8020
±0.0080 | This paper |
| 494 | g-U-Nets | 0.7960
±0.0020 | This paper |
TU Dortmund
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | MSH-GNN | 0.9910
±0.0030 | |
| 2 | ID-GNN+BEC | 0.9830
±0.0360 | |
| 3 | SupCosine | 0.9830
±0.0250 | |
| 39 | GIN | 0.9330
±0.0290 best reported | best at pub: ProtGNN: Towards Self-Explaining Graph Neural Networks (AAAI 2021) |
| 63 | GCN | 0.9220
±0.0440 best reported | best at pub: Breaking the Limits of Message Passing Graph Neural Networks (ICML 2021) best today: Depth-Adaptive Graph Neural Networks via Learnable Bakry-Émery Curvature (KDD 2025) |
| 101 | TopKPool | 0.9130
±0.0070 best reported | best at pub: Embedding Graphs on Grassmann Manifold (2022) best today: Mind the Label Shift of Augmentation-based Graph OOD Generalization (2023) |
| 141 | MINCUTPOOL | 0.9060
±0.0460 best reported | best today: Graph Classification via Reference Distribution Learning: Theory and Practice (NeurIPS 2024) |
| 158 | GraphSAGE | 0.9040
±0.0780 | |
| 190 | SortPool | 0.9000
±0.0224 best reported | best at pub: CogDL: A Comprehensive Library for Graph Deep Learning (WWW 2021) |
| 202 | GMT | 0.8344
±0.0133 | best today: Graph Classification via Reference Distribution Learning: Theory and Practice (NeurIPS 2024) Improvement within reported uncertainty; not highlighted. |
| 239 | DiffPool | 0.8940
±0.0460 best reported | best at pub: CogDL: A Comprehensive Library for Graph Deep Learning (WWW 2021) best today: Graph Classification via Reference Distribution Learning: Theory and Practice (NeurIPS 2024) |
| 367 | ASAP | 0.8740
±0.0570 best reported | best at pub: Multi-grained Semantics-aware Graph Neural Networks (2020) best today: Graph Classification via Reference Distribution Learning: Theory and Practice (NeurIPS 2024) |
| 403 | MLP | 0.8660
±0.0495 | |
| 437 | SEP-G | 0.8556
±0.0109 | This paper |
| 593 | Set2Set | 0.8084
±0.0067 best reported | best at pub: Graph Convolutional Networks with EigenPooling (KDD 2019) |
| 635 | StructPool | 0.7950
±0.0075 | prev best: Multi-grained Semantics-aware Graph Neural Networks (2020) Improvement within reported uncertainty; not highlighted. |
| 685 | SAGPool(G) | 0.7678
±0.0212 | This paper |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | WL-MLP | 0.9954 | |
| 2 | GIN | 0.9934 | |
| 3 | GCN | 0.9727 | |
| 131 | GraphSAGE | 0.8200
±0.0100 | |
| 226 | GMT | 0.7990
±0.0040 best reported | best today: Graph Classification via Reference Distribution Learning: Theory and Practice (NeurIPS 2024) Difference within reported uncertainty; not counted as underperformance. |
| 284 | StructPool | 0.7864
±0.0153 | prev best: Multi-grained Semantics-aware Graph Neural Networks (2020) Improvement within reported uncertainty; not highlighted. |
| 304 | SEP-G | 0.7835
±0.0033 | This paper |
| 355 | TopKPool | 0.7706
±0.0090 best reported | best at pub: Embedding Graphs on Grassmann Manifold (2022) |
| 368 | MINCUTPOOL | 0.7680
±0.0160 best reported | best at pub: Rethinking pooling in graph neural networks (NeurIPS 2020) best today: Higher-order Clustering and Pooling for Graph Neural Networks (2022) |
| 415 | SortPool | 0.7496
±0.0134 best reported | best at pub: CogDL: A Comprehensive Library for Graph Deep Learning (WWW 2021) best today: Graph Structure Prompt Learning: A Novel Methodology to Improve Performance of Graph Neural Networks (2024) Difference within reported uncertainty; not counted as underperformance. |
| 436 | SAGPool(G) | 0.7418
±0.0120 | This paper |
| 463 | Set2Set | 0.7170
±0.0073 best reported | best at pub: Graph Convolutional Networks with EigenPooling (KDD 2019) |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | MSH-GNN | 0.9410
±0.0330 | |
| 2 | GRDL | 0.8260
±0.0120 | |
| 3 | GMN | 0.8225 | |
| 7 | DiffPool | 0.8161
±0.0232 best reported | best at pub: Memory-Based Graph Networks (ICLR 2020) |
| 29 | ASAP | 0.7892
±0.0145 best reported | best at pub: Multi-grained Semantics-aware Graph Neural Networks (2020) |
| 31 | StructPool | 0.7884
±0.0170 best reported | best at pub: Multi-grained Semantics-aware Graph Neural Networks (2020) |
| 32 | GIN | 0.7880
±0.0410 best reported | best at pub: Breaking the Limits of Message Passing Graph Neural Networks (ICML 2021) |
| 58 | TopKPool | 0.7768 best reported | best at pub: Memory-Based Graph Networks (ICLR 2020) |
| 67 | SAGE | 0.7726
±0.0228 | |
| 114 | SEP-G | 0.7642
±0.0039 | This paper |
| 148 | GCN | 0.7600
±0.0320 best reported | best at pub: GraphiT: Encoding Graph Structure in Transformers (2021) best today: Simplifying Graph Convolutional Networks with Redundancy-Free Neighbors (2025) Difference within reported uncertainty; not counted as underperformance. |
| 149 | MLP | 0.7600
±0.0320 | |
| 152 | SortPool | 0.7598
±0.0176 best reported | best at pub: Memory-Based Graph Networks (ICLR 2020) |
| 172 | MINCUTPOOL | 0.7575
±0.0380 best reported | best at pub: Pooling Architecture Search for Graph Classification (2021) Difference within reported uncertainty; not counted as underperformance. |
| 236 | GMT | 0.7510
±0.0060 best reported | best today: Multi-Scale Harmonic Encoding for Feature-Wise Graph Message Passing (2025) Difference within reported uncertainty; not counted as underperformance. |
| 301 | Set2Set | 0.7429 best reported | best at pub: Memory-Based Graph Networks (ICLR 2020) Difference within reported uncertainty; not counted as underperformance. |
| 408 | SAGPool(G) | 0.7202
±0.0108 | This paper |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | WL-MLP | 0.9792 | |
| 2 | MSH-GNN | 0.9640
±0.0070 | |
| 3 | CoCN | 0.8722
±0.0013 | |
| 23 | GCN | 0.8260
±0.0220 best reported | best at pub: Fast Graph Representation Learning with PyTorch Geometric (ICLR 2019) best today: Scalable Expressiveness through Preprocessed Graph Perturbations (2024) Difference within reported uncertainty; not counted as underperformance. |
| 30 | GIN | 0.8220
±0.0210 best reported | best at pub: On the Equivalence between Graph Isomorphism Testing and Function Approximation with GNNs (NeurIPS 2019) best today: Scalable Expressiveness through Preprocessed Graph Perturbations (2024) |
| 33 | DiffPool | 0.8213
±0.0043 best reported | best at pub: CogDL: A Comprehensive Library for Graph Deep Learning (WWW 2021) best today: Scalable Graph Compressed Convolutions (2024) |
| 57 | TopKPool | 0.8140
±0.0094 best reported | best at pub: Embedding Graphs on Grassmann Manifold (2022) |
| 70 | SEP-G | 0.8128
±0.0015 | This paper |
| 86 | MINCUTPOOL | 0.8087
±0.0034 | prev best: DiffWire: Inductive Graph Rewiring via the Lovász Bound (LoG 2022) |
| 93 | GMT | 0.8074
±0.0054 | This paper |
| 125 | SortPool | 0.8007 best reported | best at pub: CogDL: A Comprehensive Library for Graph Deep Learning (WWW 2021) |
| 136 | GraphSAGE | 0.7970
±0.0170 | |
| 142 | Set2Set | 0.7960
±0.0230 best reported | best at pub: Fast Graph Representation Learning with PyTorch Geometric (ICLR 2019) Difference within reported uncertainty; not counted as underperformance. |
| 151 | MLP | 0.7920
±0.0240 | |
| 164 | SAGPool(G) | 0.7885
±0.0056 | This paper |
| 166 | ASAP | 0.7864
±0.0050 | This paper |
| 200 | StructPool | 0.7727
±0.0051 | This paper |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | GC-GNN (GCN) | 0.9210
±0.0810 | |
| 2 | SUGAR | 0.8403
±0.0133 | |
| 3 | Ortho-g-U-Nets | 0.8387 | |
| 15 | SAGE | 0.8088
±0.0233 | |
| 18 | DiffPool | 0.8064 best reported | best at pub: Graph U-Nets (2019) best today: Orthogonal Graph Neural Networks (AAAI 2021) |
| 23 | StructPool | 0.8010
±0.0177 best reported | best at pub: Multi-grained Semantics-aware Graph Neural Networks (2020) Difference within reported uncertainty; not counted as underperformance. |
| 32 | ASAP | 0.7950
±0.0180 best reported | best at pub: Multi-grained Semantics-aware Graph Neural Networks (2020) |
| 43 | GCN | 0.7912
±0.0307 best reported | best at pub: Hierarchical Graph Capsule Network (AAAI 2020) |
| 55 | GMT | 0.7872
±0.0059 | This paper |
| 69 | MINCUTPOOL | 0.7822
±0.0054 | prev best: Pooling Architecture Search for Graph Classification (2021) Improvement within reported uncertainty; not highlighted. |
| 74 | GIN | 0.7805
±0.0189 best reported | best at pub: Multi-grained Semantics-aware Graph Neural Networks (2020) |
| 78 | SoTA | 0.7798
±0.0057 | This paper |
| 90 | SortPool | 0.7558
±0.0072 | prev best: Multi-grained Semantics-aware Graph Neural Networks (2020) best today: On Exploring Node-feature and Graph-structure Diversities for Node Drop Graph Pooling (2023) Improvement within reported uncertainty; not highlighted. |
| 110 | TopKPool | 0.7640
±0.0232 best reported | best at pub: Embedding Graphs on Grassmann Manifold (2022) Difference within reported uncertainty; not counted as underperformance. |
| 140 | SAGPool(H) | 0.7472
±0.0149 | This paper |
| 146 | Set2Set | 0.7450 best reported | best at pub: Graph Convolutional Networks with EigenPooling (KDD 2019) |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | ID-GNN+BEC | 0.8150
±0.0240 | |
| 2 | GIN+BEC | 0.8080
±0.0330 | |
| 3 | GCN+BEC | 0.7990
±0.0340 | |
| 4 | GCN | 0.7980
±0.0230 best reported | best at pub: Contrastive Multi-View Representation Learning on Graphs (ICML 2020) best today: Depth-Adaptive Graph Neural Networks via Learnable Bakry-Émery Curvature (KDD 2025) |
| 14 | GIN | 0.7810
±0.0350 best reported | best at pub: Graph Homomorphism Convolution (ICML 2020) best today: Depth-Adaptive Graph Neural Networks via Learnable Bakry-Émery Curvature (KDD 2025) Difference within reported uncertainty; not counted as underperformance. |
| 49 | ASAP | 0.7427
±0.0397 best reported | best at pub: Pooling Architecture Search for Graph Classification (2021) Difference within reported uncertainty; not counted as underperformance. |
| 55 | SEP-G | 0.7412
±0.0056 | This paper |
| 70 | GMT | 0.7348
±0.0076 | This paper |
| 78 | DiffPool | 0.7314
±0.0070 | prev best: Pooling Architecture Search for Graph Classification (2021) Improvement within reported uncertainty; not highlighted. |
| 85 | Set2Set | 0.7290
±0.0075 | This paper |
| 94 | MINCUTPOOL | 0.7265
±0.0075 | prev best: Pooling Architecture Search for Graph Classification (2021) Improvement within reported uncertainty; not highlighted. |
| 95 | SAGPool(H) | 0.7255
±0.0128 | This paper |
| 99 | GraphSAGE | 0.7230
±0.0530 | |
| 104 | SortPool | 0.7212
±0.0112 | prev best: Graph Information Bottleneck for Subgraph Recognition (ICLR 2020) Improvement within reported uncertainty; not highlighted. |
| 108 | StructPool | 0.7206
±0.0064 | This paper |
| 117 | TopKPool | 0.7158
±0.0095 | This paper |
| # | Model | Accuracy ↑ | Paper |
|---|---|---|---|
| 1 | GraphSNN | 0.7790
±0.0360 | |
| 2 | SIN | 0.7560
±0.0320 | |
| 3 | CIN | 0.7560
±0.0370 | |
| 5 | GIN | 0.7510
±0.0510 best reported | best at pub: Empowering GNNs via Edge-Aware Weisfeiler-Leman Algorithm (TMLR 2022) |
| 38 | GCN | 0.5190 best reported | best at pub: Contrastive Multi-View Representation Learning on Graphs (ICML 2020) best today: AutoGEL: An Automated Graph Neural Network with Explicit Link Information (NeurIPS 2021) |
| 41 | SEP-G | 0.5153
±0.0065 | This paper |
| 43 | DiffPool | 0.5131
±0.0072 | prev best: Pooling Architecture Search for Graph Classification (2021) Improvement within reported uncertainty; not highlighted. |
| 45 | MINCUTPOOL | 0.5104
±0.0070 | prev best: Pooling Architecture Search for Graph Classification (2021) Improvement within reported uncertainty; not highlighted. |
| 46 | GraphSAGE | 0.5090 | |
| 48 | ASAP | 0.5078
±0.0075 | prev best: Pooling Architecture Search for Graph Classification (2021) Improvement within reported uncertainty; not highlighted. |
| 50 | GMT | 0.5066
±0.0082 | This paper |
| 55 | SAGPool(H) | 0.5023
±0.0044 | This paper |
| 56 | StructPool | 0.5023
±0.0053 | This paper |
| 57 | Set2Set | 0.5019
±0.0039 | This paper |
| 85 | TopKPool | 0.4859
±0.0072 | This paper |
| 90 | SortPool | 0.4818
±0.0083 | This paper |
