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Boosting Graph Pooling with Persistent Homology
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Proposes Topology-Invariant Pooling (TIP), which plugs persistent homology (cycle-based 1-dim PH with learnable filtration, Gumbel-softmax resampling, and a topological statistics loss) into dense graph pooling methods to improve graph classification and regression.
Problem
Naively adding persistent homology features to GNN layers yields marginal gains with low interpretability, and dense graph pooling produces fully connected coarsened graphs that lose topological structure.
Approach
Align persistent homology's filtration with the cut-off nature of graph pooling: resample the coarsened adjacency with Gumbel-softmax to make it sparse, reweight edges by 1-dimensional persistence from a learnable filtration, and add a loss that matches mean and std statistics of persistence diagrams between original and pooled graphs.
Key results
- DiffPool-TIP reaches 65.05% on ENZYMES versus 48.34% for DiffPool, and 83.75% on NCI1 versus 77.64%.
- On ZINC regression, DiffPool-TIP reduces MSE from 0.34 to 0.28, and MinCutPool-TIP and DMoNPool-TIP also improve over their base pooling methods.
- On the topology-preservation experiment, DiffPool-TIP lowers Wasserstein distance on ring from 234.57 to 8.03.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add TIP to dense pooling | DiffPool, MinCutPool, and DMoNPool · graph classification benchmarks | accuracy: DiffPool: 48.34 $\pm$ 5.14 -> 65.05 $\pm$ 4.24 (ENZYMES); 78.81 $\pm$ 3.12 -> 79.86 $\pm$ 3.12 (PROTEINS) | Helped |
| Add TIP to graph pooling | DiffPool, MinCutPool, and DMoNPool · ZINC | mean square error: 0.34$\pm$0.01 -> 0.28$\pm$0.01; 0.42$\pm$0.01 -> 0.38$\pm$0.01; 0.40$\pm$0.01 -> 0.35$\pm$0.01 | Helped |
| Add TIP to dense pooling | DiffPool, MinCutPool, and DMoNPool · Cycles | accuracy: 54.3 $\pm$ 1.1 -> 65.1 $\pm$ 2.7; 54.2 $\pm$ 2.6 -> 65.0 $\pm$ 2.9; 55.0 $\pm$ 2.7 -> 68.7 $\pm$ 2.7 | Helped |
| Add TIP to DiffPool | DiffPool · 2-Cycles | accuracy: 50.0 $\pm$ 2.2 -> 51.4 $\pm$ 2.8 | No clear effect |
| Add resampling (ablation) | TIP pooling · graph classification: NCI1, PROTEINS, ENZYMES, and IMDB-BINARY | accuracy: DiffPool NCI1: 80.82 $\pm$ 1.71 -> 83.75 $\pm$ 1.70; PROTEINS: 77.89 $\pm$ 4.07 -> 79.86 $\pm$ 3.12 | Helped |
| Add persistence injection (ablation) | TIP pooling · graph classification: NCI1, PROTEINS, ENZYMES, and IMDB-BINARY | accuracy: DiffPool NCI1: 81.99 $\pm$ 1.15 -> 83.75 $\pm$ 1.70; PROTEINS: 79.30 $\pm$ 1.26 -> 79.86 $\pm$ 3.12 | Helped |
| Add the topological loss (ablation) | TIP pooling · graph classification: NCI1, PROTEINS, ENZYMES, and IMDB-BINARY | accuracy: DiffPool NCI1: 82.33 $\pm$ 2.14 -> 83.75 $\pm$ 1.70; ENZYMES: 58.77 $\pm$ 5.15 -> 65.05 $\pm$ 4.24 | Helped |
| Add 0-dimensional topological features (ablation) | TIP pooling · graph classification: NCI1, PROTEINS, ENZYMES, and IMDB-BINARY | accuracy: DiffPool NCI1: 82.45 $\pm$ 1.40 -> 83.75 $\pm$ 1.70; PROTEINS: 79.12 $\pm$ 1.63 -> 79.86 $\pm$ 3.12 | No clear effect |
| Use learnable instead of fixed random filtration (ablation) | TIP pooling · graph classification: NCI1, PROTEINS, ENZYMES, and IMDB-BINARY | accuracy: DiffPool NCI1: 83.21 $\pm$ 1.55 -> 83.75 $\pm$ 1.70; PROTEINS: 77.91 $\pm$ 3.46 -> 79.86 $\pm$ 3.12; MinCutPool NCI1: 76 | Mixed |
| Replace the proposed topological loss with Wasserstein distance (ablation) | TIP pooling · graph classification: NCI1, PROTEINS, ENZYMES, and IMDB-BINARY | accuracy: DiffPool NCI1: 83.02 $\pm$ 1.08 (TIP-W) vs 83.75 $\pm$ 1.70 (TIP); IMDB-BINARY: 76.75 $\pm$ 3.66 vs 76.40 $\pm$ 3.13 | Mixed |
Surprising findings
- Adding 0-dimensional persistence features does not improve performance and only increases runtime, so the method uses only 1-dimensional features.
- TIP cannot distinguish graphs that differ only in number of connected components (2-Cycles task near chance accuracy).
Limitations
- Relies heavily on cycles in graphs, so may be less effective on tree-like graphs.
- Cannot discriminate graphs whose only difference is the number of connected components.
- Wall-clock training time roughly 1.5 to 2x that of base pooling methods (e.g., NCI1 DiffPool 209s vs 339s).
Experimental setup
Datasets: NCI1 NCI109 ENZYMES PROTEINS DD IMDB-BINARY IMDB-MULTI OGBG-MOLHIV ZINC ring torus grid2d
Benchmarks: NCI1 NCI109 ENZYMES PROTEINS DD IMDB-BINARY IMDB-MULTI OGBG-MOLHIV ZINC EXPWL1
Metrics: test accuracy mean square error 1-Wasserstein distance
Baselines: GCN GIN GraphSAGE TOGL GSN Graclus TopK DiffPool MinCutPool DMoNPool
Builds on: DiffPool MinCutPool DMoNPool TOGL
Closest papers by meaning
- TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration (ICML 2025 · similarity 0.89)
- Geometry-Aware Edge Pooling for Graph Neural Networks (NeurIPS 2025 · similarity 0.89)
- Graph Persistence goes Spectral (NeurIPS 2025 · similarity 0.88)
- Positional Encoding meets Persistent Homology on Graphs (ICML 2025 · similarity 0.88)
- Graphcode: Learning from multiparameter persistent homology using graph neural networks (NeurIPS 2024 · similarity 0.87)
- Compositional PAC-Bayes: Generalization of GNNs with persistence and beyond (NeurIPS 2024 · similarity 0.87)
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