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TopER: Topological Embeddings in Graph Representation Learning
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Proposes TopER, a training-free, low-dimensional graph embedding that fits a least-squares line to node/edge counts along a filtration (pivot and growth), evaluated on graph classification, clustering, and visualization.
Problem
Graph-level embeddings are typically high-dimensional and opaque, and persistent homology, though topologically informative, is computationally expensive (cubic complexity) and yields high-dimensional vectors, limiting scalability and interpretability for graph representation learning.
Approach
Instead of computing persistence diagrams or Betti numbers, TopER records the node and edge counts of each filtration subgraph and summarizes their evolution by the intercept (pivot) and slope (growth) of a least-squares line, giving a 2D descriptor per filtration function; a new Popularity filtration is also introduced.
Key results
- TopER achieves an average deviation of 1.60% from the best per-dataset accuracy across eight classification benchmarks, the best among compared models (TopoGCL 1.76%).
- TopER reaches 80.21 test AUC on OGBG-MOLHIV, second-best behind Graphormer (80.51).
- TopER is over 10 times faster than persistent homology on REDDIT-5K and matches or beats PH accuracy on six benchmarks; it processes 100K-node graphs in just over a minute.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use TopER instead of the strongest reported classifier | graph classification · BZR | accuracy: 89.00{ $\pm$5.00} -> 90.13{ $\pm$4.14} | Helped |
| Use TopER instead of the strongest reported classifier | graph classification · COX2 | accuracy: 85.53 -> 82.01{ $\pm$4.59} | Hurt |
| Use TopER instead of the strongest reported classifier | graph classification · REDDIT-B | accuracy: 91.03{ $\pm$0.22} -> 92.70{ $\pm$2.38} | Helped |
| Replace persistent homology with TopER | graph classification · IMDB-B | accuracy: 69.5{ $\pm$0.5} -> 73.2{ $\pm$3.4} | Helped |
| Replace persistent homology with TopER | graph classification · REDDIT-B | accuracy: 84.50 ± 2.51 -> 79.55 ± 2.20 | Hurt |
| Add multiple filtration functions rather than use one (ablation) | TopER · BZR | accuracy: 82.48{ $\pm$1.98} -> 86.68{ $\pm$3.81} | Helped |
| Add multiple filtration functions rather than use one (ablation) | TopER · REDDIT-B | accuracy: 81.95{ $\pm$2.74} -> 91.50{ $\pm$2.01} | Helped |
| Replace Ollivier-Ricci filtration with HKS (ablation) | TopER · REDDIT-5K | accuracy: 56.37±1.89 -> 57.33±2.02 | Helped |
| Remove Ollivier-Ricci filtration (ablation) | TopER · COX2 | accuracy: 82.01±4.59 -> 77.96±8.38 | Hurt |
| Use fewer thresholds (10 rather than 200) (ablation) | TopER · REDDIT-B | accuracy: 93.15{ $\pm$2.10} -> 90.55{ $\pm$1.96} | Hurt |
| Use fewer thresholds (10 rather than 200) (ablation) | TopER · REDDIT-5K | accuracy: 56.51{ $\pm$2.04} -> 55.99{ $\pm$1.97} | No clear effect |
Surprising findings
- Quadratic coefficients of the fit are negligible, so a linear fit suffices for the filtration growth pattern.
- TopER without Ollivier-Ricci but with Heat Kernel Signature recovers most of the performance lost when Ricci is removed.
Limitations
- Performance depends on the choice of filtration function, which may require domain knowledge.
- Runtime dominated by node functions such as closeness and Ollivier-Ricci curvature; approximations are left to future work.
- Theoretical stability results are stated for a fixed threshold set and rely on proofs deferred to the appendix with some inconsistencies.
Experimental setup
Datasets: BZR COX2 MUTAG PROTEINS IMDB-B IMDB-M REDDIT-B REDDIT-5K OGBG-MOLHIV
Benchmarks: BZR COX2 MUTAG PROTEINS IMDB-B IMDB-M REDDIT-B REDDIT-5K OGBG-MOLHIV
Metrics: accuracy AUC Silhouette Calinski-Harabasz Davies-Bouldin runtime
Baselines: DiffPool SAGPool Top-k MinCutPool GIN TopoGCL AutoGCL RGCL WWLS EPIC EMP PGOT
Builds on: persistent homology Betti curves stability Forman-Ricci curvature Ollivier-Ricci curvature
Closest papers by meaning
- TopoFormer: Topology Meets Attention for Graph Learning (ICLR 2026 · similarity 0.91)
- TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration (ICML 2025 · similarity 0.90)
- TopoGCL: Topological Graph Contrastive Learning (AAAI 2024 · similarity 0.89)
- Topological Neural Networks go Persistent, Equivariant, and Continuous (ICML 2024 · similarity 0.87)
- TMetaNet: Topological Meta-Learning Framework for Dynamic Link Prediction (ICML 2025 · similarity 0.87)
- Learning Representations for Hierarchies with Minimal Support (NeurIPS 2024 · similarity 0.87)
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