mlsanityData to 7 Oct 2026

Going beyond persistent homology using persistent homology

arXiv 2311.06152 ↗First posted 10 Nov 2023cs.LG cs.AIcited by 21

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Abstract (arXiv)

Representational limits of message-passing graph neural networks (MP-GNNs), e.g., in terms of the Weisfeiler-Leman (WL) test for isomorphism, are well understood. Augmenting these graph models with topological features via persistent homology (PH) has gained prominence, but identifying the class of attributed graphs that PH can recognize remains open. We introduce a novel concept of color-separating sets to provide a complete resolution to this important problem. Specifically, we establish the necessary and sufficient conditions for distinguishing graphs based on the persistence of their connected components, obtained from filter functions on vertex and edge colors. Our constructions expose the limits of vertex- and edge-level PH, proving that neither category subsumes the other. Leveraging these theoretical insights, we propose RePHINE for learning topological features on graphs. RePHINE efficiently combines vertex- and edge-level PH, achieving a scheme that is provably more powerful than both. Integrating RePHINE into MP-GNNs boosts their expressive power, resulting in gains over standard PH on several benchmarks for graph classification.

Key facts. Cited by 21 papers in the map (data through 7 Oct 2026). Builds on 22 of its 40 references that are in the map. First posted on arXiv 10 Nov 2023.

Builds on

22 of 40 references.

References · 22

  1. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,282
  2. 2015Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shiftcited by 6,979
  3. 2018How Powerful are Graph Neural Networks?cited by 4,355
  4. 2019Fast Graph Representation Learning with PyTorch Geometriccited by 2,414
  5. 2020Open Graph Benchmark: Datasets for Machine Learning on Graphscited by 2,148
  6. 2017Deep Setscited by 1,394
  7. 2018Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networkscited by 878
  8. 2020Benchmarking Graph Neural Networkscited by 672
  9. 2020Generalization and Representational Limits of Graph Neural Networkscited by 217
  10. 2019Topology-Preserving Deep Image Segmentationcited by 198
  11. 2019What graph neural networks cannot learn: depth vs widthcited by 195
  12. 2019PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signaturescited by 125
10 more
  1. 2017Deep Learning with Topological Signaturescited by 113
  2. 2020A Survey on The Expressive Power of Graph Neural Networkscited by 111
  3. 2021Topological Graph Neural Networkscited by 86
  4. 2019Approximation Ratios of Graph Neural Networks for Combinatorial Problemscited by 68
  5. 2019Graph Filtration Learningcited by 68
  6. 2019Learning metrics for persistence-based summaries and applications for graph classificationcited by 57
  7. 2019A Framework for Differential Calculus on Persistence Barcodescited by 29
  8. 2021Topological Relational Learning on Graphscited by 27
  9. 2023On the Expressivity of Persistent Homology in Graph Learningcited by 18
  10. 2014Using persistent homology and dynamical distances to analyze protein bindingcited by 15

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 6; 2025: 9; 2026: 6.

21 papers.

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  1. 2024Topological Neural Networks go Persistent, Equivariant, and Continuouscited by 15
  2. 2024MANTRA: The Manifold Triangulations Assemblagecited by 12
  3. 2025Enhancing Graph Representation Learning with Localized Topological Featurescited by 6
  4. 2024CliquePH: Higher-Order Information for Graph Neural Networks through Persistent Homology on Clique Graphscited by 3
  5. 2024PHLP: Sole Persistent Homology for Link Prediction - Interpretable Feature Extractioncited by 2
  6. 2025On topological descriptors for graph productscited by 2
  7. 2026The Logical Expressiveness of Topological Neural Networkscited by 2
  8. 2024TopER: Topological Embeddings in Graph Representation Learningcited by 1
  9. 2025Graph machine learning for flight delay prediction due to holding manouvercited by 1
  10. 2025TMetaNet: Topological Meta-Learning Framework for Dynamic Link Predictioncited by 1
  11. 2025Positional Encoding meets Persistent Homology on Graphscited by 1
  12. 2025Graph Persistence goes Spectralcited by 1
9 more
  1. 2025TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtrationcited by 1
  2. 2025T3former: Temporal Graph Classification with Topological Machine Learningcited by 1
  3. 2026TopoFormer: Topology Meets Attention for Graph Learningcited by 1
  4. 2024What Ails Generative Structure-based Drug Design: Expressivity is Too Little or Too Much?cited by 0
  5. 2025Frequent subgraph-based persistent homology for graph classificationcited by 0
  6. 2026Scalable Topology-Preserving Graph Coarsening: Concepts and Algorithmscited by 0
  7. 2026Contraction and Hourglass Persistence for Learning on Graphs, Simplices, and Cellscited by 0
  8. 2026Weisfeiler Lehman Test on Combinatorial Complexes: Generalized Expressive Power of Topological Neural Networkscited by 0
  9. 2026From Persistence to Survival: Hypothesis Testing, Effect Sizes and Vectorisation for Topological Featurescited by 0