# Going beyond persistent homology using persistent homology

arXiv: https://arxiv.org/abs/2311.06152 · HTML: https://mlsanity.com/paper/2311.06152/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 40 references that are in the map.
- First posted on arXiv 10 Nov 2023.

## Builds on (most cited first, 22 of 22)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (2020)
- [Generalization and Representational Limits of Graph Neural Networks](https://mlsanity.com/paper/2002.06157/) (2020)
- [What graph neural networks cannot learn: depth vs width](https://mlsanity.com/paper/1907.03199/) (2019)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [A Survey on The Expressive Power of Graph Neural Networks](https://mlsanity.com/paper/2003.04078/) (2020)
- [Topological Graph Neural Networks](https://mlsanity.com/paper/2102.07835/) (2021)
- [Approximation Ratios of Graph Neural Networks for Combinatorial Problems](https://mlsanity.com/paper/1905.10261/) (2019)
- [Graph Filtration Learning](https://mlsanity.com/paper/1905.10996/) (2019)
- [Learning metrics for persistence-based summaries and applications for graph classification](https://mlsanity.com/paper/1904.12189/) (2019)
- [A Framework for Differential Calculus on Persistence Barcodes](https://mlsanity.com/paper/1910.00960/) (2019)
- [Topological Relational Learning on Graphs](https://mlsanity.com/paper/2110.15529/) (2021)
- [On the Expressivity of Persistent Homology in Graph Learning](https://mlsanity.com/paper/2302.09826/) (2023)
- [Using persistent homology and dynamical distances to analyze protein binding](https://mlsanity.com/paper/1412.1394/) (2014)

## Built on by (most cited first, 22 of 22)

- [Topological Neural Networks go Persistent, Equivariant, and Continuous](https://mlsanity.com/paper/2406.03164/) (2024)
- [MANTRA: The Manifold Triangulations Assemblage](https://mlsanity.com/paper/2410.02392/) (2024)
- [Enhancing Graph Representation Learning with Localized Topological Features](https://mlsanity.com/paper/2501.09178/) (2025)
- [CliquePH: Higher-Order Information for Graph Neural Networks through Persistent Homology on Clique Graphs](https://mlsanity.com/paper/2409.08217/) (2024)
- [On topological descriptors for graph products](https://mlsanity.com/paper/2511.08846/) (2025)
- [PHLP: Sole Persistent Homology for Link Prediction - Interpretable Feature Extraction](https://mlsanity.com/paper/2404.15225/) (2024)
- [Graph Persistence goes Spectral](https://mlsanity.com/paper/2506.06571/) (2025)
- [The Logical Expressiveness of Topological Neural Networks](https://mlsanity.com/paper/2604.19212/) (2026)
- [TopER: Topological Embeddings in Graph Representation Learning](https://mlsanity.com/paper/2410.01778/) (2024)
- [Graph machine learning for flight delay prediction due to holding manouver](https://mlsanity.com/paper/2502.04233/) (2025)
- [TMetaNet: Topological Meta-Learning Framework for Dynamic Link Prediction](https://mlsanity.com/paper/2506.00453/) (2025)
- [Positional Encoding meets Persistent Homology on Graphs](https://mlsanity.com/paper/2506.05814/) (2025)
- [TopInG: Topologically Interpretable Graph Learning via Persistent Rationale Filtration](https://mlsanity.com/paper/2510.05102/) (2025)
- [T3former: Temporal Graph Classification with Topological Machine Learning](https://mlsanity.com/paper/2510.13789/) (2025)
- [Contraction and Hourglass Persistence for Learning on Graphs, Simplices, and Cells](https://mlsanity.com/paper/2604.17548/) (2026)
- [From Persistence to Survival: Hypothesis Testing, Effect Sizes and Vectorisation for Topological Features](https://mlsanity.com/paper/2606.11911/) (2026)
- [TopoFormer: Topology Meets Attention for Graph Learning](https://mlsanity.com/paper/2607.28259/) (2026)
- [What Ails Generative Structure-based Drug Design: Expressivity is Too Little or Too Much?](https://mlsanity.com/paper/2408.06050/) (2024)
- [Frequent subgraph-based persistent homology for graph classification](https://mlsanity.com/paper/2512.24917/) (2025)
- [Scalable Topology-Preserving Graph Coarsening: Concepts and Algorithms](https://mlsanity.com/paper/2601.22943/) (2026)
- [Weisfeiler Lehman Test on Combinatorial Complexes: Generalized Expressive Power of Topological Neural Networks](https://mlsanity.com/paper/2605.00725/) (2026)
- [Exact Topological Compliance: Generating Persistence-Equivalent Graphs](https://mlsanity.com/paper/2610.18856/) (2026)

## 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.