# On the Expressivity of Persistent Homology in Graph Learning

arXiv: https://arxiv.org/abs/2302.09826 · HTML: https://mlsanity.com/paper/2302.09826/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 91 references that are in the map.
- First posted on arXiv 20 Feb 2023.

## Builds on (most cited first, 30 of 36)

- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networks](https://mlsanity.com/paper/1810.02244/) (2018)
- [Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting](https://mlsanity.com/paper/2006.09252/) (2020)
- [A Fair Comparison of Graph Neural Networks for Graph Classification](https://mlsanity.com/paper/1912.09893/) (2019)
- [Persistence Images: A Stable Vector Representation of Persistent Homology](https://mlsanity.com/paper/1507.06217/) (2015)
- [Can Graph Neural Networks Count Substructures?](https://mlsanity.com/paper/2002.04025/) (2020)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Weisfeiler and Leman go Machine Learning: The Story so far](https://mlsanity.com/paper/2112.09992/) (2021)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (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)
- [On the Expressive Power of Geometric Graph Neural Networks](https://mlsanity.com/paper/2301.09308/) (2023)
- [Topological Graph Neural Networks](https://mlsanity.com/paper/2102.07835/) (2021)
- [The Logic of Graph Neural Networks](https://mlsanity.com/paper/2104.14624/) (2021)
- [A Stable Multi-Scale Kernel for Topological Machine Learning](https://mlsanity.com/paper/1412.6821/) (2014)
- [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)
- [GraphWorld: Fake Graphs Bring Real Insights for GNNs](https://mlsanity.com/paper/2203.00112/) (2022)
- [A Theoretical Comparison of Graph Neural Network Extensions](https://mlsanity.com/paper/2201.12884/) (2022)
- [Topological Relational Learning on Graphs](https://mlsanity.com/paper/2110.15529/) (2021)
- [Persistent homology detects curvature](https://mlsanity.com/paper/1905.13196/) (2019)
- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [Curvature Filtrations for Graph Generative Model Evaluation](https://mlsanity.com/paper/2301.12906/) (2023)
- [Neural Approximation of Graph Topological Features](https://mlsanity.com/paper/2201.12032/) (2022)

## Built on by (most cited first, 19 of 19)

- [Going beyond persistent homology using persistent homology](https://mlsanity.com/paper/2311.06152/) (2023)
- [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)
- [Topological Data Analysis for Neural Network Analysis: A Comprehensive Survey](https://mlsanity.com/paper/2312.05840/) (2023)
- [Boosting Graph Pooling with Persistent Homology](https://mlsanity.com/paper/2402.16346/) (2024)
- [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)
- [Graph Persistence goes Spectral](https://mlsanity.com/paper/2506.06571/) (2025)
- [Addressing caveats of neural persistence with deep graph persistence](https://mlsanity.com/paper/2307.10865/) (2023)
- [Towards Scalable Topological Regularizers](https://mlsanity.com/paper/2501.14641/) (2025)
- [CellCLAT: Preserving Topology and Trimming Redundancy in Self-Supervised Cellular Contrastive Learning](https://mlsanity.com/paper/2505.21587/) (2025)
- [Positional Encoding meets Persistent Homology on Graphs](https://mlsanity.com/paper/2506.05814/) (2025)
- [Heat Kernel Goes Topological](https://mlsanity.com/paper/2507.12380/) (2025)
- [Topological Data Analysis for Unsupervised Anomaly Detection and Customer Segmentation on Banking Data](https://mlsanity.com/paper/2508.14136/) (2025)
- [Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis](https://mlsanity.com/paper/2509.08350/) (2025)
- [Graph Homomorphism Distortion: A structural (pseudo)-metric for attributed graphs](https://mlsanity.com/paper/2511.03068/) (2025)
- [Contraction and Hourglass Persistence for Learning on Graphs, Simplices, and Cells](https://mlsanity.com/paper/2604.17548/) (2026)
- [Diversity Curves for Graph Representation Learning](https://mlsanity.com/paper/2605.06466/) (2026)
- [Exact Topological Compliance: Generating Persistence-Equivalent Graphs](https://mlsanity.com/paper/2610.18856/) (2026)

## Abstract (arXiv)

Persistent homology, a technique from computational topology, has recently shown strong empirical performance in the context of graph classification. Being able to capture long range graph properties via higher-order topological features, such as cycles of arbitrary length, in combination with multi-scale topological descriptors, has improved predictive performance for data sets with prominent topological structures, such as molecules. At the same time, the theoretical properties of persistent homology have not been formally assessed in this context. This paper intends to bridge the gap between computational topology and graph machine learning by providing a brief introduction to persistent homology in the context of graphs, as well as a theoretical discussion and empirical analysis of its expressivity for graph learning tasks.