# Persistence Homology of Networks: Methods and Applications

arXiv: https://arxiv.org/abs/1907.08708 · HTML: https://mlsanity.com/paper/1907.08708/

## Key facts

- Cited by 30 papers in the map (data through 7 Oct 2026).
- First posted on arXiv 19 Jul 2019.

## Built on by (most cited first, 30 of 30)

- [Dist2Cycle: A Simplicial Neural Network for Homology Localization](https://mlsanity.com/paper/2110.15182/) (2021)
- [ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discovery](https://mlsanity.com/paper/2211.03808/) (2022)
- [Multivariate Time-Series Anomaly Detection based on Enhancing Graph Attention Networks with Topological Analysis](https://mlsanity.com/paper/2408.13082/) (2024)
- [Characterizing and Measuring the Similarity of Neural Networks with Persistent Homology](https://mlsanity.com/paper/2101.07752/) (2021)
- [Graph Filtration Kernels](https://mlsanity.com/paper/2110.11862/) (2021)
- [Quantitative Performance Assessment of CNN Units via Topological Entropy Calculation](https://mlsanity.com/paper/2103.09716/) (2021)
- [EMP: Effective Multidimensional Persistence for Graph Representation Learning](https://mlsanity.com/paper/2401.13713/) (2024)
- [Persistent Homology for High-dimensional Data Based on Spectral Methods](https://mlsanity.com/paper/2311.03087/) (2023)
- [Conditional Uncertainty Quantification for Tensorized Topological Neural Networks](https://mlsanity.com/paper/2410.15241/) (2024)
- [Time-Series Forecasting via Topological Information Supervised Framework with Efficient Topological Feature Learning](https://mlsanity.com/paper/2503.23757/) (2025)
- [Reduction Algorithms for Persistence Diagrams of Networks: CoralTDA and PrunIT](https://mlsanity.com/paper/2211.13708/) (2022)
- [Analysing Multiscale Clusterings with Persistent Homology](https://mlsanity.com/paper/2305.04281/) (2023)
- [Dynamic Neural Dowker Network: Approximating Persistent Homology in Dynamic Directed Graphs](https://mlsanity.com/paper/2408.09123/) (2024)
- [On topological descriptors for graph products](https://mlsanity.com/paper/2511.08846/) (2025)
- [Have Graph -- Will Lift? The Case for Higher-Order Benchmarks](https://mlsanity.com/paper/2605.07397/) (2026)
- [Persistent Homological State-Space Estimation of Functional Human Brain Networks at Rest](https://mlsanity.com/paper/2201.00087/) (2022)
- [TopER: Topological Embeddings in Graph Representation Learning](https://mlsanity.com/paper/2410.01778/) (2024)
- [Vulnerability Detection via Topological Analysis of Attention Maps](https://mlsanity.com/paper/2410.03470/) (2024)
- [TopoDiT-3D: Topology-Aware Diffusion Transformer with Bottleneck Structure for 3D Point Cloud Generation](https://mlsanity.com/paper/2505.09140/) (2025)
- [Chordless cycle filtrations for dimensionality detection in complex networks via topological data analysis](https://mlsanity.com/paper/2509.08350/) (2025)
- [The Shape of Data: Topology Meets Analytics. A Practical Introduction to Topological Analytics and the Stability Index (TSI) in Business](https://mlsanity.com/paper/2511.13503/) (2025)
- [TopoFormer: Topology Meets Attention for Graph Learning](https://mlsanity.com/paper/2607.28259/) (2026)
- [Edge Rewiring Goes Neural: Boosting Network Resilience without Rich Features](https://mlsanity.com/paper/2110.09035/) (2021)
- [Detection of Small Holes by the Scale-Invariant Robust Density-Aware Distance (RDAD) Filtration](https://mlsanity.com/paper/2204.07821/) (2022)
- [A Multi-scale Graph Signature for Persistence Diagrams based on Return Probabilities of Random Walks](https://mlsanity.com/paper/2209.14264/) (2022)
- [A Metric Topology of Deep Learning for Data Classification](https://mlsanity.com/paper/2501.11265/) (2025)
- [Topological Spatial Graph Coarsening](https://mlsanity.com/paper/2512.24327/) (2025)
- [Frequent subgraph-based persistent homology for graph classification](https://mlsanity.com/paper/2512.24917/) (2025)
- [Denoising 3D images: robustness of persistent homology measures](https://mlsanity.com/paper/2607.24579/) (2026)
- [Signed Graph Pre-Training and Prompt Learning](https://mlsanity.com/paper/2609.25722/) (2026)

## Abstract (arXiv)

Information networks are becoming increasingly popular to capture complex relationships across various disciplines, such as social networks, citation networks, and biological networks. The primary challenge in this domain is measuring similarity or distance between networks based on topology. However, classical graph-theoretic measures are usually local and mainly based on differences between either node or edge measurements or correlations without considering the topology of networks such as the connected components or holes. In recent years, mathematical tools and deep learning based methods have become popular to extract the topological features of networks. Persistent homology (PH) is a mathematical tool in computational topology that measures the topological features of data that persist across multiple scales with applications ranging from biological networks to social networks. In this paper, we provide a conceptual review of key advancements in this area of using PH on complex network science. We give a brief mathematical background on PH, review different methods (i.e. filtrations) to define PH on networks and highlight different algorithms and applications where PH is used in solving network mining problems. In doing so, we develop a unified framework to describe these recent approaches and emphasize major conceptual distinctions. We conclude with directions for future work. We focus our review on recent approaches that get significant attention in the mathematics and data mining communities working on network data. We believe our summary of the analysis of PH on networks will provide important insights to researchers in applied network science.