# Analysing Multiscale Clusterings with Persistent Homology

arXiv: https://arxiv.org/abs/2305.04281 · HTML: https://mlsanity.com/paper/2305.04281/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 58 references that are in the map.
- First posted on arXiv 7 May 2023.

## Builds on (most cited first, 17 of 17)

- [BERTopic: Neural topic modeling with a class-based TF-IDF procedure](https://mlsanity.com/paper/2203.05794/) (2022)
- [A roadmap for the computation of persistent homology](https://mlsanity.com/paper/1506.08903/) (2015)
- [Hierarchical Block Structures and High-resolution Model Selection in Large Networks](https://mlsanity.com/paper/1310.4377/) (2013)
- [Persistence Homology of Networks: Methods and Applications](https://mlsanity.com/paper/1907.08708/) (2019)
- [Random Walks, Markov Processes and the Multiscale Modular Organization of Complex Networks](https://mlsanity.com/paper/1502.04381/) (2015)
- [The many facets of community detection in complex networks](https://mlsanity.com/paper/1611.07769/) (2016)
- [Approximating Persistent Homology for Large Datasets](https://mlsanity.com/paper/2204.09155/) (2022)
- [Axiomatic Construction of Hierarchical Clustering in Asymmetric Networks](https://mlsanity.com/paper/1301.7724/) (2013)
- [Extracting Persistent Clusters in Dynamic Data via M"obius inversion](https://mlsanity.com/paper/1712.04064/) (2017)
- [HELOC Applicant Risk Performance Evaluation by Topological Hierarchical Decomposition](https://mlsanity.com/paper/1811.10658/) (2018)
- [Hierarchical community structure in networks](https://mlsanity.com/paper/2009.07196/) (2020)
- [From Free Text to Clusters of Content in Health Records: An Unsupervised Graph Partitioning Approach](https://mlsanity.com/paper/1811.05711/) (2018)
- [From trees to barcodes and back again: theoretical and statistical perspectives](https://mlsanity.com/paper/2010.11620/) (2020)
- [Multiscale mobility patterns and the restriction of human movement](https://mlsanity.com/paper/2201.06323/) (2022)
- [Multiscale methods for signal selection in single-cell data](https://mlsanity.com/paper/2206.07760/) (2022)
- [Community as a Vague Operator: Epistemological Questions for a Critical Heuristics of Community Detection Algorithms](https://mlsanity.com/paper/2210.02753/) (2022)
- [PyGenStability: Multiscale community detection with generalized Markov Stability](https://mlsanity.com/paper/2303.05385/) (2023)

## Built on by (most cited first, 2 of 2)

- [MCbiF: Measuring Topological Autocorrelation in Multiscale Clusterings via 2-Parameter Persistent Homology](https://mlsanity.com/paper/2510.14710/) (2025)
- [Conditional Entropy of Heat Diffusion on Temporal Networks](https://mlsanity.com/paper/2605.21514/) (2026)

## Abstract (arXiv)

In data clustering, it is often desirable to find not just a single partition into clusters but a sequence of partitions that describes the data at different scales (or levels of coarseness). A natural problem then is to analyse and compare the (not necessarily hierarchical) sequences of partitions that underpin such multiscale descriptions. Here, we use tools from topological data analysis and introduce the Multiscale Clustering Filtration (MCF), a well-defined and stable filtration of abstract simplicial complexes that encodes arbitrary cluster assignments in a sequence of partitions across scales of increasing coarseness. We show that the zero-dimensional persistent homology of the MCF measures the degree of hierarchy of this sequence, and the higher-dimensional persistent homology tracks the emergence and resolution of conflicts between cluster assignments across the sequence of partitions. To broaden the theoretical foundations of the MCF, we provide an equivalent construction via a nerve complex filtration, and we show that, in the hierarchical case, the MCF reduces to a Vietoris-Rips filtration of an ultrametric space. Using synthetic data, we then illustrate how the persistence diagram of the MCF provides a feature map that can serve to characterise and classify multiscale clusterings.