Extended persistent homology of filtration of graded subgroups

Shengwen Xie; Xuezhi Zhao; Fang Sun; Shengwen Xie; Xuezhi Zhao; Fang Sun

doi:10.3934/era.2026041

Electronic Research Archive

2026, Volume 33, Issue 2: 890-909. doi: 10.3934/era.2026041

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Research article

Extended persistent homology of filtration of graded subgroups

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Received: 22 September 2025 Revised: 21 December 2025 Accepted: 19 January 2026 Published: 28 January 2026

In this work, we presented an extended persistence for filtration of graded subgroups by defining a relative homology in this setting. Our work provided a more comprehensive and flexible approach to obtaining an algebraic invariant and overcoming the limitations of the standard approach. As the application of mathematical tools in data analysis requires stability—meaning small perturbations in the input data should induce only small changes in the output—our main contribution was the development of a stability theorem for of filtration of graded subgroups. This theorem was established using an extension of the definition of interleaving, along with the rectangle measure and functor extension. We demonstrated the theorem's utility by applying it to extended persistence modules obtained from the path homology of directed graphs and the homology of hypergraphs, two important examples in topological data analysis.
- persistent homology,
- extended persistent homology,
- TDA,
- persistence modules,
- path homology,
- hypergraph
Citation: Shengwen Xie, Xuezhi Zhao, Fang Sun. Extended persistent homology of filtration of graded subgroups[J]. Electronic Research Archive, 2026, 33(2): 890-909. doi: 10.3934/era.2026041

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Abstract

In this work, we presented an extended persistence for filtration of graded subgroups by defining a relative homology in this setting. Our work provided a more comprehensive and flexible approach to obtaining an algebraic invariant and overcoming the limitations of the standard approach. As the application of mathematical tools in data analysis requires stability—meaning small perturbations in the input data should induce only small changes in the output—our main contribution was the development of a stability theorem for of filtration of graded subgroups. This theorem was established using an extension of the definition of interleaving, along with the rectangle measure and functor extension. We demonstrated the theorem's utility by applying it to extended persistence modules obtained from the path homology of directed graphs and the homology of hypergraphs, two important examples in topological data analysis.

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