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Maximal persistence in random clique complexes

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Abstract

We study the persistent homology of an Erdős–Rényi random clique complex filtration on n vertices. Here, each edge e appears independently at a uniform random time \(p_e \in [0,1]\), and the persistence of a cycle \(\sigma \) is defined as \(p_2(\sigma ) / p_1(\sigma )\), where \(p_1(\sigma )\) and \(p_2(\sigma )\) are the birth and death times of \(\sigma \). We show that if \(k \ge 1\) is fixed, then with high probability the maximal persistence of a k-cycle is of order \(n^{1/k(k+1)}\).

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Acknowledgements

We thank both anonymous referees for their corrections and helpful comments. MK also thanks Greg Malen and Andrew Newman for several helpful conversations.

Funding

Both authors gratefully acknowledge the support of NSF-DMS #2005630.

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Authors and Affiliations

  1. Department of Mathematics, The University of Jordan, Amman, 11942, Jordan

    Ayat Ababneh

  2. Department of Mathematics, The Ohio State University, Columbus, OH, USA

    Matthew Kahle

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  1. Ayat Ababneh
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Correspondence to Matthew Kahle.

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Ababneh, A., Kahle, M. Maximal persistence in random clique complexes. J Appl. and Comput. Topology (2023). https://doi.org/10.1007/s41468-023-00131-y

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  • DOI: https://doi.org/10.1007/s41468-023-00131-y

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