Probabilistic convergence and stability of random mapper graphs Brown, Adam Bobrowski, Omer Munch, Elizabeth Wang, Bei ddc:510 We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space X equipped with a continuous function f:X→R. We first give a categorification of the mapper graph and the Reeb graph by interpreting them in terms of cosheaves and stratified covers of the real line R. We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of (X,f) when it is applied to points randomly sampled from a probability density function concentrated on (X,f). Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of (X,f), a constructible R-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of (X,f) to the mapper of a super-level set of a probability density function concentrated on (X,f). Finally, building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b), we show that, with high probability, we can recover the mapper of the super-level set given a sufficiently large sample. Our work is the first to consider the mapper construction using the theory of cosheaves in a probabilistic setting. It is part of an ongoing effort to combine sheaf theory, probability, and statistics, to support topological data analysis with random data. Springer Nature 2021 info:eu-repo/semantics/article doc-type:article text http://purl.org/coar/resource_type/c_2df8fbb1 https://research-explorer.ista.ac.at/record/9111 https://research-explorer.ista.ac.at/download/9111/9112 Brown A, Bobrowski O, Munch E, Wang B. Probabilistic convergence and stability of random mapper graphs. <i>Journal of Applied and Computational Topology</i>. 2021;5(1):99-140. doi:<a href="https://doi.org/10.1007/s41468-020-00063-x">10.1007/s41468-020-00063-x</a> eng info:eu-repo/semantics/altIdentifier/doi/10.1007/s41468-020-00063-x info:eu-repo/semantics/altIdentifier/issn/2367-1726 info:eu-repo/semantics/altIdentifier/e-issn/2367-1734 info:eu-repo/semantics/altIdentifier/arxiv/1909.03488 info:eu-repo/semantics/openAccess