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Probabilistic Convergence and Stability of Random Mapper Graphs

We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space $\mathbb{X}$ equipped with a continuous function $f: \mathbb{X} \rightarrow \mathbb{R}$. 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 $\mathbb{R}$. We then introduce a variant of the classic mapper graph of Singh et al.~(2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of $(\mathbb{X}, f)$ when it is applied to points randomly sampled from a probability density function concentrated on $(\mathbb{X}, f)$. Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (2018), we first show that the mapper graph of $(\mathbb{X}, f)$, a constructible $\mathbb{R}$-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of $(\mathbb{X},f)$ to the mapper of a super-level set of a probability density function concentrated on $(\mathbb{X}, f)$. Finally, building on the approach of Bobrowski et al.~(2017), 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.