# Structure and Stability of the One-Dimensional Mapper

## Abstract

Given a continuous function \(f:X\rightarrow \mathbb {R}\) and a cover \(\mathcal {I}\) of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover \(f^{-1}(\mathcal {I})\). Despite its success in applications, little is known about the structure and stability of this construction from a theoretical point of view. As a pixelized version of the Reeb graph of *f*, it is expected to capture a subset of its features (branches, holes), depending on how the interval cover is positioned with respect to the critical values of the function. Its stability should also depend on this positioning. We propose a theoretical framework that relates the structure of the Mapper to the one of the Reeb graphs, making it possible to predict which features will be present and which will be absent in the Mapper given the function and the cover, and for each feature, to quantify its degree of (in-)stability. Using this framework, we can derive guarantees on the structure of the Mapper, on its stability, and on its convergence to the Reeb graph as the granularity of the cover \(\mathcal {I}\) goes to zero.

## Keywords

Topological data analysis Mapper Reeb graph Topological persistence## Mathematics Subject Classification

55U10 68U05## Notes

### Acknowledgements

This work was supported by ERC grant Gudhi (ERC-2013-ADG-339025) and by ANR project TopData (ANR-13-BS01-0008). The authors would like to thank the anonymous referees for their constructive criticism and in particular for suggesting the connection to zigzag persistence, which had been overlooked in preliminary versions of this article. The second author acknowledges the support of ICERM and Brown University, as part of this work was carried out, while he was participating in the ICERM program *Topology in Motion* during the Fall of 2016.

## Supplementary material

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