Any Graph is a Mapper Graph

TL;DR

This paper shows that for any given graph, it is possible to choose Mapper parameters to produce a Mapper graph isomorphic to it, enabling targeted graph construction in topological data analysis.

Contribution

We demonstrate that any graph can be realized as a Mapper graph by appropriately selecting parameters, addressing an inverse problem in topological data analysis.

Findings

Any graph can be constructed as a Mapper graph with suitable parameters

The paper provides explicit constructions for such Mapper graphs

This enables tailored graph generation in topological data analysis

Abstract

The Mapper algorithm is a popular tool for visualization and data exploration in topological data analysis. We investigate an inverse problem for the Mapper algorithm: Given a dataset $X$ and a graph $G$, does there exist a set of Mapper parameters such that the output Mapper graph of $X$ is isomorphic to $G$? We provide constructions that affirmatively answer this question. Our results demonstrate that it is possible to engineer Mapper parameters to generate a desired graph.