Multiscale 2-Mapper -- Exploratory Data Analysis Guided by the First Betti Number

Abstract: The Mapper algorithm is a fundamental tool in exploratory topological data analysis for identifying connectivity and topological clustering in data. Derived from the nerve construction, Mapper graphs can contain additional information about clustering density when considering the higher-dimensional skeleta. To observe two-dimensional features, and capture one-dimensional topology, we construct 2-Mapper. A common issue in using Mapper algorithms is parameter choice. We develop tools to choose 2-Mapper parameters that reflect persistent Betti-1 information. Computationally, we study how cover choice affects 2-Mapper and analyze this through a computational Multiscale Mapper algorithm. We test our constructions on three-dimensional shape data, including the Klein bottle.