The Mapper algorithm is one of several core techniques in Topological Data Analysis, and a major part of the power it derives in applications is its relatively high interpretability. However, for a formally reliable notion of interpretability, we would rely on some version of the Nerve lemma. Failing to check that the refined cover generating the Mapper complex can generate arbitrarily large changes to the topology of the complex, which is a cause to doubt specific interpretations of the complex.

In order to provide a statistical certification of the quality of a Mapper complex analysis, we would need to test every subset of the data associated with one of the Mapper simplices for a lack of topological structure. This places us solidly in the domain of multiple hypothesis testing, and a naïve approach is likely to lead to erroneous conclusions. We propose a method based on a generative model for feature-less data and simulation testing across all Mapper simplices as an aggregate. This produces a method with controlled family-wise rejection errors, as we can demonstrate in validation simulations.