Is magnitude 'generically continuous' for finite metric spaces?

TL;DR

This paper studies the stability of the magnitude invariant in finite metric spaces, revealing it is nowhere continuous but potentially generically stable, which is important for applications in topological data analysis.

Contribution

It demonstrates the lack of continuity of magnitude in finite metric spaces and provides evidence for its generic stability, highlighting its relevance for practical applications.

Findings

Magnitude is nowhere continuous on the Gromov-Hausdorff space.

Evidence suggests magnitude may be generically continuous.

Generic stability of magnitude could be key for applications.

Abstract

Magnitude is a real-valued invariant of metric spaces which, in the finite setting, can be understood as recording the 'effective number of points' in a space as the scale of the metric varies. Motivated by applications in topological data analysis, this paper investigates the stability of magnitude: its continuity properties with respect to the Gromov-Hausdorff topology. We show that magnitude is nowhere continuous on the Gromov-Hausdorff space of finite metric spaces. Yet, we find evidence to suggest that it may be 'generically continuous', in the sense that generic Gromov-Hausdorff limits are preserved by magnitude. We make the case that, in fact, 'generic stability' is what matters for applicability.