Stability and extension of steady and ranging persistence

Abstract. Persistent homology is a topological data analysis tool that has been widely generalized, extending its scope beyond the field of topology. Among its extensions, steady and ranging persistence were developed to study a wide variety of graph properties. Precisely, given a feature of interest on graphs, it is possible to build two types of persistence (steady and ranging persistence) that follow the evolution of the feature along graph filtrations. This study extends steady and ranging persistence to other objects using category theory and investigates the stability of such persistence. In particular, a characterization of the features that induce balanced steady and ranging persistence is provided. The main results of this study are illustrated using a practical implementation for hypergraphs.

Examples were conducted with hypergraph filtrations derived from Shakespeare plays. The code can be found here. Here is the scene hypergraph filtration of the play King Lear at \(t=5\) and \(t=8\):

Here is the steady persistence diagram of the mean originality feature, computed on the aforementionned hypergraph filtration:

Yann-Situ Gazull, "Stability and extension of steady and ranging persistence." Journal of Applied and Computational Topology, 2025.
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