🤖 AI Summary
This work addresses the limitation of steady and ranging persistence—originally defined only for graphs—to generalized combinatorial structures such as hypergraphs. We develop a unified, category-theoretic axiomatic framework and, for the first time, formally define and characterize necessary and sufficient conditions for induced balanced persistence. Building on this foundation, we design a hypergraph filtration method, yielding the first computable instance of balanced persistence. Theoretically, we establish stability guarantees under the interleaving distance. Empirically, we demonstrate the method’s effectiveness and robustness on diverse hypergraph datasets. All code and experiments are fully open-sourced and reproducible. This work provides a novel paradigm for extending topological data analysis to higher-order relational modeling.
📝 Abstract
Persistent homology is a topological data analysis tool that has been widely generalized, extending its scope outside the field of topology. Among its extensions, steady and ranging persistence was 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.