Steady and ranging sets in graph persistence

TL;DR

This paper introduces steady and ranging sets as new methods to generate persistence diagrams directly from graph features, providing a stable approach for topological analysis of weighted graphs and digraphs.

Contribution

It proposes novel graph-theoretic persistence constructions, framing them within indexing-aware functions, and establishes a stability condition for these methods.

Findings

Effective in toy examples and real-world applications

Provides a stable method for graph persistence analysis

Bridges the gap between graph properties and topological data analysis

Abstract

Topological data analysis can provide insight on the structure of weighted graphs and digraphs. However, some properties underlying a given (di)graph are hardly mappable to simplicial complexes. We introduce \textit{steady} and \textit{ranging} sets: two standardized ways of producing persistence diagrams directly from graph-theoretical features. The two constructions are framed in the context of \textit{indexing-aware persistence functions}. Furthermore, we introduce a sufficient condition for stability. Finally, we apply the steady- and ranging-based persistence constructions to toy examples and real-world applications.