Generalised persistence functions (gp-functions) are defined on (R, ≤)-indexed diagrams in a given category. A sufficient condition for stability is also introduced. In the category of graphs, a standard way of producing gp-functions is proposed: steady and ranging sets for a given feature. The example of steady and ranging hubs is studied in depth; their meaning is investigated in three concrete networks.

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 steady and ranging sets: two standardized ways of producing persistence diagrams directly from graph-theoretical features. The two constructions are framed in the context of 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.