Steady and ranging sets in graph persistence

Generalised persistence functions (gp-functions) are defined on $(\mathbb{R}, \le)$-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.