Steady and ranging sets in graph persistence

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 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.

Exploring Graph and Digraph Persistence

“…The difference between any of the categorical persistence functions introduced in [7] and the functions of the present subsection (presented originally in [8]) is that the former come from a functor defined on Gr, while the latter strictly depend on the filtration, thus descending from a functor defined on (R, ≤). Definition 4.…”

“…Definition 4. [8] (Def. 5) Let p be a map assigning to each filtered (di)graph (G, f ) a categorical persistence function p (G, f ) on (R, ≤), such that p (G, f ) = p (G , f ) whenever an isomorphism between (G, f ) and (G , f ) compatible with the functions f , f exists.…”

“…Definition 8. [8] (Def. 9) We define the maximal feature mF associated with F as follows: for any X ⊆ (V ∪ E), mF (X) = true if and only if F (X) = true and there is no Y ⊆ (V ∪ E) such that X ⊂ Y and F (Y) = true.…”

“…These generalizations come in many flavors, which we shall briefly describe in Section 2.1. Here, we shall consider the rank-based persistence framework and its developments and applications detailed in [6][7][8]. This framework revolves around an axiomatic approach that yields usable definitions of persistence in categories such as graphs and quivers, without requiring the usage of auxiliary topological constructions.…”

“…In the current manuscript, following the approach described in [8,9], we provide strategies to compute rank-based persistence on weighted graphs (either directed or undirected). Weighted (di)graphs are widely used in many real-world scenarios, ranging from the analysis of interactions in social networks to context-based recommendation.…”