Exploring Graph and Digraph Persistence

Abstract

Among the various generalizations of persistent topology, that based on rank functions and leading to indexing-aware functions appears to be particularly suited to catching graph-theoretical properties without the need for a simplicial construction and a homology computation. This paper defines and studies “simple” and “single-vertex” features in directed and undirected graphs, through which several indexing-aware persistence functions are produced, within the scheme of steady and ranging sets. The implementation of the “sink” feature and its application to trust networks provide an example of the ease of use and meaningfulness of the method.