A notion of homotopy for directed graphs and their flag complexes

Directed graphs can be studied by their associated directed flag complex. The homology of this complex has been successful in applications as a topological invariant for digraphs. Through comparison with path homology theory, we derive a homotopy-like equivalence relation on digraph maps such that equivalent maps induce identical maps on the homology of the directed flag complex. Thus, we obtain an equivalence relation on digraphs such that equivalent digraphs have directed flag complexes with isomorphic homology. With the help of these relations, we can prove a generic stability theorem for the persistent homology of the directed flag complex of filtered digraphs. In particular, we show that the persistent homology of the directed flag complex of the shortest-path filtration of a weighted directed acyclic graph is stable to edge subdivision. In contrast, we also discuss some important instabilities that are not present in persistent path homology. We also derive similar equivalence relations for ordered simplicial complexes at large. Since such complexes can alternatively be viewed as simplicial sets, we verify that these two perspectives yield identical relations.