The reachability homology of a directed graph
Richard Hepworth, Emily Roff
TL;DR
This work introduces reachability homology, RH_*(G), as the target of the magnitude-path spectral sequence, defined via the reachability complex RC_*(G) generated by tuples encoding directed reachability. RH satisfies strong homological properties: homotopy invariance under long homotopy, a Künneth theorem for both the box and strong products, and excision plus Mayer–Vietoris for long cofibrations, making RH a robust homology theory for directed graphs. The authors establish a concrete chain-level Eilenberg–Zilber equivalence for RC(G) ⊗ RC(H) with RC(G □ H) and RC(G H), enabling algebraic Künneth decompositions with Tor terms. They also develop a framework of long cofibrations and Dwyer morphisms to prove an excision theorem and MV sequence for RH, tying RH to both the magnitude and path homologies while providing a unifying lens on the MPSS. Overall, RH serves as a principled, computable invariant that clarifies the interplay between magnitude and path homology and informs the global structure of the magnitude-path spectral sequence.
Abstract
The last decade has seen the development of path homology and magnitude homology -- two homology theories of directed graphs, each satisfying classic properties such as Kunneth and Mayer-Vietoris theorems. Recent work of Asao has shown that magnitude homology and path homology are related, appearing in different pages of a certain spectral sequence. Here we study the target of that spectral sequence, which we call reachability homology. We prove that it satisfies appropriate homotopy invariance, Kunneth, excision, and Mayer-Vietoris theorems, these all being stronger than the corresponding properties for either magnitude or path homology.