Bigraded path homology and the magnitude-path spectral sequence

TL;DR

This paper explores the properties and computational aspects of the magnitude-path spectral sequence, introducing bigraded path homology as a new invariant that refines existing graph homology theories.

Contribution

It establishes that each page of the spectral sequence functions as a homology theory, develops a homotopy framework based on bigraded path homology, and computes these invariants for key graph families.

Findings

Every page of the spectral sequence satisfies excision and Kunneth theorems.

Bigraded path homology distinguishes graphs beyond ordinary path homology.

Complete computations of the spectral sequence for directed and bi-directed cycles.

Abstract

Two important invariants of directed graphs, namely magnitude homology and path homology, have recently been shown to be intimately connected: there is a 'magnitude-path spectral sequence' or 'MPSS' in which magnitude homology appears as the first page, and in which path homology appears as an axis of the second page. In this paper we study the homological and computational properties of the spectral sequence, and in particular of the full second page, which we now call 'bigraded path homology'. We demonstrate that every page of the MPSS deserves to be regarded as a homology theory in its own right, satisfying excision and Kunneth theorems (along with a homotopy invariance property already established by Asao), and that magnitude homology and bigraded path homology also satisfy Mayer-Vietoris theorems. We construct a homotopy theory of graphs (in the form of a cofibration category structure) in which weak equivalences are the maps inducing isomorphisms on bigraded path homology, strictly refining an existing structure based on ordinary path homology. And we provide complete computations of the MPSS for two important families of graphs - the directed and bi-directed cycles - which demonstrate the power of both the MPSS, and bigraded path homology in particular, to distinguish graphs that ordinary path homology cannot.