  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.
