  Weighted digraphs are used to model a variety of natural systems and can
exhibit interesting structure across a range of scales. In order to understand
and compare these systems, we require stable, interpretable, multiscale
descriptors. To this end, we propose grounded persistent path homology (GrPPH)
- a new, functorial, topological descriptor that describes the structure of an
edge-weighted digraph via a persistence barcode. We show there is a choice of
circuit basis for the graph which yields geometrically interpretable
representatives for the features in the barcode. Moreover, we show the barcode
is stable, in bottleneck distance, to both numerical and structural
perturbations. GrPPH arises from a flexible framework, parametrised by a choice
of digraph chain complex and a choice of filtration; for completeness, we also
investigate replacing the path homology complex, used in GrPPH, by the directed
flag complex.
