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Weighted digraphs are used to model a variety of natural systems and canexhibit interesting structure across a range of scales. In order to understandand compare these systems, we require stable, interpretable, multiscaledescriptors. To this end, we propose grounded persistent path homology (GrPPH)- a new, functorial, topological descriptor that describes the structure of anedge-weighted digraph via a persistence barcode. We show there is a choice ofcircuit basis for the graph which yields geometrically interpretablerepresentatives for the features in the barcode. Moreover, we show the barcodeis stable, in bottleneck distance, to both numerical and structuralperturbations. GrPPH arises from a flexible framework, parametrised by a choiceof digraph chain complex and a choice of filtration; for completeness, we alsoinvestigate replacing the path homology complex, used in GrPPH, by the directedflag complex.

Grounded persistent path homology: a stable, topological descriptor for weighted digraphs