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

TL;DR

Grounded persistent path homology (GrPPH) provides a stable, interpretable, multiscale topological descriptor for weighted directed graphs, enabling better understanding and comparison of complex systems across various scales.

Contribution

This paper introduces GrPPH, a novel functorial topological descriptor for weighted digraphs, with stability guarantees and geometrically interpretable features, expanding the toolkit for graph analysis.

Findings

Barcodes are stable under perturbations.

Circuit basis yields interpretable features.

Framework is flexible with different complexes.

Abstract

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.