Grounded persistent path homology: a stable, topological descriptor for weighted digraphs
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2022
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915007894126592 |
|---|---|
| author | Chaplin, Thomas Harrington, Heather A. Tillmann, Ulrike |
| author_facet | Chaplin, Thomas Harrington, Heather A. Tillmann, Ulrike |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_11274 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Grounded persistent path homology: a stable, topological descriptor for weighted digraphs Chaplin, Thomas Harrington, Heather A. Tillmann, Ulrike Algebraic Topology 55N31 (Primary), 05C20 (Secondary), 05C22 (Secondary) G.2.2 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. |
| title | Grounded persistent path homology: a stable, topological descriptor for weighted digraphs |
| topic | Algebraic Topology 55N31 (Primary), 05C20 (Secondary), 05C22 (Secondary) G.2.2 |
| url | https://arxiv.org/abs/2210.11274 |