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Persistent homology in graph power filtrations

The persistence of homological features in simplicial complex representations of big datasets in R(n) resulting from Vietoris–Rips or Čech filtrations is commonly used to probe the topological structure of such datasets. In this paper, the notion of homological persistence in simplicial complexes ob...

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Autores principales: Parks, Allen D., Marchette, David J.
Formato: Online Artículo Texto
Lenguaje:English
Publicado: The Royal Society 2016
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5098965/
https://www.ncbi.nlm.nih.gov/pubmed/27853540
http://dx.doi.org/10.1098/rsos.160228
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author Parks, Allen D.
Marchette, David J.
author_facet Parks, Allen D.
Marchette, David J.
author_sort Parks, Allen D.
collection PubMed
description The persistence of homological features in simplicial complex representations of big datasets in R(n) resulting from Vietoris–Rips or Čech filtrations is commonly used to probe the topological structure of such datasets. In this paper, the notion of homological persistence in simplicial complexes obtained from power filtrations of graphs is introduced. Specifically, the rth complex, r ≥ 1, in such a power filtration is the clique complex of the rth power G(r) of a simple graph G. Because the graph distance in G is the relevant proximity parameter, unlike a Euclidean filtration of a dataset where regional scale differences can be an issue, persistence in power filtrations provides a scale-free insight into the topology of G. It is shown that for a power filtration of G, the girth of G defines an r range over which the homology of the complexes in the filtration are guaranteed to persist in all dimensions. The role of chordal graphs as trivial homology delimiters in power filtrations is also discussed and the related notions of ‘persistent triviality’, ‘transient noise’ and ‘persistent periodicity’ in power filtrations are introduced.
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spelling pubmed-50989652016-11-16 Persistent homology in graph power filtrations Parks, Allen D. Marchette, David J. R Soc Open Sci Mathematics The persistence of homological features in simplicial complex representations of big datasets in R(n) resulting from Vietoris–Rips or Čech filtrations is commonly used to probe the topological structure of such datasets. In this paper, the notion of homological persistence in simplicial complexes obtained from power filtrations of graphs is introduced. Specifically, the rth complex, r ≥ 1, in such a power filtration is the clique complex of the rth power G(r) of a simple graph G. Because the graph distance in G is the relevant proximity parameter, unlike a Euclidean filtration of a dataset where regional scale differences can be an issue, persistence in power filtrations provides a scale-free insight into the topology of G. It is shown that for a power filtration of G, the girth of G defines an r range over which the homology of the complexes in the filtration are guaranteed to persist in all dimensions. The role of chordal graphs as trivial homology delimiters in power filtrations is also discussed and the related notions of ‘persistent triviality’, ‘transient noise’ and ‘persistent periodicity’ in power filtrations are introduced. The Royal Society 2016-10-26 /pmc/articles/PMC5098965/ /pubmed/27853540 http://dx.doi.org/10.1098/rsos.160228 Text en © 2016 The Authors. http://creativecommons.org/licenses/by/4.0/ Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.
spellingShingle Mathematics
Parks, Allen D.
Marchette, David J.
Persistent homology in graph power filtrations
title Persistent homology in graph power filtrations
title_full Persistent homology in graph power filtrations
title_fullStr Persistent homology in graph power filtrations
title_full_unstemmed Persistent homology in graph power filtrations
title_short Persistent homology in graph power filtrations
title_sort persistent homology in graph power filtrations
topic Mathematics
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5098965/
https://www.ncbi.nlm.nih.gov/pubmed/27853540
http://dx.doi.org/10.1098/rsos.160228
work_keys_str_mv AT parksallend persistenthomologyingraphpowerfiltrations
AT marchettedavidj persistenthomologyingraphpowerfiltrations