MAXIMALLY PERSISTENT CYCLES IN RANDOM GEOMETRIC COMPLEXES

• 成果类型: Article
• 署名作者: Bobrowski, Omer; Kahle, Matthew; Skraba, Primoz
• 署名单位: Duke University; University System of Ohio; Ohio State University; Slovenian Academy of Sciences & Arts (SASA); Jozef Stefan Institute; University of Primorska; Technion Israel Institute of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/16-AAP1232
• 发表日期: 2017
• 页码: 2032-2060
• 关键词: topology; connectivity; HOMOLOGY; points; graph

摘要

We initiate the study of persistent homology of random geometric simplicial complexes. Our main interest is in maximally persistent cycles of degree-k in persistent homology, for a either the Cech or the Vietoris-Rips filtration built on a uniform Poisson process of intensity n in the unit cube [0, 1](d). This is a natural way of measuring the largest k-dimensional hole in a random point set. This problem is in the intersection of geometric probability and algebraic topology, and is naturally motivated by a probabilistic view of topological inference. We show that for all d >= 2 and 1 <= k <= d - 1 the maximally persistent cycle has (multiplicative) persistence of order Theta((log n/log log n)(1/k)), with high probability, characterizing its rate of growth as n -> infinity. The implied constants depend on k, d and on whether we consider the Vietoris-Rips or Cech filtration.