On the vanishing of homology in random Čech complexes. Issue 1 (7th November 2016)

• Record Type: Journal Article
• Authors: Bobrowski, Omer; Weinberger, Shmuel
• Is Part Of: Random structures & algorithms. Volume 51:Issue 1(2017)
• Page Start: 14
• Page End: 51
• Publication Date: 2016-11-07
• Subjects: random topology -- simplicial complexes -- topological data analysis -- homology; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20697
• Languages: English
• ISSNs: 1042-9832

Abstract

ABSTRACT: We compute the homology of random Čech complexes over a homogeneous Poisson process on the d ‐dimensional torus, and show that there are, coarsely, two phase transitions. The first transition is analogous to the Erdős ‐Rényi phase transition, where the Čech complex becomes connected. The second transition is where all the other homology groups are computed correctly (almost simultaneously). Our calculations also suggest a finer measurement of scales, where there is a further refinement to this picture and separation between different homology groups. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 14–51, 2017