2026-07-26T02:16:06Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:20015 2024-06-06T06:21:16Z ddc:004 open_access

Strange Random Topology of the Circle Lim, Uzu Topological data analysis persistent homology stochastic topology A paradigm in topological data analysis asserts that persistent homology should be computed to recover the homology of a data manifold. But could there be more to persistent homology? In this paper I bound probabilities that a random m Čech complex built on a circle attains high-dimensional topology. This builds on the known result that any nerve complex of circular arcs has the homotopy type of a bouquet of spheres. We observe a phase transition going from one 1-sphere, bouquet of 2-spheres, one 3-sphere, bouquet of 4-spheres, and so on. Furthermore, the even-dimensional Betti numbers become arbitrarily large over shrinking intervals. Our main tool is an exact computation of the expected Euler characteristic, combined with constraints on homotopy types. The systematic behaviour we observe cannot be regarded as a "topological noise", and calls for deeper investigations from the TDA community. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Uzu Lim 2024 Is Part Of LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2024.70 urn:nbn:de:0030-drops-200150 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.70 eng https://creativecommons.org/licenses/by/4.0/legalcode