On the topology of random complexes built over stationary point processes

DOI10.1214/14-AAP1075zbMath1328.60123arXiv1211.0061OpenAlexW3105553535MaRDI QIDQ894808

Robert J. Adler, Dhandapani Yogeshwaran

Publication date: 24 November 2015

Published in: The Annals of Applied Probability (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1211.0061

zbMATH Keywords

Betti numberscomponent countsstationary point processesVietoris-Rips complexesrandom geometric complexesČech complexesMorse critical points

Mathematics Subject Classification ID

Geometric probability and stochastic geometry (60D05) Stationary stochastic processes (60G10) Planar graphs; geometric and topological aspects of graph theory (05C10) Simplicial sets and complexes in algebraic topology (55U10) Point processes (e.g., Poisson, Cox, Hawkes processes) (60G55) Critical points of functions and mappings on manifolds (58K05) Combinatorial aspects of simplicial complexes (05E45)

Related Items (21)

Limit theory for U-statistics under geometric and topological constraints with rare events ⋮ Topology of random geometric complexes: a survey ⋮ Limit theorems for persistence diagrams ⋮ Limit theorems for Betti numbers of extreme sample clouds with application to persistence barcodes ⋮ Homological connectivity in random Čech complexes ⋮ Testing goodness of fit for point processes via topological data analysis ⋮ A limit theorem for persistence diagrams of random filtered complexes built over marked point processes ⋮ Bootstrapping persistent Betti numbers and other stabilizing statistics ⋮ Large deviation principle for geometric and topological functionals and associated point processes ⋮ A remark on the convergence of Betti numbers in the thermodynamic regime ⋮ Topological data analysis of noise: uniform unimodal distributions ⋮ Statistical Topology and the Random Interstellar Medium ⋮ Random Čech complexes on Riemannian manifolds ⋮ Strong law of large numbers for Betti numbers in the thermodynamic regime ⋮ Thresholds for vanishing of `isolated' faces in random Čech and Vietoris-Rips complexes ⋮ Limit theorems for process-level Betti numbers for sparse and critical regimes ⋮ Random geometric complexes in the thermodynamic regime ⋮ Topological crackle of heavy-tailed moving average processes ⋮ On limit theorems for persistent Betti numbers from dependent data ⋮ Unnamed Item ⋮ Convergence of persistence diagram in the sparse regime

Uses Software

• spatstat
• javaPlex

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