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December 2015 On the topology of random complexes built over stationary point processes
D. Yogeshwaran, Robert J. Adler
Ann. Appl. Probab. 25(6): 3338-3380 (December 2015). DOI: 10.1214/14-AAP1075

Abstract

There has been considerable recent interest, primarily motivated by problems in applied algebraic topology, in the homology of random simplicial complexes. We consider the scenario in which the vertices of the simplices are the points of a random point process in $\mathbb{R}^{d}$, and the edges and faces are determined according to some deterministic rule, typically leading to Čech and Vietoris–Rips complexes. In particular, we obtain results about homology, as measured via the growth of Betti numbers, when the vertices are the points of a general stationary point process. This significantly extends earlier results in which the points were either i.i.d. observations or the points of a Poisson process. In dealing with general point processes, in which the points exhibit dependence such as attraction or repulsion, we find phenomena quantitatively different from those observed in the i.i.d. and Poisson cases. From the point of view of topological data analysis, our results seriously impact considerations of model (non)robustness for statistical inference. Our proofs rely on analysis of subgraph and component counts of stationary point processes, which are of independent interest in stochastic geometry.

Citation

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D. Yogeshwaran. Robert J. Adler. "On the topology of random complexes built over stationary point processes." Ann. Appl. Probab. 25 (6) 3338 - 3380, December 2015. https://doi.org/10.1214/14-AAP1075

Information

Received: 1 November 2012; Revised: 1 September 2014; Published: December 2015
First available in Project Euclid: 1 October 2015

zbMATH: 1328.60123
MathSciNet: MR3404638
Digital Object Identifier: 10.1214/14-AAP1075

Subjects:
Primary: 05E45, 60D05, 60G55
Secondary: 05C10, 55U10, 58K05

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.25 • No. 6 • December 2015
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