The Annals of Applied Probability

On the topology of random complexes built over stationary point processes

D. Yogeshwaran and Robert J. Adler

Full-text: Open access

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.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 6 (2015), 3338-3380.

Dates
Received: November 2012
Revised: September 2014
First available in Project Euclid: 1 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1443703776

Digital Object Identifier
doi:10.1214/14-AAP1075

Mathematical Reviews number (MathSciNet)
MR3404638

Zentralblatt MATH identifier
1328.60123

Subjects
Primary: 60G55: Point processes 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05E45: Combinatorial aspects of simplicial complexes
Secondary: 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 55U10: Simplicial sets and complexes 58K05: Critical points of functions and mappings

Keywords
Point process random geometric complex Čech Vietoris–Rips component counts Betti numbers Morse critical points negative association

Citation

Yogeshwaran, D.; Adler, Robert J. On the topology of random complexes built over stationary point processes. Ann. Appl. Probab. 25 (2015), no. 6, 3338--3380. doi:10.1214/14-AAP1075. https://projecteuclid.org/euclid.aoap/1443703776


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