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June 2020 Optimal Cheeger cuts and bisections of random geometric graphs
Tobias Müller, Mathew D. Penrose
Ann. Appl. Probab. 30(3): 1458-1483 (June 2020). DOI: 10.1214/19-AAP1534

Abstract

Let $d\geq 2$. The Cheeger constant of a graph is the minimum surface-to-volume ratio of all subsets of the vertex set with relative volume at most 1/2. There are several ways to define surface and volume here: the simplest method is to count boundary edges (for the surface) and vertices (for the volume). We show that for a geometric (possibly weighted) graph on $n$ random points in a $d$-dimensional domain with Lipschitz boundary and with distance parameter decaying more slowly (as a function of $n$) than the connectivity threshold, the Cheeger constant (under several possible definitions of surface and volume), also known as conductance, suitably rescaled, converges for large $n$ to an analogous Cheeger-type constant of the domain. Previously, García Trillos et al. had shown this for $d\geq 3$ but had required an extra condition on the distance parameter when $d=2$.

Citation

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Tobias Müller. Mathew D. Penrose. "Optimal Cheeger cuts and bisections of random geometric graphs." Ann. Appl. Probab. 30 (3) 1458 - 1483, June 2020. https://doi.org/10.1214/19-AAP1534

Information

Received: 1 May 2018; Revised: 1 September 2019; Published: June 2020
First available in Project Euclid: 29 July 2020

MathSciNet: MR4133378
Digital Object Identifier: 10.1214/19-AAP1534

Subjects:
Primary: 05C80, 60D05
Secondary: 62H30

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 3 • June 2020
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