# Optimal Cheeger cuts and bisections of random geometric graphs

Mathew Penrose, Tobias Müller

Research output: Contribution to journalArticle

### Abstract

Let d ≥ 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, Garca Trillos et al had shown this for d ≥ 3 but had required an extra condition on the distance parameter when d = 2.
Original language English 33 Annals of Applied Probability Accepted/In press - 14 Sep 2019

### Cite this

In: Annals of Applied Probability, 14.09.2019.

Research output: Contribution to journalArticle

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abstract = "Let d ≥ 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, Garca Trillos et al had shown this for d ≥ 3 but had required an extra condition on the distance parameter when d = 2.",
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AU - Müller, Tobias

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AB - Let d ≥ 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, Garca Trillos et al had shown this for d ≥ 3 but had required an extra condition on the distance parameter when d = 2.

UR - https://arxiv.org/abs/1805.08669

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JF - Annals of Applied Probability

SN - 1050-5164

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