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December 2015 A geometric Achlioptas process
Tobias Müller, Reto Spöhel
Ann. Appl. Probab. 25(6): 3295-3337 (December 2015). DOI: 10.1214/14-AAP1074

Abstract

The random geometric graph is obtained by sampling $n$ points from the unit square (uniformly at random and independently), and connecting two points whenever their distance is at most $r$, for some given $r=r(n)$. We consider the following variation on the random geometric graph: in each of $n$ rounds in total, a player is offered two random points from the unit square, and has to select exactly one of these two points for inclusion in the evolving geometric graph.

We study the problem of avoiding a linear-sized (or “giant”) component in this setting. Specifically, we show that for any $r\ll(n\log\log n)^{-1/3}$ there is a strategy that succeeds in keeping all component sizes sublinear, with probability tending to one as $n\to\infty$. We also show that this is tight in the following sense: for any $r\gg(n\log\log n)^{-1/3}$, the player will be forced to create a component of size $(1-o(1))n$, no matter how he plays, again with probability tending to one as $n\to\infty$. We also prove that the corresponding offline problem exhibits a similar threshold behaviour at $r(n)=\Theta(n^{-1/3})$.

These findings should be compared to the existing results for the (ordinary) random geometric graph: there a giant component arises with high probability once $r$ is of order $n^{-1/2}$. Thus, our results show, in particular, that in the geometric setting the power of choices can be exploited to a much larger extent than in the classical Erdős–Rényi random graph, where the appearance of a giant component can only be delayed by a constant factor.

Citation

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Tobias Müller. Reto Spöhel. "A geometric Achlioptas process." Ann. Appl. Probab. 25 (6) 3295 - 3337, December 2015. https://doi.org/10.1214/14-AAP1074

Information

Received: 1 April 2013; Revised: 1 May 2014; Published: December 2015
First available in Project Euclid: 1 October 2015

zbMATH: 1326.05143
MathSciNet: MR3404637
Digital Object Identifier: 10.1214/14-AAP1074

Subjects:
Primary: 05C80, 60D05
Secondary: 60K35

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.25 • No. 6 • December 2015
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