The Annals of Applied Probability

A geometric Achlioptas process

Tobias Müller and Reto Spöhel

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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.

Article information

Ann. Appl. Probab., Volume 25, Number 6 (2015), 3295-3337.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Random geometric graph Achlioptas process


Müller, Tobias; Spöhel, Reto. A geometric Achlioptas process. Ann. Appl. Probab. 25 (2015), no. 6, 3295--3337. doi:10.1214/14-AAP1074.

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