A geometric Achlioptas process

TL;DR

This paper investigates a two-choice process in random geometric graphs, demonstrating a phase transition in the ability to prevent giant components based on the connection radius, with implications for understanding the power of choices in spatial networks.

Contribution

It introduces a new two-choice process in geometric graphs and establishes a sharp threshold for the emergence of giant components, highlighting the significant impact of choices in spatial network formation.

Findings

Successful avoidance of giant components for r << (n log log n)^{-1/3}

Inevitable giant component formation for r >> (n log log n)^{-1/3}

Offline problem exhibits similar threshold behavior at r(n)=Θ(n^{-1/3})

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 Erd\H{o}s-R\'{e}nyi random graph, where the appearance of a giant component can only be delayed by a constant factor.