Two-point concentration in random geometric graphs

Müller, Tobias

doi:10.1007/s00493-008-2283-3

Cited by 24 publications

(29 citation statements)

References 14 publications

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“…Here we have used a switch to polar coordinates to get the second line; we used (17) to get the third line; the change of variables y = nrz to get the sixth line; and in the last line we use k ≥ 2. Now we turn attention to R sde .…”

Section: And;mentioning

confidence: 99%

“…Random geometric graphs have been the subject of a considerable research effort in the last two decades or so. As a result, detailed information is now known on aspects such as (k-)connectivity [20,21], the largest component [22], the chromatic number and clique number [16,17] and the simple random walks on the graph [7]. A good overview of the results prior to 2003 can be found in the monograph [22].…”

Section: Introductionmentioning

confidence: 99%

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Maker‐breaker games on random geometric graphs

Beveridge

Dudek

Frieze

et al. 2014

Random Struct Algorithms

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In a Maker-Breaker game on a graph G, Breaker and Maker alternately claim edges of G. Maker wins if, after all edges have been claimed, the graph induced by his edges has some desired property. We consider four Maker-Breaker games played on random geometric graphs. For each of our four games we show that if we add edges between n points chosen uniformly at random in the unit square by order of increasing edge-length then, with probability tending to one as n → ∞, the graph becomes Maker-win the very moment it satisfies a simple necessary condition. In particular, with high probability, Maker wins the connectivity game as soon as the minimum degree is at least two; Maker wins the Hamilton cycle game as soon as the minimum degree is at least four; Maker wins the perfect matching game as soon as the minimum degree is at least two and every edge has at least three neighbouring vertices; and Maker wins the H-game as soon as there is a subgraph from a finite list of "minimal graphs". These results also allow us to give precise expressions for the limiting probability that G(n, r) is Maker-win in each case, where G(n, r) is the graph on n points chosen uniformly at random on the unit square with an edge between two points if and only if their distance is at most r.

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Section: And;mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Maker‐breaker games on random geometric graphs

Beveridge

Dudek

Frieze

et al. 2014

Random Struct Algorithms

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show abstract

“…Random geometric graphs have been the subject of a considerable research effort in the last two decades. As a result, detailed information is now known on various aspects such as (k-)connectivity [18,19], the largest component [20], the chromatic number and clique number [16,15], the (non-)existence of Hamilton cycles [3,17] and the simple random walk on the graph [8]. A good overview of the results prior to 2003 can be found in the monograph [20].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…The following Lemma is a simplification of Lemma A.1 in [16], where slightly more is proved. Lemma 3.1 ([16]).…”

Section: Proof Of the Lower Bound In Theorem 11mentioning

confidence: 99%

The acquaintance time of (percolated) random geometric graphs

Müller

Prałat

2015

European Journal of Combinatorics

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Abstract. In this paper, we study the acquaintance time AC(G) defined for a connected graph G. We focus on G(n, r, p), a random subgraph of a random geometric graph in which n vertices are chosen uniformly at random and independently from [0,1] 2 , and two vertices are adjacent with probability p if the Euclidean distance between them is at most r. We present asymptotic results for the acquaintance time of G(n, r, p) for a wide range of p = p(n) and r = r(n). In particular, we show that with high probability AC(G) = Θ(r −2 ) for G ∈ G(n, r, 1), the "ordinary" random geometric graph, provided that πnr 2 − ln n → ∞ (that is, above the connectivity threshold). For the percolated random geometric graph G ∈ G(n, r, p), we show that with high probability AC(G) = Θ(r −2 p −1 ln n), provided that pnr 2 ≥ n 1/2+ε and p < 1 − ε for some ε > 0.

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“…Random geometric graphs have been the subject of considerable research effort over the past decades, and quite precise results are now known for this model on aspects such as connectivity, Hamilton cycles, the clique number, the chromatic number and random walks on the graph. (See, e.g., [4,9,20,22,25].) A comprehensive overview of the results prior to 2003 can be found in the monograph [24].…”

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confidence: 99%

A geometric Achlioptas process

Müller¹,

Spöhel²

2015

Ann. Appl. Probab.

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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 ≪ (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 → ∞. We also show that this is tight in the following sense: for any r ≫ (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 → ∞. We also prove that the corresponding offline problem exhibits a similar threshold behaviour at r(n) = Θ(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.

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Two-point concentration in random geometric graphs

Cited by 24 publications

References 14 publications

Maker‐breaker games on random geometric graphs

Maker‐breaker games on random geometric graphs

The acquaintance time of (percolated) random geometric graphs

A geometric Achlioptas process

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