The Annals of Applied Probability

Monotone properties of random geometric graphs have sharp thresholds

Ashish Goel, Sanatan Rai, and Bhaskar Krishnamachari

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Random geometric graphs result from taking n uniformly distributed points in the unit cube, [0,1]d, and connecting two points if their Euclidean distance is at most r, for some prescribed r. We show that monotone properties for this class of graphs have sharp thresholds by reducing the problem to bounding the bottleneck matching on two sets of n points distributed uniformly in [0,1]d. We present upper bounds on the threshold width, and show that our bound is sharp for d=1 and at most a sublogarithmic factor away for d≥2. Interestingly, the threshold width is much sharper for random geometric graphs than for Bernoulli random graphs. Further, a random geometric graph is shown to be a subgraph, with high probability, of another independently drawn random geometric graph with a slightly larger radius; this property is shown to have no analogue for Bernoulli random graphs.

Article information

Ann. Appl. Probab. Volume 15, Number 4 (2005), 2535-2552.

First available in Project Euclid: 7 December 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 5C80 90B10: Network models, deterministic

Geometric random graphs sharp thresholds wireless networks


Goel, Ashish; Rai, Sanatan; Krishnamachari, Bhaskar. Monotone properties of random geometric graphs have sharp thresholds. Ann. Appl. Probab. 15 (2005), no. 4, 2535--2552. doi:10.1214/105051605000000575.

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