The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 15, Number 4 (2005), 2535-2552.
Monotone properties of random geometric graphs have sharp thresholds
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.
Ann. Appl. Probab. Volume 15, Number 4 (2005), 2535-2552.
First available in Project Euclid: 7 December 2005
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 5C80 90B10: Network models, deterministic
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. http://projecteuclid.org/euclid.aoap/1133965771.