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ALGORITHMICA
2007

Random Geometric Graph Diameter in the Unit Ball

11 years 12 months ago
Random Geometric Graph Diameter in the Unit Ball
The unit ball random geometric graph G = Gd p(λ, n) has as its vertices n points distributed independently and uniformly in the unit ball in Rd, with two vertices adjacent if and only if their ℓp-distance is at most λ. Like its cousin the Erd˝os-R´enyi random graph, G has a connectivity threshold: an asymptotic value for λ in terms of n, above which G is connected and below which G is disconnected. In the connected zone, we determine upper and lower bounds for the graph diameter of G. Specifically, almost always, diamp(B)(1− o(1))/λ ≤ diam(G) ≤ diamp(B)(1 + O((ln ln n/ ln n)1/d))/λ, where diamp(B) is the ℓp-diameter of the unit ball B. We employ a combination of methods from probabilistic combinatorics and stochastic geometry.
Robert B. Ellis, Jeremy L. Martin, Catherine H. Ya
Added 08 Dec 2010
Updated 08 Dec 2010
Type Journal
Year 2007
Where ALGORITHMICA
Authors Robert B. Ellis, Jeremy L. Martin, Catherine H. Yan
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