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.

• Type: Journal
• Year: 2007
• Where: ALGORITHMICA
• Authors: Robert B. Ellis, Jeremy L. Martin, Catherine H. Yan
• Keywords: ALGORITHMICA 2007, Erd˝os-R´enyi Random Graph, Random Geometric Graph, Unit Ball