Abstract
The unit ball random geometric graph \(G=G^d_p(\lambda,n)\) has as its vertices n points distributed independently and uniformly in the unit ball in \({\Bbb R}^d\), with two vertices adjacent if and only if their ℓp-distance is at most λ. Like its cousin the Erdos-Renyi 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, \({\rm diam}_p({\bf B})(1-o(1))/\lambda\leq {\rm diam}(G) \leq {\rm diam}_p({\bf B})(1+O((\ln \ln n/{\rm ln}\,n)^{1/d}))/\lambda\), where \({\rm diam}_p({\bf B})\) is the ℓp-diameter of the unit ball B. We employ a combination of methods from probabilistic combinatorics and stochastic geometry.
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Ellis, R., Martin, J. & Yan, C. Random Geometric Graph Diameter in the Unit Ball. Algorithmica 47, 421–438 (2007). https://doi.org/10.1007/s00453-006-0172-y
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DOI: https://doi.org/10.1007/s00453-006-0172-y