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Random Geometric Graph Diameter in the Unit Ball | | Robert B. Ellis
; Jeremy L. Martin
; Catherine Yan
; | | Date: |
14 Dec 2004 | | Subject: | Combinatorics; Probability MSC-class: 05C80, 60D05 | math.CO math.PR | | 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 $d$-dimensional unit ball, with two vertices adjacent if and only if their $l_p$-distance is at most $lambda$. Like its cousin the Erdos-Renyi random graph, $G$ has a connectivity threshold: an asymptotic value for $lambda$ in terms of $n$, above which $G$ is connected and below which $G$ is disconnected (and in fact has isolated vertices in most cases). In the disconnected zone, we discuss the number of isolated vertices. In the connected zone, we determine upper and lower bounds for the graph diameter of $G$. We employ a combination of methods from probabilistic combinatorics and stochastic geometry. | | Source: | arXiv, math.CO/0501214 | | Services: | Forum | Review | PDF | Favorites | |
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