Distance graphs: from random geometric graphs to Bernoulli graphs and between

14 years 19 days ago

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A random geometric graph G(n, r) is a graph resulting from placing n points uniformly at random on the unit area disk, and connecting two points iff their Euclidean distance is at most the radius r(n). Recently, this class of graphs have received much attention as a model for wireless networks. The Bernoulli graph B(n, p) is a random graph in which each edge is chosen independently with edge probability p(n). The critical parameter for connectivity played a major role in the study of both G(n, r) and B(n, p), and in what may seem surprising, it has been shown that both graphs have closely related critical connectivity thresholds for the radius and the edge probability. In particular, if r2 = p = log n+n n then both graphs are connected w.h.p. iff n + and disconnected w.h.p. iff n -. To explain the similarities in the connectivity thresholds, we introduce an extension of the random geometric graphs: the random distance graph, D(n, g): A graph resulting from placing n points uniformly ...

Chen Avin

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