Infinite Random Geometric Graphs |
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Authors: | Anthony Bonato Jeannette Janssen |
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Affiliation: | 1. Department of Mathematics, Ryerson University, Toronto, ON, M5B 2K3, Canada 2. Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H 3J5, Canada
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Abstract: | We introduce a new class of countably infinite random geometric graphs, whose vertices V are points in a metric space, and vertices are adjacent independently with probability p ? (0, 1){p in (0, 1)} if the metric distance between the vertices is below a given threshold. For certain choices of V as a countable dense set in mathbbRn{mathbb{R}^n} equipped with the metric derived from the L ∞-norm, it is shown that with probability 1 such infinite random geometric graphs have a unique isomorphism type. The isomorphism type, which we call GR n , is characterized by a geometric analogue of the existentially closed adjacency property, and we give a deterministic construction of GR n . In contrast, we show that infinite random geometric graphs in mathbbR2{mathbb{R}^{2}} with the Euclidean metric are not necessarily isomorphic. |
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