Infinite Random Geometric Graphs |
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Authors: | Anthony Bonato Jeannette Janssen |
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Institution: | 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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Keywords: | |
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