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 \({p \in (0, 1)}\) if the metric distance between the vertices is below a given threshold. For certain choices of V as a countable dense set in \({\mathbb{R}^n}\) 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 \({\mathbb{R}^{2}}\) with the Euclidean metric are not necessarily isomorphic.
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The authors gratefully acknowledge support from NSERC and MITACS.
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Bonato, A., Janssen, J. Infinite Random Geometric Graphs. Ann. Comb. 15, 597–617 (2011). https://doi.org/10.1007/s00026-011-0111-8
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DOI: https://doi.org/10.1007/s00026-011-0111-8