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arXiv · 0908.2590

Infinite random geometric graphs

Anthony Bonato·Jeannette Janssen
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Abstract

We introduce a new class of countably infinite random geometric graphs, whose vertices are points in a metric space, and vertices are adjacent independently with probability p if the metric distance between the vertices is below a given threshold. If the vertex set is a countable dense set in R^n equipped with the metric derived from the L_{\infty}-norm, then 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 R^2 with the Euclidean metric are not necessarily isomorphic.

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Anthony Bonato, Jeannette Janssen. 2009-08-18. Infinite random geometric graphs. https://arxiv.org/abs/0908.2590

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