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arXiv · 1311.3897

Connectivity of soft random geometric graphs

Mathew D. Penrose
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Abstract

Consider a graph on $n$ uniform random points in the unit square, each pair being connected by an edge with probability $p$ if the inter-point distance is at most $r$. We show that as $n\to\infty$ the probability of full connectivity is governed by that of having no isolated vertices, itself governed by a Poisson approximation for the number of isolated vertices, uniformly over all choices of $p,r$. We determine the asymptotic probability of connectivity for all $(p_n,r_n)$ subject to $r_n=O(n^{-\varepsilon})$, some $\varepsilon >0$. We generalize the first result to higher dimensions and to a larger class of connection probability functions.

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Mathew D. Penrose. 2016-04-06. Connectivity of soft random geometric graphs. https://doi.org/10.1214/15-aap1110

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