# CONNECTIVITY OF SOFT RANDOM GEOMETRIC GRAPHS

@article{Penrose2016CONNECTIVITYOS,
title={CONNECTIVITY OF SOFT RANDOM GEOMETRIC GRAPHS},
author={Mathew D. Penrose},
journal={Annals of Applied Probability},
year={2016},
volume={26},
pages={986-1028}
}
• M. Penrose
• Published 15 November 2013
• Mathematics
• Annals of Applied Probability
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…
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