
06590329
j
06590329
Penrose, Mathew D.
Connectivity of soft random geometric graphs.
Ann. Appl. Probab. 26, No. 2, 9861028 (2016).
2016
Institute of Mathematical Statistics (IMS), Beachwood, OH/Bethesda, MD
EN
random graph
stochastic geometry
random connection model
connectivity
isolated points
continuum percolation
doi:10.1214/15AAP1110
euclid:aoap/1458651826
Summary: Consider a graph on $n$ uniform random points in the unit square, each pair being connected by an edge with probability $p$ if the interpoint 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.