Open Access
April 2016 Connectivity of soft random geometric graphs
Mathew D. Penrose
Ann. Appl. Probab. 26(2): 986-1028 (April 2016). DOI: 10.1214/15-AAP1110


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. "Connectivity of soft random geometric graphs." Ann. Appl. Probab. 26 (2) 986 - 1028, April 2016.


Received: 1 April 2014; Revised: 1 January 2015; Published: April 2016
First available in Project Euclid: 22 March 2016

zbMATH: 1339.05369
MathSciNet: MR3476631
Digital Object Identifier: 10.1214/15-AAP1110

Primary: 05C40 , 05C80 , 60D05 , 60K35

Keywords: connectivity , continuum percolation , isolated points , random connection model , random graph , Stochastic geometry

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.26 • No. 2 • April 2016
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