The Annals of Applied Probability

Connectivity of soft random geometric graphs

Mathew D. Penrose

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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.

Article information

Ann. Appl. Probab., Volume 26, Number 2 (2016), 986-1028.

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

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 05C40: Connectivity 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Random graph stochastic geometry random connection model connectivity isolated points continuum percolation


Penrose, Mathew D. Connectivity of soft random geometric graphs. Ann. Appl. Probab. 26 (2016), no. 2, 986--1028. doi:10.1214/15-AAP1110.

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