The Annals of Applied Probability

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1458651826

Digital Object Identifier
doi:10.1214/15-AAP1110

Mathematical Reviews number (MathSciNet)
MR3476631

Zentralblatt MATH identifier
1339.05369

Subjects
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]

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

Citation

Penrose, Mathew D. Connectivity of soft random geometric graphs. Ann. Appl. Probab. 26 (2016), no. 2, 986--1028. doi:10.1214/15-AAP1110. https://projecteuclid.org/euclid.aoap/1458651826


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