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2022 Giant component of the soft random geometric graph
Mathew D. Penrose
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Electron. Commun. Probab. 27: 1-10 (2022). DOI: 10.1214/22-ECP491

Abstract

Consider a 2-dimensional soft random geometric graph G(λ,s,ϕ), obtained by placing a Poisson(λs2) number of vertices uniformly at random in a square of side s, with edges placed between each pair x,y of vertices with probability ϕ(xy), where ϕ:R+[0,1] is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph G(λ,s,ϕ) in the large-s limit with (λ,ϕ) fixed. We prove that the proportion of vertices in the largest component converges in probability to the percolation probability for the corresponding random connection model, which is a random graph defined similarly for a Poisson process on the whole plane. We do not cover the case where λ equals the critical value λc(ϕ).

Funding Statement

Supported by EPSRC grant EP/T028653/1.

Citation

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Mathew D. Penrose. "Giant component of the soft random geometric graph." Electron. Commun. Probab. 27 1 - 10, 2022. https://doi.org/10.1214/22-ECP491

Information

Received: 23 April 2022; Accepted: 9 October 2022; Published: 2022
First available in Project Euclid: 26 October 2022

arXiv: 2204.10219
Digital Object Identifier: 10.1214/22-ECP491

Subjects:
Primary: 60C05 , 60D05 , 60K35

Keywords: continuum percolation , random connection model , soft random geometric graph

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