Giant component of the soft random geometric graph

2022/04/21 by Mathew D. Penrose, Penrose, Mathew D. · 1 citation
Mathematics · Computer Science · #Stochastic processes and statistical mechanics #Geometry and complex manifolds #Data Management and Algorithms

paper · pdf · doi:10.48550/arxiv.2204.10219

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(‖x-y‖), where ϕ: \bf 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 λ_c(ϕ).

Cited by

• Supercritical phase of the random connection model

Related

• On the first and second largest components in the percolated Random Geometric Graph
• Continuum AB percolation and AB random geometric graphs
• Distribution of components in the k-nearest neighbour random geometric graph for k below the connectivity threshold
• Anatomy of the giant component: The strictly supercritical regime
• The asymptotic size of the largest component in random geometric graphs with some applications