Giant component of the soft random geometric graph

TL;DR

This paper studies the large-scale behavior of a 2D soft random geometric graph, showing that the largest component's size converges to the percolation probability as the domain size grows, under fixed parameters.

Contribution

It establishes the asymptotic proportion of vertices in the largest component converges to the percolation probability for the associated infinite model, extending understanding of phase transitions in such graphs.

Findings

Proportion of vertices in the largest component converges to the percolation probability.

Asymptotic behavior analyzed in the large domain limit with fixed parameters.

Results apply to the non-critical regime, excluding the critical point.

Abstract

Consider a 2-dimensional soft random geometric graph $G(\lambda,s,\phi)$, obtained by placing a Poisson($\lambda s^2$) number of vertices uniformly at random in a square of side $s$, with edges placed between each pair $x,y$ of vertices with probability $\phi(\|x-y\|)$, where $\phi: {\bf R}_+ \to [0,1]$ is a finite-range connection function. This paper is concerned with the asymptotic behaviour of the graph $G(\lambda,s,\phi)$ in the large-$s$ limit with $(\lambda,\phi)$ 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 $\lambda$ equals the critical value $\lambda_c(\phi)$.