Scaling of Components in Critical Geometric Random Graphs on 2-dim Torus

TL;DR

This paper investigates the critical phase of a 2D torus inhomogeneous random graph, deriving a diffusion approximation for the largest component size, and shows it shares scaling properties with Erdős-Rényi graphs.

Contribution

It provides the first diffusion approximation for the largest component in the critical phase of this inhomogeneous random graph model, extending understanding of its scaling behavior.

Findings

Largest component size scales with (N^2)^{2/3} in the critical phase

Model shares the same class as Erdős-Rényi graphs regarding component scaling

Diffusion approximation for component size in the critical regime

Abstract

We consider random graphs on the set of $N^2$ vertices placed on the discrete $2$-dimensional torus. The edges between pairs of vertices are independent, and their probabilities decay with the distance $\rho$ between these vertices as $(N\rho)^{-1}$. This is an example of an inhomogeneous random graph which is not of rank 1. The reported previously results on the sub- and super-critical cases of this model exhibit great similarity to the classical Erd\H{o}s-R\'{e}nyi graphs. Here we study the critical phase. A diffusion approximation for the size of the largest connected component rescaled with $(N^2)^{-2/3}$ is derived. This completes the proof that in all regimes the model is within the same class as Erd\H{o}s-R\'{e}nyi graph with respect to scaling of the largest component.