Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1

TL;DR

This paper establishes a diffusion approximation for component sizes in critical inhomogeneous random graphs of rank 1, revealing that the largest component scales as n^{2/3} and depends on the third moment of the vertex type distribution.

Contribution

It proves a diffusion limit for component sizes in inhomogeneous random graphs at criticality, linking the sizes to excursions of a reflecting Brownian motion with specific drift and diffusion.

Findings

Component sizes follow a joint distribution of Brownian excursions.

Largest component size scales as n^{2/3}.

Finite third moment of vertex types is necessary for the diffusion limit.

Abstract

Consider the random graph on $n$ vertices $1, ..., n$. Each vertex $i$ is assigned a type $X_i$ with $X_1, ..., X_n$ being independent identically distributed as a nonnegative discrete random variable $X$. We assume that ${\bf E} X^3<\infty$. Given types of all vertices, an edge exists between vertices $i$ and $j$ independent of anything else and with probability $\min \{1, \frac{X_iX_j}{n}(1+\frac{a}{n^{1/3}}) \}$. We study the critical phase, which is known to take place when ${\bf E} X^2=1$. We prove that normalized by $n^{-2/3}$ the asymptotic joint distributions of component sizes of the graph equals the joint distribution of the excursions of a reflecting Brownian motion $B^a(s)$ with diffusion coefficient $\sqrt{{\bf E}X{\bf E}X^3}$ and drift $a-\frac{{\bf E}X^3}{{\bf E}X}s$. This shows that finiteness of ${\bf E}X^3$ is the necessary condition for the diffusion limit. In particular, we conclude that the size of the largest connected component is of order $n^{2/3}$.