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

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 $n^{2/3}$.