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

Consider the random graph on $n$ vertices $1,\mathrm{.}\mathrm{.}\mathrm{.},n$. Each vertex $i$ is assigned a type $X_i$ with $X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n$ being independent identically distributed as a nonnegative discrete random variable $X$. We assume that ${\bf E} X^3<\infty$\mathrm{.}Giventypesofallvertices,anedgeexistsbetweenvertices$i$and$j$independentofanythingelseandwithprobability$\min {1, \frac{X_iX_j}{n}(1+\frac{a}{n^{1/3}}) }$. We study the critical phase, which is known to take place when $\mathbf{E}{X}^2=1$. We prove that normalized by $n^{-2\mathrm{/}3}$ 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 $\sqrt{\mathbf{E}X\mathbf{E}{X}^3}$ and drift $a-\frac{\mathbf{E}{X}^3}{\mathbf{E}X}s$. This shows that finiteness of $\mathbf{E}{X}^3$ is the necessary condition for the diffusion limit. In particular, we conclude that the size of the largest connected component is of order $n^{2\mathrm{/}3}$.