First-order asymptotics for the structure of the inhomogeneous random graph

Abstract: In the inhomogeneous random graph model, each vertex $i\in{1,\ldots,n}$ is assigned a weight $W_i\sim\text{Unif}(0,1)$, and an edge between any two vertices $i,j$ is present with probability $k(W_i,W_j)/\lambda_n\in[0,1]$, where $k$ is a positive, symmetric function and $\lambda_n$ is a scaling parameter that controls the graph density. When $\lambda_n=1$ (resp.~$\lambda_n=O(n)$) the typical resulting graph is dense (resp.~sparse). The goal of this paper is the study of structural properties of \textit{large} inhomogeneous random graphs. We focus our attention on graph functions that grow sufficiently slowly as the graph size increases. Under some additional technical assumptions, we show that the first-order asymptotic behavior of all such properties is the same for the inhomogeneous random graph and for the Erd\H{o}s-R\'enyi random graph. Our proof relies on two couplings between the inhomogeneous random graph and appropriately constructed Erd\H{o}s-R\'enyi random graphs. We demonstrate our method by obtaining asymptotics for two structural properties of the inhomogeneous random graph which were previously unknown. In the sparse regime, we find the leading-order term for the chromatic number. In the dense regime, we find the asymptotics of the so-called $\gamma$-quasi-clique number.