The asymptotic variance of the giant component of configuration model random graphs

Abstract

For a supercritical configuration model random graph, it is well known that, subject to mild conditions, there exists a unique giant component, whose size $R_{n}$ is $O(n)$, where $n$ is the total number of vertices in the random graph. Moreover, there exists $0<\rho \leq 1$ such that $R_{n}/n\stackrel{p}{\longrightarrow}\rho$ as $n\rightarrow \infty$. We show that for a sequence of well behaved configuration model random graphs with a deterministic degree sequence satisfying $0<\rho <1$; there exists $\sigma^{2}>0$, such that $\operatorname{var}(\sqrt{n}(R_{n}/n-\rho))\rightarrow \sigma^{2}$ as $n\rightarrow \infty$. Moreover, an explicit, easy to compute, formula is given for $\sigma^{2}$. This provides a key stepping stone for computing the asymptotic variance of the size of the giant component for more general random graphs.