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

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