Optimal recovery of correlated Erd\H{o}s-R\'enyi graphs

For two unlabeled graphs $G_1,G_2$ independently sub-sampled from an Erd\H{o}s-R\'enyi graph $\mathbf G(n,p)$ by keeping each edge with probability $s$, we aim to recover \emph{as many as possible} of the corresponding vertex pairs. We establish a connection between the recoverability of vertex pairs and the balanced load allocation in the true intersection graph of $G_1$ and $G_2$. Using this connection, we analyze the partial recovery regime where $p = n^{-\alpha + o(1)}$ for some $\alpha \in (0, 1]$ and $nps^2 = \lambda = O(1)$. We derive upper and lower bounds for the recoverable fraction in terms of $\alpha$ and the limiting load distribution $\mu_\lambda$ (as introduced in \cite{AS16}). These bounds coincide asymptotically whenever $\alpha^{-1}$ is not an atom of $\mu_\lambda$. Therefore, for each fixed $\lambda$, our result characterizes the asymptotic optimal recovery fraction for all but countably many $\alpha \in (0, 1]$.