Limit Laws for the Distance to Fr\'echet Means of Random Graphs

This paper investigates the Fr\'echet mean of the Erd\H{o}s-R\'enyi random graph $G_{n,p}$ with respect to the Frobenius distance on graph Laplacians, a metric that captures global structural information beyond local edge flips. We first characterize the Fr\'echet mean set as consisting of quasi-regular graphs (i.e., graphs where all vertex degrees differ by at most one). We then analyze the asymptotic behavior of the Frobenius distance $F_n=d_{\mathrm{F}}(G_{n,p},R)$ as $n\to\infty$, where $R$ is any Fr\'echet mean. Closed-form expressions for the mean and variance of $F_n^2$ are derived, which are invariant to the choice of $R$. Leveraging these results, we establish several weak convergence laws for the Frobenius distance over all regimes of $p \in (0,1)$ as $n \to \infty$. Finally, under the scaling condition $n^2 p(1-p) \to \infty$ we prove the asymptotic normality of this distance, which exhibits a phase transition governed by the growth rate of $np(1-p)$. Our results reveal how metric selection fundamentally shapes Fr\'echet mean geometry in random graphs.