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

TL;DR

This paper studies the asymptotic behavior of the Frobenius distance to Fréchet means in Erdős-Rényi random graphs, revealing phase transitions and invariance properties.

Contribution

It characterizes the Fréchet mean set for Erdős-Rényi graphs and derives limit laws for the Frobenius distance, including asymptotic normality under specific scaling.

Findings

The Fréchet mean set consists of quasi-regular graphs.

Derived closed-form expressions for the mean and variance of the Frobenius distance.

Established weak convergence laws and asymptotic normality with phase transition.

Abstract

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.