Fluctuations of the giant of Poisson random graphs

TL;DR

The paper extends functional central limit theorems for the giant from dynamic Erdős–Rényi to dynamic rank-one inhomogeneous random graphs ${\mathcal G}_n({\bf w},\lambda)$, under weak convergence of the weight distribution ($W_n\Rightarrow W$) and convergence of its second moment. By encoding the giant via simultaneous breadth-first walks and leveraging weighted empirical process limits (Shorack) alongside Limic’s excursion representation, the authors establish a joint functional CLT for the giant’s size and volume in the supercritical regime (λ>λ_crit, with $λ_{\text{crit}}=1/\mathbb{E}[W^2]$). The limit is a two-dimensional centered Gaussian process $X(λ)$ with continuous paths, explicitly expressed in terms of two base Gaussian processes $\Psi_0,\Psi_1$, and the weight distribution through their covariance $\mathbb{E}[\Psi_p(s)\Psi_q(t)]=\mathbb{E}[W^{p+q} e^{-W s}(1-e^{-W t})]$. This work broadens the CLT toolbox for giants to inhomogeneous models and demonstrates a robust approach via simultaneous breadth-first exploration, with potential implications for barely supercritical behavior and other finite-type inhomogeneous networks.

Abstract

Enriquez, Faraud, and Lemaire (2023) have established process-level fluctuations for the giant of the dynamic Erdős-Rényi random graph above criticality and show that the limit is a centered Gaussian process with continuous sample paths. A random walk proof was recently obtained by Corujo, Limic and Lemaire (2024). We show that a similar result holds for rank-one inhomogeneous models whenever the empirical weight distribution converges to a limit and its second moment converges as well.

Theorems & Definitions (18)

• Theorem 1.1: Enriquez, Faraud and Lemaire EFL.23
• Theorem 1.3
• Theorem 2.1: Limic Limic.19
• Corollary 2.2
• proof
• Theorem 2.3: Convergence of weighted empirical processes
• Remark 2.4
• Theorem 2.5
• proof
• Theorem 3.1