Fluctuations of the giant of Poisson random graphs

TL;DR

This paper extends the understanding of fluctuations of the giant component in inhomogeneous random graphs, showing that under certain conditions, the process-level fluctuations converge to a Gaussian process similar to the Erdős-Rényi case.

Contribution

It generalizes fluctuation results from Erdős-Rényi graphs to rank-one inhomogeneous models with converging empirical weight distributions.

Findings

Fluctuations converge to a Gaussian process.

Results apply to models with converging empirical weight distributions.

Extends previous work on dynamic Erdős-Rényi graphs.

Abstract

Enriquez, Faraud, and Lemaire (2023) have established process-level fluctuations for the giant of the dynamic Erd\H{o}s-R\'{e}nyi 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.