A short proof of a central limit theorem for the order of the giant component and $k$-core

TL;DR

This paper introduces a simple method using the Efron--Stein inequality to prove central limit theorems for global graph parameters like the giant component and $k$-core in sparse random graphs.

Contribution

It presents a new, straightforward approach for establishing CLTs for global graph properties based on local approximations and stability analysis.

Findings

Proves CLT for the size of the giant component.

Establishes CLT for the $k$-core in sparse graphs.

Simplifies previous proofs using combinatorial analysis.

Abstract

In this note we outline a new and simple approach to proving central limit theorems for various 'global' graph parameters which have robust 'local' approximations, using the Efron--Stein inequality, which relies on a combinatorial analysis of the stability of these approximations under resampling an edge. As an application, we give short proofs of a central limit theorem for the order of the giant component and of the $k$-core for sparse random graphs.