A moment-generating formula for Erd\H{o}s-R\'enyi component sizes

TL;DR

This paper introduces a simple formula for the distribution of component sizes in Erdős-Rényi graphs, enabling elementary proofs of key properties like susceptibility and the giant component's size.

Contribution

It provides a novel, straightforward formula for component size distribution in Erdős-Rényi graphs, simplifying proofs of existing results.

Findings

Elementary proof of susceptibility in subcritical graphs

Central limit theorem for giant component size

Characterization of component size distribution

Abstract

We derive a simple formula characterizing the distribution of the size of the connected component of a fixed vertex in the Erd\H{o}s-R\'enyi random graph which allows us to give elementary proofs of some results of Federico, van der Hofstad, den Hollander and Hulshof as well as Janson and Luczak about the susceptibility in the subcritical graph and the central limit theorem of Pittel for the size of the giant component in the supercritical graph.