Component sizes of the random graph outside the scaling window

TL;DR

This paper offers straightforward proofs characterizing the size of the largest component in Erdős-Rényi random graphs outside the critical scaling window, where the edge probability slightly exceeds 1/n.

Contribution

It provides simplified proofs for the behavior of the largest component in G(n,p) outside the critical window, extending understanding beyond previous complex methods.

Findings

Largest component size outside the scaling window is characterized by simple proofs.

Behavior of the largest component is described when p={1+ε(n)}/n with ε(n)→0 and ε(n)n^{1/3}→∞.

Results clarify the phase transition in Erdős-Rényi graphs outside the critical window.

Abstract

We provide simple proofs describing the behavior of the largest component of the Erdos-Renyi random graph G(n,p) outside of the scaling window, p={1+\eps(n) \over n} where \eps(n) tends to 0, but \eps(n)n^{1/3} tends to \infty.