Anatomy of the giant component: The strictly supercritical regime

TL;DR

This paper provides a detailed description of the structure of the giant component in Erdős-Rényi graphs for fixed supercritical parameters, extending previous work on the critical window.

Contribution

It introduces a contiguous model for the supercritical giant component, detailing its structure through a sequence of probabilistic transformations.

Findings

Describes the kernel as a random multigraph with a given degree sequence.

Models the 2-core by replacing edges with geometric-length paths.

Attaches Poisson Galton-Watson trees to vertices to form the giant component.

Abstract

In a recent work of the authors and Kim, we derived a complete description of the largest component of the Erd\H{o}s-R\'enyi random graph $G(n,p)$ as it emerges from the critical window, i.e. for $p = (1+\epsilon)/n$ where $\epsilon^3 n \to\infty$ and $\epsilon=o(1)$, in terms of a tractable contiguous model. Here we provide the analogous description for the supercritical giant component, i.e., the largest component of $G(n,p)$ for $p = \lambda/n$ where $\lambda>1$ is fixed. The contiguous model is roughly as follows: Take a random degree sequence and sample a random multigraph with these degrees to arrive at the kernel; Replace the edges by paths whose lengths are i.i.d. geometric variables to arrive at the 2-core; Attach i.i.d. Poisson Galton-Watson trees to the vertices for the final giant component. As in the case of the emerging giant, we obtain this result via a sequence of contiguity arguments at the heart of which are Kim's Poisson-cloning method and the Pittel-Wormald local limit theorems.