The phase transition in the configuration model

TL;DR

This paper investigates the precise nature of the phase transition for the emergence of a giant component in random graphs with fixed degree sequences, extending known results from simpler models.

Contribution

It introduces a unified method to analyze the phase transition in configuration models, providing detailed results across the entire transition window.

Findings

Precise characterization of the phase transition in configuration models.

Extension of methods from Erdős–Rényi graphs to fixed degree sequences.

Matching known results for simpler models with more complex degree constraints.

Abstract

Let $G=G(d)$ be a random graph with a given degree sequence $d$, such as a random $r$-regular graph where $r\ge 3$ is fixed and $n=|G|\to\infty$. We study the percolation phase transition on such graphs $G$, i.e., the emergence as $p$ increases of a unique giant component in the random subgraph $G[p]$ obtained by keeping edges independently with probability $p$. More generally, we study the emergence of a giant component in $G(d)$ itself as $d$ varies. We show that a single method can be used to prove very precise results below, inside and above the `scaling window' of the phase transition, matching many of the known results for the much simpler model $G(n,p)$. This method is a natural extension of that used by Bollobas and the author to study $G(n,p)$, itself based on work of Aldous and of Nachmias and Peres; the calculations are significantly more involved in the present setting.