Phase transitions of extremal cuts for the configuration model

TL;DR

This paper investigates phase transitions in extremal cuts for the configuration model, revealing similarities with Erdős-Rényi graphs and expanding understanding of structural properties in random graph models.

Contribution

It demonstrates phase transitions in the $k$-section width and Max-Cut for the configuration model, extending known phenomena from Erdős-Rényi graphs to more general degree distributions.

Findings

Phase transitions depend on degree distribution parameters.

Transitions mirror those in Erdős-Rényi graphs.

Results apply to asymptotic degree distributions.

Abstract

The $k$-section width and the Max-Cut for the configuration model are shown to exhibit phase transitions according to the values of certain parameters of the asymptotic degree distribution. These transitions mirror those observed on Erd\H{o}s-R\'enyi random graphs, established by Luczak and McDiarmid (2001), and Coppersmith et al. (2004), respectively.