Diameters and mixing times for giant components of random graphs with given degrees

TL;DR

This paper analyzes the properties of giant components in random graphs with specified degree sequences, providing bounds on their diameter, mixing times, and the size of smaller components.

Contribution

It establishes high-probability bounds on the diameter and mixing times of giant components in random graphs with given degrees, and demonstrates the tightness of these bounds.

Findings

Giant components are almost surely unique in such graphs.

Bounds on diameter and mixing time are established and shown to be tight.

Size and diameter bounds for smaller components are also provided.

Abstract

A sequence $D = \{d_1,...d_n\}$ is a feasible degree sequence if there is a graph on $\{1,...,n\}$ such that $i$ has degree $d_i$. For such a sequence, $G(D)$ is a graph chosen uniformly at random from those with the given degree sequence. We consider sequences $\{D_\ell\}_{\ell \geq 1}$ of feasible degree sequences which have a giant component. We show that with high probability this giant component is unique, and bound its diameter and the mixing time of the random walk on it. We also bound the size and diameter of the other components and show that many of these bounds are tight.