Largest component of subcritical random graphs with given degree sequence

TL;DR

This paper analyzes the size of the largest component in subcritical random graphs with prescribed degree sequences, providing tight bounds for the configuration model and uniform model, including cases with infinite variance degrees.

Contribution

It offers new asymptotically tight bounds for the largest component in subcritical regimes, improving prior results and applying to degree sequences with infinite variance.

Findings

Upper bounds for the configuration model are asymptotically tight.

Weaker bounds for the uniform model are tight up to logarithmic factors.

Results apply to degree sequences with infinite variance.

Abstract

We study the size of the largest component of two models of random graphs with prescribed degree sequence, the configuration model (CM) and the uniform model (UM), in the (barely) subcritical regime. For the CM, we give upper bounds that are asymptotically tight for certain degree sequences. These bounds hold under mild conditions on the sequence and improve previous results of Hatami and Molloy on the barely subcritical regime. For the UM, we give weaker upper bounds that are tight up to logarithmic terms but require no assumptions on the degree sequence. In particular, the latter result applies to degree sequences with infinite variance in the subcritical regime.