Large deviations of the giant component in scale-free inhomogeneous random graphs

TL;DR

This paper investigates the probabilities of unusually large connected components in inhomogeneous scale-free random graphs, revealing that such events are driven by vertices with linearly increasing degrees and establishing related distributional limits.

Contribution

It introduces a large deviation principle for the size of the largest component in inhomogeneous random graphs with regularly varying degree distributions, highlighting the role of high-degree vertices.

Findings

Large deviation principle with logarithmic speed for the largest component

Rare events involve many vertices with linear degrees

Distributional limits of weight and component size under rare events

Abstract

We study large deviations of the size of the largest connected component in a general class of inhomogeneous random graphs with iid weights, parametrized so that the degree distribution is regularly varying. We derive a large-deviation principle with logarithmic speed: the rare event that the largest component contains linearly more vertices than expected is caused by the presence of constantly many vertices with linear degree. Conditionally on this rare event, we prove distributional limits of the weight distribution and component-size distribution.