Scaling limits for critical inhomogeneous random graphs with finite third moments

TL;DR

This paper determines the limiting distribution of the largest component sizes in critical inhomogeneous random graphs with degree exponent greater than 4, showing convergence to inhomogeneous Brownian motion excursions.

Contribution

It extends previous results by identifying the scaling limits for component sizes in a new regime with finite third moments, using martingale techniques.

Findings

Component sizes converge to inhomogeneous Brownian motion excursion lengths.

Results extend the understanding of critical behavior in inhomogeneous random graphs.

Uses martingale convergence and concentration methods.

Abstract

We identify the scaling limits for the sizes of the largest components at criticality for inhomogeneous random graphs when the degree exponent $\tau$ satisfies $\tau>4$. We see that the sizes of the (rescaled) components converge to the excursion lengths of an inhomogeneous Brownian motion, extending results of \cite{Aldo97}. We rely heavily on martingale convergence techniques, and concentration properties of (super)martingales. This paper is part of a programme to study the critical behavior in inhomogeneous random graphs of so-called rank-1 initiated in \cite{Hofs09a}.