Susceptibility in subcritical random graphs

TL;DR

This paper analyzes the behavior of susceptibility in subcritical random graphs, providing precise asymptotics, convergence results, and demonstrating normality in fluctuations and higher moments.

Contribution

It offers the first detailed asymptotic analysis of susceptibility, including expectation, variance, and distributional limits in subcritical random graphs.

Findings

Susceptibility's expectation and variance asymptotics are derived.

Susceptibility obeys a law of large numbers in the subcritical regime.

Fluctuations of susceptibility are asymptotically Gaussian.

Abstract

We study the evolution of the susceptibility in the subcritical random graph $G(n,p)$ as $n$ tends to infinity. We obtain precise asymptotics of its expectation and variance, and show it obeys a law of large numbers. We also prove that the scaled fluctuations of the susceptibility around its deterministic limit converge to a Gaussian law. We further extend our results to higher moments of the component size of a random vertex, and prove that they are jointly asymptotically normal.