An elementary approach to component sizes in critical random graphs

Abstract

In this article we introduce a simple tool to derive polynomial upper bounds for the probability of observing unusually large maximal components in some models of random graphs when considered at criticality. Specifically, we apply our method to a model of random intersection graph, a random graph obtained through $p$-bond percolation on a general $d$-regular graph, and a model of inhomogeneous random graph.

arXiv comment16 pages - minor revision. A very similar version accepted by Journal of Applied Probability