The Average Size of Giant Components Between the Double-Jump

Abstract

We study the sizes of connected components according to their excesses during a random graph process built with $n$ vertices. The considered model is the continuous one defined in Janson 2000. An $\ell$-component is a connected component with $\ell$ edges more than vertices. $\ell$ is also called the \textit{excess} of such component. As our main result, we show that when $\ell$ and $\frac{n}{\ell }$ are both large, the expected number of vertices that ever belong to an $\ell$-component is about ${12}^{1/3}{\ell }^{1/3}{n}^{2/3}$. We also obtain limit theorems for the number of creations of $\ell$-components.

A paraître dans Algorithmica