The Average Size of Giant Components Between the Double-Jump

TL;DR

This paper analyzes the sizes of connected components with a given excess in a continuous random graph process, revealing how the expected size scales with excess and number of vertices during the double-jump phase.

Contribution

It provides new asymptotic estimates for the size and creation count of -components in the double-jump phase of a continuous random graph process.

Findings

Expected size of -components scales as ^{1/3} n^{2/3} when  and n/ are large.

Limit theorems describe the distribution of -component creation events.

Results extend understanding of component evolution during the double-jump in random graphs.

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 ${n \over \ell}$ 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.