The Average Size of Giant Components Between the Double-Jump

Details The Average Size of Giant Components Between the Double-Jump The Average Size of Giant Components Between the Double-Jump

2006-07-12

arxiv.org/abs/cs/0607057v1

Algorithmica Issue sp\'{e}ciale "Analysis of Algorithms" (2006) A para\^{i}tre

Computer Science

Discrete Mathematics

A para\^{i}tre dans Algorithmica

Scientific paper

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.

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