Concentration of measure for the number of isolated vertices in the Erd\H{o}s-R\'{e}nyi random graph by size bias couplings

TL;DR

This paper establishes concentration inequalities for the number of isolated vertices in Erdős-Rényi graphs using size bias couplings, providing bounds on tail probabilities under certain asymptotic conditions.

Contribution

It introduces a novel application of size bias couplings to derive concentration bounds for isolated vertices in Erdős-Rényi graphs, extending previous methods.

Findings

Exponential tail bounds for the number of isolated vertices.

Left tail inequality applicable for all graph sizes and probabilities.

Results valid under the asymptotic regime where np approaches a positive constant.

Abstract

A concentration of measure result is proved for the number of isolated vertices $Y$ in the Erd\H{o}s-R\'{e}nyi random graph model on $n$ edges with edge probability $p$. When $\mu$ and $\sigma^2$ denote the mean and variance of $Y$ respectively, $P((Y-\mu)/\sigma\ge t)$ admits a bound of the form $e^{-kt^2}$ for some constant positive $k$ under the assumption $p \in (0,1)$ and $np\rightarrow c \in (0,\infty)$ as $n \rightarrow \infty$. The left tail inequality $$ P(\frac{Y-\mu}{\sigma}\le -t)&\le& \exp(-\frac{t^2\sigma^2}{4\mu}) $$ holds for all $n \in {2,3,...},p \in (0,1)$ and $t \ge 0$. The results are shown by coupling $Y$ to a random variable $Y^s$ having the $Y$-size biased distribution, that is, the distribution characterized by $E[Yf(Y)]=\mu E[f(Y^s)] $ for all functions $f$ for which these expectations exist.