Upper Tails for Cliques

TL;DR

This paper derives tight bounds on the upper tail probabilities for the number of cliques in Erdős-Rényi random graphs, revealing precise exponential decay rates depending on graph parameters.

Contribution

It provides a sharp probabilistic bound for the upper tails of clique counts in Erdős-Rényi graphs, extending understanding of large deviations in random graph theory.

Findings

Upper tail probability bounds are tight up to constants.

Exponential decay rates depend on graph size and edge probability.

Results apply for clique sizes greater than one.

Abstract

With $\xi_{k}=\xi_{k}^{n,p}$ the number of copies of $K_k$ in the usual (Erd\H{o}s-R\'enyi) random graph $G(n,p)$, $p\geq n^{-2/(k-1)}$ and $\eta>0$, we show when $k>1$ $$\Pr(\xi_k> (1+\eta)\E \xi_k) < \exp [-\gO_{\eta,k} \min\{n^2p^{k-1}\log(1/p), n^kp^{\binom{k}{2}}\}].$$ This is tight up to the value of the constant in the exponent.