Counting cliques in a random graph

TL;DR

This paper derives an asymptotic formula for the expected number of cliques in Erdős-Rényi random graphs, revealing how clique counts scale with graph size and edge probability.

Contribution

It provides a precise asymptotic expression for the expected number of cliques in G(n,p), advancing understanding of clique distribution in random graphs.

Findings

Expected number of cliques scales as n^{(1/(-2 log p))(log n - 2 log log n + O(1))}

Results improve theoretical understanding of clique counts in Erdős-Rényi graphs

Analytical formula applicable for large n and various p values.

Abstract

We show that the expected number of cliques in the Erd\H{o}s-R\'enyi random graph $G(n,p)$ is $n^{\frac1{-2\log p}(\log n-2\log\log n+O(1))}$.