Dense Subgraphs in Random Graphs

TL;DR

This paper characterizes the size of dense subgraphs with a given edge density in random graphs, showing they concentrate on two specific sizes and developing new techniques for analyzing overlaps.

Contribution

It provides a precise probabilistic description of dense subgraph sizes in Erdős–Rényi graphs, extending understanding beyond cliques to quasi-cliques with new analytical methods.

Findings

Dense subgraph size concentrates on two integers

Explicit formula for the typical size involving logarithms

New techniques for handling overlaps in quasi-cliques

Abstract

For a constant $\gamma \in[0,1]$ and a graph $G$, let $\omega_{\gamma}(G)$ be the largest integer $k$ for which there exists a $k$-vertex subgraph of $G$ with at least $\gamma\binom{k}{2}$ edges. We show that if $0<p<\gamma<1$ then $\omega_{\gamma}(G_{n,p})$ is concentrated on a set of two integers. More precisely, with $\alpha(\gamma,p)=\gamma\log\frac{\gamma}{p}+(1-\gamma)\log\frac{1-\gamma}{1-p}$, we show that $\omega_{\gamma}(G_{n,p})$ is one of the two integers closest to $\frac{2}{\alpha(\gamma,p)}\big(\log n-\log\log n+\log\frac{e\alpha(\gamma,p)}{2}\big)+\frac{1}{2}$, with high probability. While this situation parallels that of cliques in random graphs, a new technique is required to handle the more complicated ways in which these "quasi-cliques" may overlap.