Quasi-cliques in inhomogeneous random graphs

TL;DR

This paper extends the understanding of quasi-cliques in random graphs by showing their size remains concentrated in inhomogeneous models, with explicit formulas depending on the maximum edge probability.

Contribution

It introduces a simple method to analyze quasi-cliques in inhomogeneous random graphs and provides explicit formulas based on the largest edge probability.

Findings

Quasi-clique size is concentrated in inhomogeneous graphs.

Explicit expression for quasi-clique size depending on maximum edge probability.

Extension of known results from Erdős-Rényi to inhomogeneous models.

Abstract

Given a graph $G$ and a constant $\gamma \in [0,1]$, let $\omega^{(\gamma)}(G)$ be the largest integer $r$ such that there exists an $r$-vertex subgraph of $G$ containing at least $\gamma \binom{r}{2}$ edges. It was recently shown that $\omega^{(\gamma)}(G)$ is highly concentrated when $G$ is an Erd\H{o}s-R\'enyi random graph (Balister, Bollob\'as, Sahasrabudhe, Veremyev, 2019). This paper provides a simple method to extend that result to a setting of inhomogeneous random graphs, showing that $\omega^{(\gamma)}(G)$ remains concentrated on a small range of values even if $G$ is an inhomogeneous random graph. Furthermore, we give an explicit expression for $\omega^{(\gamma)}(G)$ and show that it depends primarily on the largest edge probability of the graph $G$.