On the clique number of noisy random geometric graphs

TL;DR

This paper investigates the impact of noise on the clique number of random geometric graphs by analyzing a perturbed model where edges are randomly added or removed, providing tight bounds across various parameters.

Contribution

It introduces a new noisy perturbation model for random geometric graphs and derives asymptotically tight bounds on their clique number under different regimes.

Findings

Derived tight bounds for clique number in noisy geometric graphs

Analyzed effects of edge addition and removal probabilities

Provided asymptotic behavior across multiple parameter regimes

Abstract

Let $G_n$ be a random geometric graph, and then for $q,p \in [0,1)$ we construct a "$(q,p)$-perturbed noisy random geometric graph" $G_n^{q,p}$ where each existing edge in $G_n$ is removed with probability $q$, while and each non-existent edge in $G_n$ is inserted with probability $p$. We give asymptotically tight bounds on the clique number $\omega\left(G_n^{q,p}\right)$ for several regimes of parameter.