On the upper tail of star counts in random graphs

TL;DR

This paper analyzes the probability of observing significantly more r-stars than expected in sparse random graphs, providing asymptotic results for the upper tail problem in this setting.

Contribution

It offers the first asymptotic characterization of the upper tail for subgraph counts in sparse Erdős–Rényi graphs with irregular structures.

Findings

Asymptotic formulas for upper tail probabilities of r-star counts

Extension of upper tail analysis to sparse irregular graphs

Solution to a longstanding problem in sparse subgraph count probabilities

Abstract

Let $X$ count the number of $r$-stars in the random binomial graph $\mathbb{G}(n,p)$. We determine, for fixed $r$ and $\varepsilon > 0$, the asymptotics of $\log \mathbb{P}(X \ge (1 + \varepsilon)\mathbb{E} X)$ assuming only $\mathbb{E} X \to \infty$ and $p \to 0$ thus giving a first class of irregular graphs for which the upper tail problem for subgraph counts (stated by Janson and Ruci\'nski in 2004) is solved in the sparse setting.