Upper tails and independence polynomials in random graphs

TL;DR

This paper extends the understanding of upper tail probabilities in Erdős–Rényi random graphs for any fixed graph H, revealing that the rate function is governed by the independence polynomial of H, and providing explicit asymptotic formulas.

Contribution

It generalizes previous results from cliques to arbitrary fixed graphs H, linking the large deviation rate to the independence polynomial of H.

Findings

Derived explicit asymptotics for upper tail probabilities for any fixed graph H.

Connected the large deviation rate function to the independence polynomial of H.

Provided formulas for the rate function in terms of graph parameters and the independence polynomial.

Abstract

The upper tail problem in the Erd\H{o}s--R\'enyi random graph $G\sim\mathcal{G}_{n,p}$ asks to estimate the probability that the number of copies of a graph $H$ in $G$ exceeds its expectation by a factor $1+\delta$. Chatterjee and Dembo showed that in the sparse regime of $p\to 0$ as $n\to\infty$ with $p \geq n^{-\alpha}$ for an explicit $\alpha=\alpha_H>0$, this problem reduces to a natural variational problem on weighted graphs, which was thereafter asymptotically solved by two of the authors in the case where $H$ is a clique. Here we extend the latter work to any fixed graph $H$ and determine a function $c_H(\delta)$ such that, for $p$ as above and any fixed $\delta>0$, the upper tail probability is $\exp[-(c_H(\delta)+o(1))n^2 p^\Delta \log(1/p)]$, where $\Delta$ is the maximum degree of $H$. As it turns out, the leading order constant in the large deviation rate function, $c_H(\delta)$, is governed by the independence polynomial of $H$, defined as $P_H(x)=\sum i_H(k) x^k $ where $i_H(k)$ is the number of independent sets of size $k$ in $H$. For instance, if $H$ is a regular graph on $m$ vertices, then $c_H(\delta)$ is the minimum between $\frac12 \delta^{2/m}$ and the unique positive solution of $P_H(x) = 1+\delta$.