Upper tail bounds for irregular graphs

TL;DR

This work analyzes upper-tail large deviations for counts of irregular subgraphs H in sparse Erdős–Rényi graphs G(n,p). It reveals a two-regime picture: a Poisson regime where upper-tail probabilities behave like those of Poisson counts with diverging mean, and a localized regime where rare events are driven by planted hub structures, with a precise variational/entropic description. For the r-armed star K_{1,r}, it further identifies an intermediate localized regime with mean-field tightness for the log-probability and derives the typical conditioned structures, including high-degree vertices and near-complete bipartite cores. The results are unified via variational principles, independence-polynomial computations on H^{*}, and a mean-field framework that extends to irregular graphs, offering sharp asymptotics and structural insights for conditioned sparse graphs.

Abstract

We consider the upper tail large deviations of subgraph counts for irregular graphs $\mathrm{H}$ in $\mathbb{G}(n,p)$, the sparse Erdős-Rényi graph on $n$ vertices with edge connectivity probability $p \in (0,1)$. For $n^{-1/Δ} \ll p \ll 1$, where $Δ$ is the maximum degree of $\mathrm{H}$, we derive the upper tail large deviations for any irregular graph $\mathrm{H}$. On the other hand, we show that for $p$ such that $1 \ll n^{v_{\mathrm{H}}} p^{e_{\mathrm{H}}} \ll (\log n)^{α^{*}_{\mathrm{H}}/\left(α^{*}_{\mathrm{H}}-1\right)}$, where $v_{\mathrm{H}}$ and $e_{\mathrm{H}}$ denote the number of vertices and edges of $\mathrm{H}$, and $α^*_{\mathrm{H}}$ denotes the fractional independence number, the upper tail large deviations of the number of unlabelled copies of $\mathrm{H}$ in $\mathbb{G}(n,p)$ is given by that of a sequence of Poisson random variables with diverging mean, for any strictly balanced graph $\mathrm{H}$. Restricting to the $r$-armed star graph we further prove a localized behavior in the intermediate range of $p$ (left open by the above two results) and show that the mean-field approximation is asymptotically tight for the logarithm of the upper tail probability. This work further identifies the typical structures of $\mathbb{G}(n,p)$ conditioned on upper tail rare events in the localized regime.

Theorems & Definitions (54)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.6
• Theorem 1.7
• Theorem 1.8: Mean-field approximation
• Conjecture 1.9
• Theorem 2.1: HMS