We consider the upper tail large deviations of subgraph counts for irregular graphs $\mathrm{ H}$ in $\mathbb{ G}(n,p)$, the sparse Erd\H{ o}s-R\'enyi graph on $n$ vertices with edge connectivity probability $p \in (0,1)$. For $n^{ -1/\Delta} \ll p \ll 1$, where $\Delta$ 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 $1 \ll n^{ v_{ \mathrm{ H}}} p^{ e_{ \mathrm{ H}}} \ll (\log n)^{ \alpha^{ *}_{ \mathrm{ H}}/\left(\alpha^{ *}_{ \mathrm{ H}}-1\right)}$, where $v_{ \mathrm{ H}}$ and $e_{ \mathrm{ H}}$ denote the number of vertices and edges of $\mathrm{ H}$, and $\alpha^*_{ \mathrm{ H}}$ 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.
Categories: math.PR math.CO