Large deviations for the largest singular value of sparse non-Hermitian matrices

We prove a large deviation principle for the largest singular value of sparse non-Hermitian random matrices, or directed Erd\H{o}s-R\'enyi networks in the constant average degree regime $p =\frac{d}{n}$ where $d$ is fixed. Entries are assumed to have Weibull distributions with tail decaying at rate $e^{-t^{\alpha}}$ for $\alpha >0$. While the law of large number results agree with the largest eigenvalue of sparse Hermitian matrices given in (Ganguly and Nam, '22) and (Ganguly, Hiesmayr and Nam, '22), large deviation results are surprisingly simpler, exhibiting a single transition at $\alpha =2$. The rate function for undirected networks with $0<\alpha \leq 2$ involved a transition at $\alpha =1$ and a complicated variational formula due to the emergence of cliques with large edge-weights (Ganguly, Hiesmayr and Nam, '22). For directed networks, we introduce a clique reduction technique which reformulates the problem for undirected networks with maximum clique size $2$, and the rate function is greatly simplified. For $\alpha >2$, both the law of large numbers and large deviation results are identical to the sparse Hermitian case. Our results easily generalize to rectangular i.i.d. ensembles.