  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.
