  Let $\mathcal A$ be the adjacency matrix of the Erd\H{o}s-R\'{e}nyi directed
graph $\mathscr G(N,p)$. We denote the eigenvalues of $\mathcal A$ by
$\lambda_1^{\cal A},...,\lambda^{\cal A}_N$, and $|\lambda_1^{\cal
A}|=\max_i|\lambda_i^{\cal A}|$. For $N^{-1+o(1)}\leq p\leq 1/2$, we show that
\[ \max_{i=2,3,...,N} \bigg|\frac{\lambda_i^{\mathcal A}}{\sqrt{Np(1-p)}}\bigg|
=1+O(N^{-1/2+o(1)}) \] with very high probability. In addition, we prove that
near the unit circle, the local eigenvalue statistics of ${\mathcal
A}/\sqrt{Np(1-p)}$ coincide with those of the real Ginibre ensemble. As a
by-product, we also show that all non-trivial eigenvectors of $\mathcal A$ are
completely delocalized.
  For Hermitian random matrices, it is known that the edge statistics are
sensitive to the sparsity: in the very sparse regime, one needs to remove many
noise random variables (which affect both the mean and the fluctuation) to
recover the Tracy-Widom distribution. Our results imply that, compared to their
analogues in the Hermitian case, the edge statistics of non-Hermitian sparse
random matrices are more robust.
  The edge of non-Hermitian matrices possesses the cusp singularity, which was
believed to be a technical difficulty of the model. Our first novelty is the
observation that the cusp singularity is in fact an advantage instead of an
obstacle, and when used properly, it can make the computation easier for
non-Hermitian matrices. The second novelty is an use of integration by parts
formula inside the Girko's Hermitization, which completely avoids the study of
the Green function at larger spectral scales. The third novelty is the
self-similarity of the self-consistent equations of certain Green functions,
which eliminates the effect of large expectations of the matrix entries.
