  We study the task of clustering in directed networks. We show that using the
eigenvalue/eigenvector decomposition of the adjacency matrix is simpler than
all common methods which are based on a combination of data regularization and
SVD truncation, and works well down to the very sparse regime where the edge
density has constant order. Our analysis is based on a Master Theorem
describing sharp asymptotics for isolated eigenvalues/eigenvectors of sparse,
non-symmetric matrices with independent entries. We also describe the limiting
distribution of the entries of these eigenvectors; in the task of digraph
clustering with spectral embeddings, we provide numerical evidence for the
superiority of Gaussian Mixture clustering over the widely used k-means
algorithm.
