  Consider a directed network with $K_{r}$ row communities and $K_{c}$ column
communities. Previous works found that modeling directed networks in which all
nodes have overlapping property requires $K_{r}=K_{c}$ for identifiability. In
this paper, we propose an overlapping and nonoverlapping model to study
directed networks in which row nodes have overlapping property while column
nodes do not. The proposed model is identifiable when $K_{r}\leq K_{c}$.
Meanwhile, we provide one identifiable model as extension of ONM to model
directed networks with variation in node degree. Two spectral algorithms with
theoretical guarantee on consistent estimations are designed to fit the models.
A small scale of numerical studies are used to illustrate the algorithms.
