  This paper considers the problem of modeling and estimating community
memberships of nodes in a directed network where every row (column) node is
associated with a vector determining its membership in each row (column)
community. To model such directed network, we propose directed degree corrected
mixed membership (DiDCMM) model by considering degree heterogeneity. DiDCMM is
identifiable under popular conditions for mixed membership network when
considering degree heterogeneity. Based on the cone structure inherent in the
normalized version of the left singular vectors and the simplex structure
inherent in the right singular vectors of the population adjacency matrix, we
build an efficient algorithm called DiMSC to infer the community membership
vectors for both row nodes and column nodes. By taking the advantage of DiMSC's
equivalence algorithm which returns same estimations as DiMSC and the recent
development on row-wise singular vector deviation, we show that the proposed
algorithm is asymptotically consistent under mild conditions by providing error
bounds for the inferred membership vectors of each row node and each column
node under DiDCMM. The theory is supplemented by a simulation study.
