  Mixed membership problem for undirected network has been well studied in
network analysis recent years. However, the more general case of mixed
membership for directed network in which nodes can belong to multiple
communities remains a challenge. Here, we propose an interpretable and
identifiable model: directed mixed membership stochastic blockmodel (DiMMSB)
for directed mixed membership networks. DiMMSB allows that row nodes and column
nodes of the adjacency matrix can be different and these nodes may have
distinct community structure in a directed network. We also develop an
efficient spectral algorithm called DiSP designed based on simplex structures
inherent in the left and right singular vectors of the population adjacency
matrix to estimate the mixed memberships for both row nodes and column nodes in
a directed network. We show that DiSP is asymptotically consistent under mild
conditions by providing error bounds for the inferred membership vectors of
each row node and each column node using delicate spectral analysis. Numerical
results on computer-generated directed mixed membership networks support our
theoretical findings and show that our DiSP outperforms its competitor in both
error rates and run-time. Applications of DiSP to real-world directed networks
demonstrate the advantages of DiSP in studying the asymmetric structure of
directed networks.
