  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 remains a challenge. Here, we propose an
interpretable model: bipartite mixed membership stochastic blockmodel (BiMMSB
for short) for directed mixed membership networks. BiMMSB 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 BiMPCA to estimate the mixed memberships
for both row nodes and column nodes in a directed network. We show that the
approach is asymptotically consistent under BiMMSB. We demonstrate the
advantages of BiMMSB with applications to a small-scale simulation study, the
directed Political blogs network and the Papers Citations network.
