  Modeling and estimating mixed memberships for overlapping unipartite
un-weighted networks has been well studied in recent years. However, to our
knowledge, there is no model for a more general case, the overlapping bipartite
weighted networks. To close this gap, we introduce a novel model, the Bipartite
Mixed Membership Distribution-Free (BiMMDF) model. Our model allows an
adjacency matrix to follow any distribution as long as its expectation has a
block structure related to node membership. In particular, BiMMDF can model
overlapping bipartite signed networks and it is an extension of many previous
models, including the popular mixed membership stochastic blcokmodels. An
efficient algorithm with a theoretical guarantee of consistent estimation is
applied to fit BiMMDF. We then obtain the separation conditions of BiMMDF for
different distributions. Furthermore, we also consider missing edges for sparse
networks. The advantage of BiMMDF is demonstrated in extensive synthetic
networks and eight real-world networks.
