Bipartite Mixed Membership Distribution-Free Model. A novel model for community detection in overlapping bipartite weighted networks

Modeling and estimating mixed memberships for un-directed un-weighted networks in which nodes can belong to multiple communities has been well studied in recent years. However, for a more general case, the bipartite weighted networks in which nodes can belong to multiple communities, row nodes can be different from column nodes, and all elements of adjacency matrices can be any finite real values, to our knowledge, there is no model for such bipartite weighted networks. To close this gap, this paper introduces a novel model, the Bipartite Mixed Membership Distribution-Free (BiMMDF) model. As a special case, bipartite signed networks with mixed memberships can also be generated from BiMMDF. Our model enjoys its advantage by allowing all elements of an adjacency matrix to be generated from any distribution as long as the expectation adjacency matrix has a block structure related to node memberships under BiMMDF. The proposed model can be viewed as 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. In particular, for a standard bipartite weighted network with two row (and column) communities, to make the algorithm's error rates small with high probability, separation conditions are obtained when adjacency matrices are generated from different distributions under BiMMDF. The behavior differences of different distributions on separation conditions are verified by extensive synthetic bipartite weighted networks generated under BiMMDF. Experiments on real-world directed weighted networks illustrate the advantage of the algorithm in studying highly mixed nodes and asymmetry between row and column communities.