Enforcing sparsity in complex networks through a probabilistic tensor factorization approach

Detecting community structures in network data is one of the central objectives in the analysis of complex networks. Inferential approaches for community detection based on probabilistic tensor factorization of the adjacency 3-way tensor are among the methodologies used for identifying communities. In particular, this approach includes models such as the stochastic block model and its variants. The most widely used variants of the stochastic blockmodeling approach are the classic stochastic blockmodel and the mixed-membership stochastic blockmodel. The former assumes that the communities are non overlapping, meaning that each node belongs solely to one community. As a result, the inferred matrices are sparse. The latter allows for overlapping communities, which means that a node can belong to more than one community and, as a result, loses some sparsity. Our model, Soft Membership Pruning, starts with the mixed-membership model and, by means of a budget parameter, constrains the membership vector for some of the nodes to be in the unit simplex. Consequently, the selected nodes belong to only one community. The introduction of hard membership nodes to the mixed membership model is equivalent to enforcing sparsity in the entries of the inferred membership matrices. Varying the budget parameter allows for a different ratio between hard and soft communities and offers a simple, controlled method to enforce sparsity in community detection. The model presented in this work is a first step toward introducing sparsity in community detection based on probabilistic tensor factorization.