Group Sparse β-Model for Network

Sparsity, homogeneity, and heterogeneity are three important characteristics of many real-life networks. The recently proposed Sparse β-Model divides nodes into core ones and peripheral ones to accommodate sparsity, but the parameters of core nodes are assumed to be of similar magnitude, which may not be in line with applications. In this article, we propose the Group Sparse β-Model that splits the core nodes into groups and assumes different orders of magnitude of parameters in different groups, accounting for the heterogeneity among core nodes. When the groups are known, we provide consistent and asymptotically normal moment estimators of the parameters that control the global and local density. Based on that, consistency and asymptotic normality of the maximum likelihood estimators of the remaining parameters are derived. We also establish finite-sample error bounds results. When the groups are unknown, a ratio method is proposed to detect groups, which is computationally efficient. Simulations show competitive results and the analysis of a corporate inter-relationships network illustrates the usefulness of the proposed model.