Estimating sparse networks with hubs

Graphical modelling techniques based on sparse estimation have been applied to infer complex networks in many fields, including biology and medicine, engineering, finance and social sciences. One structural feature of some of these networks that poses a challenge for statistical inference is the presence of a small number of strongly interconnected nodes, which are called hubs. For example, in microbiome research hubs or microbial taxa play a significant role in maintaining stability of the microbial community structure. In this paper, we investigate the problem of estimating sparse networks in which there are a few highly connected hub nodes. Methods based on L1-regularization have been widely used for performing sparse estimation in the graphical modelling context. However, while these methods encourage sparsity, they do not take into account structural information of the network. We introduce a new method for estimating networks with hubs that exploits the ability of (inverse) covariance estimation methods to include structural information about the underlying network. Our method is a weighted lasso approach with novel row/column sum weights, which we refer to as the hubs weighted graphical lasso. A practical advantage of the new method is that it leads to an optimization problem that is solved using the efficient graphical lasso algorithm that is already implemented in the R package glasso (Friedman et al., 2019). We establish large sample properties of the method when the number of parameters diverges with the sample size. We then show via simulations that the method outperforms competing methods and illustrate its performance with an application to microbiome data.