  Graphical modelling techniques based on sparse selection 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
the networks in such applications that poses a challenge for statistical
inference is the presence of a small number of strongly interconnected nodes in
a network 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
selection 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 selection methods to include
structural information about the underlying network. Our proposed method is a
weighted lasso approach with novel row/column sum weights, which we refer to as
the hubs weighted graphical lasso. We establish large sample properties of the
method when the number of parameters diverges with the sample size, and
evaluate its finite sample performance via extensive simulations. We illustrate
the method with an application to microbiome data.
