# Estimating Sparse Networks with Hubs

## TL;DR

This paper introduces a novel weighted graphical lasso method tailored for estimating sparse networks with hub nodes, effectively capturing structural features like highly interconnected hubs in various complex systems.

## Contribution

The paper proposes a new hubs weighted graphical lasso that incorporates structural information for better estimation of networks with hubs, along with theoretical properties and empirical validation.

## Key findings

- The method accurately identifies hub nodes in simulated networks.
- It outperforms standard L1-regularization methods in finite sample scenarios.
- Application to microbiome data reveals biologically relevant hub structures.

## Abstract

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.