Analyzing and learning sparse and scale-free networks using Gaussian graphical models

Abstract

In this paper, we consider the problem of fitting a sparse precision matrix to multivariate Gaussian data. The zero elements in the precision matrix correspond to conditional independencies between variables. We focus on the estimation of a class of sparse precision matrix which represents the scale-free networks. It has been demonstrated that some of the important networks display features similar to scale-free graphs. We propose a new log-likelihood formulation, which promotes the sparseness of the precision matrix as well as the topological structure of scale-free networks. To optimize this new energy formulation, the alternating direction method of multipliers form is used with the general \(L_1\)-regularized loss optimization. We tested our proposed method on various databases. Our proposed method exhibits better estimation performance with various number of samples, N, and different selection of sparsity parameter, \(\rho \).