Learning sparse and scale-free networks

Gaussian networks study undirected interactions between random variables, through the estimation of the precision matrices. Recently, it has been demonstrated that some of the important networks display features similar to scale-free graphs. There have been few works on the learning of the sparse Gaussian graphical models aiming to preserve properties of networks which are believed to be scale-free or have dominating hubs. We prefer to name both networks as `scale-free´ networks for simplicity in this paper. We propose a new log-likelihood formulation, which promotes the sparseness of the precision matrix and features of scale-free graphical topology. We used the alternating direction method of multipliers (ADMM) form, which is used for the convex optimization, to solve the general L₁ regularized loss optimization. Our proposed method exhibits better estimation performance on various data sets and various number of samples, N. Also, the proposed method and some of the state of the arts methods are tested under various penalty constants to validate the robustness.