Structure learning of sparse directed acyclic graphs incorporating the scale-free property

Directed acyclic graphs have been widely used to model the causal relationships among variables. Many existing works focus on l1 based methods to induce sparsity. However, in addition to sparsity, studies on networks show that many real networks are scale-free, that is, the degree of the network follows a power-law. To capture the scale-free property, in this paper we propose a novel penalized likelihood method by employing a log 1-norm group penalty which is the composite of the well-known log-type and lasso-type penalty functions. We then design an efficient coordinate descent algorithm to solve the resulting nonconvex problem. Moreover, we establish the estimation consistency of the estimator under the setting where the error variances are fixed at an identical constant. Numerical studies are also conducted to demonstrate the merits of our method.