Graphical model is an effective method to analyze the network structure, in which directed acyclic graphs have been widely used to model the causal relationships among variables. While many real networks are scale-free, that is, the degree of the network follows a power-law. The paper considers the problem of structure learning in directed acyclic graphs incorporating the scale-free prior. Specifically, we assume the order of nodes is known in advance. To capture the scale-free property, we propose a novel regularization model with a penalty which is the composite of the Log-type and $l_q (0<q<1)$-type penalty functions to solve the non-convex model and to analyze the convergence of the algorithm. Experiments show that the proposed method performs well for both the simulation study and real data applications.