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We consider the problem of sparse estimation of undirected graphical models via the L1 regularization. The ordinary lasso encourages the sparsity on all edges equally likely, so that all nodes tend to have small degrees. On the other hand, many real-world networks are often scale-free, where some nodes have a large number of edges. In such cases, a penalty that induces structured sparsity, such as a log penalty, performs better than the ordinary lasso. In practical situations, however, it is difficult to determine an optimal penalty among the ordinary lasso, log penalty, or somewhere in between. In this paper, we introduce a new class of penalty that is based on the exponentiation of the minimax concave penalty. The proposed penalty includes both the lasso and the log penalty, and the gap between these two penalties is bridged by a tuning parameter. We apply cross-validation to select an appropriate value of the tuning parameter. Monte Carlo simulations are conducted to investigate the performance of our proposed procedure. The numerical result shows that the proposed method can perform better than the existing log penalty and the ordinary lasso.

ESTIMATING SCALE-FREE NETWORKS VIA THE EXPONENTIATION OF MINIMAX CONCAVE PENALTY