  Directed acyclic graphs (DAGs) are commonly used to represent causal
relationships among random variables in graphical models. Applications of these
models arise in the study of physical, as well as biological systems, where
directed edges between nodes represent the influence of components of the
system on each other. The general problem of estimating DAGs from observed data
is computationally NP-hard, Moreover two directed graphs may be observationally
equivalent. When the nodes exhibit a natural ordering, the problem of
estimating directed graphs reduces to the problem of estimating the structure
of the network. In this paper, we propose a penalized likelihood approach that
directly estimates the adjacency matrix of DAGs. Both lasso and adaptive lasso
penalties are considered and an efficient algorithm is proposed for estimation
of high dimensional DAGs. We study variable selection consistency of the two
penalties when the number of variables grows to infinity with the sample size.
We show that although lasso can only consistently estimate the true network
under stringent assumptions, adaptive lasso achieves this task under mild
regularity conditions. The performance of the proposed methods is compared to
alternative methods in simulated, as well as real, data examples.
