Title: On Low Rank Directed Acyclic Graphs and Causal Structure Learning. Abstract: Despite several important advances in recent years, learning causal structures represented by directed acyclic graphs (DAGs) remains a challenging task in high dimensional settings when the graphs to be learned are not sparse.  In this paper, we propose to exploit a low rank assumption regarding the (weighted) adjacency matrix of a DAG causal model to mitigate this problem. We demonstrate how to adapt existing methods for causal structure learning to take advantage of this assumption and establish several useful results relating interpretable graphical conditions to the low rank assumption. In particular, we show that the maximum rank is highly related to hubs, suggesting that scale-free networks which are frequently encountered in real applications tend to be low rank. We also provide empirical evidence for the utility of our low rank adaptations, especially on relatively large and dense graphs. Not only do they outperform existing algorithms when the low rank condition is satisfied, the performance is also  competitive even though the rank of the underlying  DAG may not be as low as is assumed.