Directed acyclic graph (DAG) models are widely used to discover causal relationships among random variables. However, most existing DAG learning algorithms are not directly applicable to heavy-tailed data which are commonly observed in finance and other fields. In this article, we propose a two-step efficient algorithm based on topological layers, referred as TopHeat, to learn linear DAGs with heavy-tailed error distributions which include Pareto, Fréchet, log-normal, Cauchy distributions, and so on. First, we reconstruct the topological layers hierarchically in a top-down fashion based on the new reconstruction criteria for heavy-tailed DAGs without assuming the popularly-employed faithfulness condition. Second, we recover the directed edges via the modified conditional independence testing for heavy-tailed distributions. We theoretically demonstrate the consistency of the exact DAG structures. Monte Carlo simulations validate the outstanding finite-sample performance of the proposed algorithm compared with competing methods. In the real data analysis, we analyze the exchange rates among 17 countries and uncover the source of financial contagion and the pathways, which indicates that the financial risk contagion effect became increasingly stable among European countries as the euro was introduced.
