Directed acyclic graphs are widely used to describe the causal effects among random variables, and the inference of those causal effects has become an popular topic in statistics and machine learning, and has wide applications in neuroinformatics, bioinformatics and so on. However, most studies focus on the estimation or inference of the directional relations among continuous random variables, those among discrete random variables have not gained much attentions. In this article we focus on the inference of directed linkages and directed pathways in a Poisson directed graphical model. We employ likelihood ratio tests subject to non-convex acyclicity constraints, and derive the asymptotic distributions of the test statistic under the null hypothesis is true in high-dimensional situations. The power analysis and simulations suggest that the tests achieve the desired objectives of inference. An analysis of a basketball statistics dataset of NBA players during 2016-2017 season illustrates the utility of the proposed method to infer directed linkages and directed pathways in player's statistics network.