Since the end of the 20th century, the study of the network science has been advanced rapidly due to the appearance of the Internet, the World Wide Web, and other technical developments. Thanks to these progresses in technical and science, we are able to examine many real-world systems under the view of complex networks. And many of them share special characteristics of complex networks such as the scale-free, modular structure or directionality of these systems. Recent studies imitated these real-world systems have shown remarkable resemblance between the dynamics appeared in these complex networks with real world phenomena, such as the resemblance of the Griffiths phases with the criticality of real brain networks. The study of the interaction between the complex network structures and dynamics may help us to have a deeper understanding into these real-world systems. We decided to further our study in these interaction between the directionality of the complex networks with the appearance of the Griffiths phases and the modularity structure with explosive synchronization.