Synchronization of directed higher-order networks via pinning control

Higher -order interaction is prevalent in biology, contagion, and the social sciences, leading to more complex dynamic behaviors. Investigating how the higher -order interaction influences the synchronization process and how to control the higher -order structure to achieve global synchronization is a critical issue. This paper discusses a general directed higher -order network, where sufficient conditions are derived for achieving synchronization via pinning controllers. To optimize synchronization while saving control costs, which nodes are controlled preferentially among the limited number of controllers available are determined. The priority control strategy for directed higher -order networks is proposed by a defined multi-indegree. Based on the theoretical analysis of directed higher -order networks, a common phenomenon in higher -order networks is explained, that is, the small higher -order interaction coupling strength always substantially improves synchronization capability. It provides help for further understanding the impact of higher -order interactions on the synchronization region. Furthermore, the adaptive coupling strength controllers are designed to synchronize directed higher -order networks, which avoids repeatedly verifying the above sufficient conditions. The theoretical results are verified by the scale -free network of Chen systems and the email network of Lorenz systems.