Pinning synchronization of a complex network: Nodes, edges and higher-order edges

\n\nIn recent years, the interdisciplinary study of complex networks has become increasingly important in fields ranging from biology and physics to sociology and mathematics. This paper focuses on pinning control, an approach essential for achieving coordinated behavior in dynamic networks. We explore recent advancements in pinning control strategies, explaining theoretical frameworks and simulation techniques. Additionally, we discuss the significance of certain structures within networks across different orders. Finally, we conclude with a summary of key insights and propose our outlook on future research.\n