How to enhance adaptive synchronization capability via higher-order network structures?

How to reduce the control costs and then enhance synchronization capability is one of the current highly anticipated issues. Based on Lyapunov stability theory, this paper derives a sufficient condition for ensuring adaptive synchronization in higher-order dynamic networks, where the first-order interactions and the second-order interactions are represented by edges and 2-simplices, respectively. Three different kinds of higher-order networks are constructed, named Erdős-Rényi simplicial complex (ERSC), Watts-Strogatz simplicial complex (WSSC), and Barabási-Albert simplicial complex (BASC). Subsequently, the first- and the second-order degree distributions of these networks are then discussed. Finally, numerical simulations validate the theoretical analyses and yield the following key findings: Smaller scale or smaller coupling strengths can save control costs and reduce energy consumption across all three networks. Increasing the number of 2-simplices, ERSC demonstrates the best adaptive synchronization capability. When varying the coupling strengths or the number of 2-simplices, WSSC has stronger robustness than ERSC and BASC.