Connectivity and Synchronization of Multi-agent Systems

Collective behaviors of multi-agent systems are important starting points in the study of complex systems, and Vicsek´s model is a typical model of multi-agent systems. In addition to its simplicity, it preserves some key features of complex systems, such as dynamic behaviors, local interactions and changing neighborhood graph. Jadbabaie et al. analyzed the synchronization of a linearized Vicsek´s model, and showed that if a certain connectivity condition is satisfied, then all agents will move in the same direction eventually. An unresolved key problem is when such a connectivity is satisfied. In this paper, we will first show that each agent´s heading will converge in Vicsek´s model, and will then give a sufficient condition imposed only on the agents´ initial states and parameters to make Vicsek´s model synchronized. Finally, we will give a counterexample to show that connectivity of the neighbor graph is not sufficient for synchronization of the Vicsek´s model if the initial headings are allowed to be in [-π, π). This reveals an essential difference between the linearized Vicsek´s model and the original Vicsek´s model.