Synchronization of a Class of Multi-agent Systems with Large Population

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Multi-agent systems arise from diverse fields of natural and artificial systems, and a typical case is that each agent has the tendency to behave as other agents do in its neighborhood described by a disk or ball. This is actually reflected by the well-known Vicsek model. Since this model is of fundamental importance in understanding the multi-agent systems, it has attracted much attention from researchers in recent years. In this paper, we will present a comprehensive theoretical analysis of the nonlinear Vicsek model in a random framework, without imposing any connectivity conditions on the system trajectories as did in most of the previous investigations. To be precise, we will show that for any givenmodel parameters, i.e., the interaction radius r and the agents’ moving velocity v , the overall system will synchronize as long as the population size is large enough, which justifies the phenomenon observed previously in simulations by Vicsek et al. (1995). The proof is based on the recent work of Tang and Guo (2007) for linearized Vicsek model, and involves the use of spectral graph theory and multi-array martingale estimation theory.