Convergence of a class of multi-agent systems in probabilistic framework

In this paper, we will study how locally interacting agents lead to synchronization of the overall system for a basic class of multi-agent systems that are described by a simplification of the well-known Vicsek model. This model looks simple, but the rigorous theoretical analysis appears to be quite complicated, because there are strong nonlinear interactions among the update laws of the agents´ positions and headings. In fact, most of the existing theoretical analyzes hinge on certain connectivity conditions on the global behavior of the agents´ trajectories (or on the neighborhood graphs of the underlying dynamical systems), which are quite hard to verify in general. In this paper, by working in a probabilistic framework, we will give a complete and rigorous proof for the following fact observed in simulation: the overall multi-agent system will synchronize with large probability for large population. The proof is carried out by analyzing both the dynamical properties of the nonlinear system evolution and the asymptotic properties of the spectrum of random geometric graphs.