The Smallest Possible Interaction Radius for Synchronization of Self-Propelled Particles
Ge Chen, Zhixin Liu, Lei Guo · SIAM Review · 2014
A central and fundamental issue in the theory of complex systems is to understand how local rules lead to collective behavior of the whole system. This paper will investigate a typical collective behavior (synchronization) of a self-propelled particle system modeled by the nearest neighbor rules. While connectivity of the dynamic neighbor graphs associated with the underlying systems is crucial for synchronization, it is widely known that the verification of such dynamical connectivity is at the core of theoretical analysis. Ideally, conditions used for synchronization should be imposed on the model parameters and the initial states of the particles. One crucial model parameter is the interaction radius, and we are interested in the following natural and basic question: What is the smallest interaction radius for synchronization? In this paper, we will show that, in a certain sense, the smallest possible interaction radius approximately equals $\sqrt{\log n/(\pi n)}$, with $n$ being the population size, which coincides with the critical radius for connectivity of static random geometric graphs known in the literature.