The collective behavior of the Cucker-Smale model on the infinite graphs
Wang Xinyu, Xue Xiaoping
Abstract
In this paper, we study the collective behavior of the Cucker-Smale model on two types of infinite graphs. The first type of graph is a fully connected infinite graph. We obtain that when the initial value is unbounded with respect to the position, the corresponding collective behavior (the consistency of velocity) still emerges. We call it the formation behavior. The second type of graph is an infinite graph with locally finite connections. We first study the locally finite infinite graphs with the infimum spectrum of the Laplace operator strictly positive and obtain sufficient conditions to guarantee the emergence of the flocking behavior. Then, we study the locally finite infinite graphs with Poincar$\acute{\rm~e}$ inequality. With the help of the geometric properties of the graph, we obtain the ultra-contractive properties of the time-varying graph, and apply them to the Cucker-Smale model to obtain the nonlinear stability (the flocking behavior of small initial values).