Clustering and Spectral Analysis of the Infinite Cucker–Smale Model

We study clustering dynamics for the infinite Cucker–Smale (ICS) model and its connection to the spectrum of the graph Laplacian. For the ICS model, we overcome the challenge of estimating the velocities of particles and derive a system of dissipative differential inequalities (SDDI) in terms of infinite norms. As in the finite ensemble, we show that mono-cluster flocking emerges exponentially fast, and additionally establish a sufficient framework for algebraic multi-cluster flocking. Moreover, for the CS model with a finite system size, we offer a complete spectral characterization of multi-cluster flocking. Specifically, the emergence of n-cluster behavior corresponds to the limit of the n-th eigenvalue of the time-varying Laplacian approaching zero. In contrast, the lower bound of the (n+1)-th eigenvalue remains strictly positive. Furthermore, we extend this framework to the ICS model by characterizing weak n-cluster flocking via spectral asymptotics, where the n-th eigenvalue tends to zero, while both the (n+1)-th eigenvalue and the infimum of the essential spectrum remain strictly positive. Our results bridge spectral analysis and clustering dynamics, providing indirect evidence for the fast emergence of mono-cluster flocking and the slow relaxation of multi-cluster patterns in both finite and infinite particle systems.