On the critical exponent of the one-dimensional Cucker-Smale model on a general graph

We study a critical exponent of the flocking behavior to the one-dimensional 1D Cucker-Smale (C-S) model with a regular inverse power law communication on a general network with a spanning tree. For this, we propose a new nonlinear functional which can control the velocity diameter and decays exponentially fast as time goes on. As an application of the time-evolution of the nonlinear functional, we show that the C-S model on a line exhibits a unique critical exponent for unconditional flocking on a general network so that this improves an earlier result [S.-Y. Ha and J.-G. Liu, A simple proof of Cucker-Smale flocking dynamics and mean field limit, Commun. Math. Sci. 7 (2009) 297-325.] on the all-to-all network. Our result also resolves the critical exponent conjecture posed in Cucker-Dong's work [On the critical exponent for flocks under hierarchical leadership, Math. Models Methods Appl. Sci. 19 (2009) 1391-1404] for 1D setting. Emergent behavior of the C-S model is independent of the special structure of the underlying network, as long as it contains a spanning tree.