EMERGENT BEHAVIORS OF THE MOTSCH-TADMOR MODEL ON INFINITE GRAPHS

We study flocking dynamics for the Motsch-Tadmor (MT) model over infinite graphs. For flocking dynamics, we consider two network topologies, namely the sender network and hierarchical leadership network, and we provide sufficient frameworks leading to asymptotic mono-cluster flocking for these network topologies. In the case of the sender network, the index set is discrete and unbounded. This produces some technical difficulty in estimating relative velocities. We first overcome the challenge of estimating relative velocities by establishing a system of differential inequalities for infinite norms. Then, we derive a mono-cluster flocking estimate using the nonlinear functional approach proposed by Ha and Liu [A simple proof of the Cucker-Smale flocking dynamics and mean-field limit, Commun. Math. Sci. 1 (2009) 297–325]. In particular, for the hierarchical leadership network, we show that the MT model exhibits slow-flocking dynamics, which gives an algebraic relaxation to the flocking state if the initial velocity fluctuations around the leader’s velocity are summable.