Emergence of stochastic flocking for the discrete Cucker–Smale model with randomly switching topologies

Abstract

We study emergent dynamics of the discrete Cucker-Smale (in short, DCS) model with randomly switching network topologies. For this, we provide a sufficient framework leading to the stochastic flocking with probability one. Our sufficient framework is formulated in terms of an admissible set of network topologies realized by digraphs and probability density function for random switching times. As examples for the law of switching times, we use the Poisson process and the geometric process and show that these two processes satisfy the required conditions in a given framework so that we have a stochastic flocking with probability one. As a corollary of our flocking analysis, we improve the earlier result [J.-G. Dong, S.-Y. Ha, J. Jung and D. Kim: On the stochastic flocking of the Cucker-Smale flock with randomly switching topologies. arXiv:1911.07390.] on the continuous C-S model.