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

TL;DR

This paper establishes a probabilistic framework for the emergence of flocking behavior in the discrete Cucker-Smale model with randomly switching network topologies, demonstrating conditions under which flocking occurs with probability one.

Contribution

It introduces a new sufficient framework for stochastic flocking in the discrete Cucker-Smale model with switching topologies, extending previous results to the discrete case.

Findings

Flocking occurs with probability one under certain switching conditions.

Poisson and geometric processes satisfy the framework's conditions.

The results improve upon earlier continuous model findings.

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.