On the stochastic flocking of the Cucker-Smale flock with randomly switching topologies

TL;DR

This paper studies the stochastic flocking behavior of the Cucker-Smale model under randomly switching network topologies, providing conditions for emergent flocking despite randomness in network switching times and topologies.

Contribution

It introduces a probabilistic framework for analyzing flocking with random switching times and topologies, extending previous deterministic models.

Findings

Flocking can be achieved under random switching with high probability.

Derived conditions relate system parameters and communication weights to flocking behavior.

Established a priori conditions ensuring flocking despite network randomness.

Abstract

We present an emergent stochastic flocking dynamics of the Cucker-Smale (CS) ensemble under randomly switching topologies. The evolution of the CS ensemble with randomly switching topologies involves two random components (switching times and choices of network topologies at switching instant). First, we allow switching times for the network topology to be random so that the successive increments are i.i.d. processes following the common probability distribution. Second, at each switching instant, we choose a network topology randomly from a finite set of admissible network topologies whose union contains a spanning tree. Even for the fixed deterministic network topology, the CS ensemble may not exhibit a mono-cluster flocking depending on the initial data and the decay mode of the communication weight functions measuring the degree of interactions between particles. For the flocking dynamics of the CS ensemble with these two random components, we first use a priori conditions on the network topologies and uniform boundedness of position diameter, and derive the flocking estimates via matrix theory together with a priori conditions, and then replace the a priori condition for the position diameter by some suitable condition on the system parameters and communication weight. The a priori condition on the network topology will be guaranteed by the suitable spanning tree time-blocks with probability one.