Consensus dynamics of behavioral swarm models with random batch interactions and external noises

We study the consensus dynamics of discrete-time behavioral swarm (DBS) models with random batch interactions and external noises. The proposed models describe behavioral swarms where each particle is characterized by its activity (level), spatial position, and heading angle. Interactions among particles are governed by the random batch method (RBM), which significantly reduces computational complexity by restricting communication to dynamically formed batches of the whole swarm. We provide several sufficient frameworks for stochastic consensus in noise-free and noisy environments, that is, under mild assumptions on system parameters, the swarm achieves almost sure alignment in both activity and heading angle, while maintaining bounded spatial dispersion. Numerical simulations validate the theoretical findings, illustrating that random batch interactions yield consensus behaviors comparable to all-to-all communication while significantly reducing computational cost. The results provide a scalable framework for analyzing and simulating large-scale swarm systems with applications to the modeling of collective behaviors and decentralized controls.