A Discrete Model of Collective Marching on Rings

TL;DR

This paper models and analyzes how autonomous agents on a ring-shaped environment reach consensus in their direction of motion, inspired by locust swarm experiments, providing bounds on convergence time and effects of random behavior.

Contribution

The work introduces a discrete stochastic model of collective motion on rings, proving convergence to local and global consensus, and deriving asymptotic bounds on stabilization time.

Findings

Agents eventually reach local consensus on each track.

Global consensus occurs with small probability of erratic jumps.

Theoretical results are supported by numerical simulations.

Abstract

We study the collective motion of autonomous mobile agents on a ringlike environment. The agents' dynamics is inspired by known laboratory experiments on the dynamics of locust swarms. In these experiments, locusts placed at arbitrary locations and initial orientations on a ring-shaped arena are observed to eventually all march in the same direction. In this work we ask whether, and how fast, a similar phenomenon occurs in a stochastic swarm of simple agents whose goal is to maintain the same direction of motion for as long as possible. The agents are randomly initiated as marching either clockwise or counterclockwise on a wide ring-shaped region, which we model as $k$ "narrow" concentric tracks on a cylinder. Collisions cause agents to change their direction of motion. To avoid this, agents may decide to switch tracks so as to merge with platoons of agents marching in their direction. We prove that such agents must eventually converge to a local consensus about their direction of motion, meaning that all agents on each narrow track must eventually march in the same direction. We give asymptotic bounds for the expected amount of time it takes for such convergence or "stabilization" to occur, which depends on the number of agents, the length of the tracks, and the number of tracks. We show that when agents also have a small probability of "erratic", random track-jumping behaviour, a global consensus on the direction of motion across all tracks will eventually be reached. Finally, we verify our theoretical findings in numerical simulations.