The Convergence of Bird Flocking

TL;DR

This paper analyzes the time complexity of flocking models, proving bounds on convergence and stability, and introduces new techniques with broad applicability in understanding collective behavior in bird flocking models.

Contribution

It establishes tight bounds on convergence time for a standard flocking model and introduces novel analytical methods applicable to similar networked systems.

Findings

Fragmentation ceases within single exponential time.

Flocking network converges after an iterated exponential number of steps.

The bounds on convergence time are proven to be optimal.

Abstract

We bound the time it takes for a group of birds to reach steady state in a standard flocking model. We prove that (i) within single exponential time fragmentation ceases and each bird settles on a fixed flying direction; (ii) the flocking network converges only after a number of steps that is an iterated exponential of height logarithmic in the number of birds. We also prove the highly surprising result that this bound is optimal. The model directs the birds to adjust their velocities repeatedly by averaging them with their neighbors within a fixed radius. The model is deterministic, but we show that it can tolerate a reasonable amount of stochastic or even adversarial noise. Our methods are highly general and we speculate that the results extend to a wider class of models based on undirected flocking networks, whether defined metrically or topologically. This work introduces new techniques of broader interest, including the "flight net," the "iterated spectral shift," and a certain "residue-clearing" argument in circuit complexity.