Flocking: no leader, three rules

In 1987 Craig Reynolds, then at Symbolics, needed flocks for computer animation and found that he did not have to script them. Each simulated bird — he called them boids — steers by three rules, using only the neighbours inside a short radius in front of it:

• Separation: steer away from neighbours that are too close.
• Alignment: steer towards the average heading of the neighbours.
• Cohesion: steer towards the average position of the neighbours.

Each rule asks for a change of velocity, and the bird blends them with the weights in the controls panel, turning by no more than a fixed amount per step. Here each bird sees 270° around itself, the edges of the screen wrap round, and everyone decides from the same snapshot before anyone moves.

The Order readout is the polarisation: add up every bird’s heading as a unit arrow, and divide the length of the total by the number of birds. Random headings give a value near 0; a flock flying in formation gives a value near 1. It is the quantity Tamás Vicsek and colleagues used in 1995 to show that this kind of system has a phase transition: turn the noise up past a critical level and the order collapses, much as a magnet loses its magnetisation above the Curie temperature.

Try it: choose School, then drag the Noise slider slowly to the right and watch the Order readout. Flocks counts separate groups of at least five birds, where birds belong to the same group if a chain of close neighbours joins them.

• Turn each rule off in turn. Without separation birds pile on top of each other; without alignment you get a swarm that holds together but goes nowhere; without cohesion small groups drift apart and never rejoin.
• Add a hawk or two. Fleeing birds pass the panic on to their neighbours through alignment, and the flock splits and flows round the predator.
• Shrink the vision slider. The flock breaks into many small groups, because birds can no longer see far enough to stay together.

Real starlings do not follow these rules exactly. Studying photographs of murmurations over Rome, Michele Ballerini and colleagues found in 2008 that each starling tracks a fixed number of neighbours — about six or seven — rather than everyone within a fixed distance. That makes a flock cohesive whether it is dense or sparse, which a fixed radius does not. The broad lesson survives the details, though: coordinated motion on this scale needs no leader and no shared plan.

• C. W. Reynolds (1987), Flocks, herds and schools: A distributed behavioral model, SIGGRAPH ’87.
• T. Vicsek et al. (1995), Novel Type of Phase Transition in a System of Self-Driven Particles, Phys. Rev. Lett.
• M. Ballerini et al. (2008), Interaction ruling animal collective behavior depends on topological rather than metric distance, PNAS.