Vicsek model: k-nearest-neighbor (topological) vs fixed-radius (metric) interactions
The Vicsek model aligns bird velocities with neighbors. Metric: interact with
all birds within radius r (interaction set shrinks in sparse regions). Topological:
always interact with the k nearest birds (interaction set is density-independent). Starling flocks
use topological rules (k≈7), which makes the flock robust to density fluctuations and predator attacks.
The order parameter Φ = |⟨e^{iθ}⟩| measures collective alignment.