Vicsek Model — Topological (Metric-free) Interactions

Metric-free (topological) Vicsek model: unlike the original metric model where particles interact within a fixed radius r, here each particle aligns with its k nearest neighbors regardless of distance. Biological motivation: starling flocks use k≈7 neighbors (Ballerini et al 2008). This makes the interaction range adaptive, stabilizing the ordered phase against density fluctuations. The order parameter φ = |⟨e^{iθ}⟩| transitions from 0 (disordered gas) to 1 (perfect flock) as noise η decreases. Nematic alignment allows headless orientation — relevant for cell migration layers.