Fig. 5: Emergent bands in k-nearest neighbors interaction (kNN).
From: Topological flocking models in spatially heterogeneous environments
a Typical macroscopic configurations observed in a system with k-nearest neighbor interaction (kNN, k = 6, where k is the number of neighboring objects), for different obstacle densities ρo, and noise intensities η. Black dots are particles and red dots are obstacles. In obstacle-free environments (ρo = 0), we observe rather homogeneous structures with no bands. As we introduce obstacles, band-like structures emerge; at a fixed noise, by increasing obstacle density, bands become sharper. For a fixed obstacle density, the sharpest bands are observed at low noise values. Snapshots specified by I, II, and III have been used to calculate 1D band profiles in panel (c). b Density modulation parameter β in different regions of phase space specified with obstacle density ρo and noise intensity η corresponding to the system in panel (a). 1D band profiles related to I, II, and III are shown in (c). c 1D band profiles—1D particle density ρL along the moving direction L (x or y)—corresponding to the specified regions of panels (a, b) (regions I, II, and III). d β vs obstacle density ρo, showing the promoting effect of obstacles in band formation, corresponding to η = 0.25 in panel (b). e Density modulation parameter β for k = 1.
