“…Vicsek model, [8] a minimal model to depict the phase transition, has been extended in various aspects since its appearance, such as from polar to apolar particles, [10,11] from intrinsic scalar noise to vectorial noise, [9] and extensions of interactions to include, e.g., cohesion [12] and disordered environments. [14,15] One particular kind of variants is to change the space where SSPs reside from two-dimensional Euclidean space to lattices [16,17] or general manifolds [18] and/or from continuous directional space to discrete ones. [19,20] In Vicsek model and its variants, it is long accepted that the collective motion is optimized with vanishing noise and maximal particle density.…”