Emergent Behavior on Flocks
summary
Proves that agents with velocities updated by weighted differences converge to a common velocity and keep bounded spread; weights use aij = const/(const + distance^{2 beta}), written compactly with Laplacian/adjacency matrices and analyzed as ODEs and a scaling PDE (Cucker-Smale PDE). Main claim: beta < 1/2 unconditional flocking; beta = 1/2 needs small initial velocity spread; beta > 1/2 needs small initial dispersion-or-velocity condition. Shows exponential convergence in the ODE model and sparked ~250 follow-up papers.
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