  Swarming phenomena are ubiquitous in various physical, biological, and social
systems, where simple local interactions between individual units lead to
complex global patterns. A common feature of diverse swarming phenomena is that
the units exhibit either convergent or divergent evolution in their behaviors,
i.e., becoming increasingly similar or distinct, respectively. The associated
dynamics changes across time, leading to complex consequences on a global
scale. In this study, we propose a generalized Laplacian dynamics model to
describe both convergent and divergent swarm behaviors, where the trends of
convergence and divergence compete with each other and jointly determine the
evolution of global patterns. We empirically observe non-trivial
phase-transition-like phenomena between the convergent and divergent evolution
phases, which are controlled by local interaction properties. We also propose a
conjecture regarding the underlying phase transition mechanisms and outline the
main theoretical difficulties for testing this conjecture. Overall, our
framework may serve as a minimal model of swarm behaviors and their intricate
phase transition dynamics.
