Collective Dynamics in the Vicsek and Vectorial Network Models Beyond Uniform Additive Noise

Abstract

In this work, we analyze the coordination of interacting individuals in two nonlinear dynamical models that are subject to a new form of noise. Specifically, we propose extensions both to the classical Vicsek model, whereby each individual averages the orientation of its geographically proximal neighbors, and to the vectorial network model, in which the selection of neighbors is random and independent of the group geometric configuration. In the traditional forms of these models, the update rule for the individuals’ orientations is affected by additive uniform noise. Motivated by biological groups in which individuals’ turn rates exhibit sporadic and large changes, we extend the uniform additive noise model to a turn rate stochastic process. Through comprehensive numerical simulations, we demonstrate the impact of such occasional large deviations (intensity and frequency), along with the role of the neighbors’ selection process, on the coordination of the group. In addition, we establish a closed-form expression for the group polarization for the vectorial network model in the vicinity of an ordered state.