Abstract

Two similar Laplacian-based models for swarms with informed agents are proposed and analyzed analytically and numerically. In these models, each individual adjusts its velocity to match that of its neighbors and some individuals are given a preferred heading direction towards which they accelerate if there is no local velocity consensus. The convergence to a collective group swarming state with constant velocity is analytically proven for a range of parameters and initial conditions. Using numerical computations, the ability of a small group of informed individuals to accurately guide a swarm of uninformed agents is investigated. The results obtained in one of our two models are analogous to those found for more realistic and complex algorithms for describing biological swarms, namely, that the fraction of informed individuals required to guide the whole group is small, and that it becomes smaller for swarms with more individuals. This observation in our simple system provides insight into the possibly robust dynamics that contribute to biologically effective collective leadership and decision-making processes. In contrast with the more sophisticated models mentioned above, we can describe conditions under which convergence to consensus is ensured.