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Doubly Stochastic Pairwise Interactions for Agreement and Alignment
Random pairwise encounters often occur in large populations or groups of mobile agents, and various types of local interactions that happen at encounters account for emergent global phenomena. In particular, in the fields of swarm robotics, sociobiology, and social dynamics, several types of local pairwise interactions were proposed and analyzed leading to spatial gathering, clustering, agreement, or coordinated motion in teams of robotic agents, in animal herds, or in human societies. We here propose a very simple stochastic interaction at encounters that leads to agreement or geometric alignment in swarms of simple agents and analyze the process of converging to consensus. Consider a group of agents whose “states" evolve in time by pairwise interactions: the state of an agent is either a real value (a randomly initialized position within an interval) or a vector that is either unconstrained (e.g., the location of the agent in the plane) or constrained to have unit length (e.g., the direction of the agent's motion). The interactions are doubly stochastic in the sense that, at discrete time steps, pairs of agents are randomly selected and their new states are independently and uniformly set at random in (local) domains or intervals defined by the states of the interacting pair. We show that such processes lead, in finite expected time (measured by the number of interactions that occurred) to agreement in case of unconstrained states and alignment when the states are unit vectors.
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