The geometry of a probabilistic consensus of opinion algorithm

We consider the problem of a group of agents whose objective is to asymptotically reach agreement of opinion. The agents exchange information subject to a communication topology modeled by a time varying graph. The agents use a probabilistic algorithm under which at each time instant an agent updates its state by probabilistically choosing from its current state/opinion and the ones of its neighbors. We show that under some minimal assumptions on the communication topology (infinitely often connectivity and bounded intercommunication time between agents), the agents reach agreement with probability one. We show that this algorithm has the same geometric properties as the linear consensus algorithm in R". More specifically, we show that the probabilistic update scheme of an agent is equivalent to choosing a point from the (generalized) convex hull of its current state and the states of its neighbors; convex hull defined on a particular convex metric space where the states of the agents live and for which a detailed description is given.