The consensus problem for opinion dynamics with local average random interactions

We study the consensus formation for an agents based model, generalizing that originally proposed by Krause \cite{Kr}, by allowing the communication channels between any couple of agents to be switched on or off randomly, at each time step, with a probability law depending on the proximity of the agents' opinions. Namely, we consider a system of agents sharing their opinions according to the following updating protocol. At time $t+1$ the opinion $X_{i}\left( t+1\right) \in\left[ 0,1\right]$ of any agent $i$ is updated at the weighted average of the opinions of the agents communicating with it at time $t.$ The weights model the confidence level an agent assigns to the opinions of the other agents and are kept fixed by the system dynamics, but the set of agents communicating with any agent $i$ at time $t+1$ is randomly updated in such a way that the agent $j$ can be chosen to belong to this set independently of the other agents with a probability that is a non increasing function of $\left\vert X_{i}\left( t\right) -X_{j}\left(t\right) \right\vert .$ This condition models the fact that a communication among the agents is more likely to happen if their opinions are close. We prove that if the agent's communication graph at time one, conditionally on the initial believes' configuration, is sufficiently connected, the system reaches consensus at geometric rate, i.e., more precisely, as the time tends to infinity the agents' opinions will reach the same value geometrically fast. We also discuss the consensus formation for a system of infinitely many agents. In particular we analyze the evolution of the empirical average of the agents' opinions in the limit as the size of the system tends to infinity and characterize its fixed points in terms of agents' consensus proving that this is reached geometrically fast with the same rate computed for the finite system.