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We study the consensus formation for an agents based model, generalizing thatoriginally proposed by Krause \cite{Kr}, by allowing the communication channelsbetween any couple of agents to be switched on or off randomly, at each timestep, with a probability law depending on the proximity of the agents'opinions. Namely, we consider a system of agents sharing their opinionsaccording to the following updating protocol. At time $t+1$ the opinion$X_{i}\left( t+1\right) \in\left[ 0,1\right] $ of agent $i$ is updated at theweighted average of the opinions of the agents communicating with it at time$t.$ The weights model the confidence level an agent assign to the opinions ofthe other agents and are kept fixed by the system dynamics, but the set ofagents communicating with any agent $i$ at time $t+1$ is randomly updated insuch a way that the agent $j$ can be chosen to belong to this set independentlyof the other agents with a probability that is a non increasing function of$\left\vert X_{i}\left( t\right) -X_{j}\left( t\right) \right\vert .$ Thiscondition models the fact that a communication among the agents is more likelyto happen if their opinions are close. We prove that the system reachesconsensus, i.e. as the time tends to infinity the agents' opinions will reachthe same value exponentially fast.

The consensus problem for opinion dynamics with local average random interactions