Distributed average consensus over random networks

In this paper, the distributed consensus problem of multi-agent networked systems is considered where agents make decisions using local information in the presence of random communication topologies. This problem is included in the framework given in [31]-[32] that allows random interconnection topologies to have distributions possibly depending on each other or time. It is shown that the random Krasnoselskii-Mann iterative algorithm converges almost surely and in mean square to the average of initial states of the agents under suitable assumptions. The algorithm does not require the distribution of random interconnection topologies or B-connectivity assumption for convergence. Therefore, it applies to asynchronous updates or/and unreliable communication protocols. We also show that the algorithm converges for synchronous updates when the weighted graph matrix is periodic and irreducible. It is shown that the agents interact among themselves to approach the consensus subspace in such a way that the projection of their states onto the consensus subspace at each time is equal to the average of their initial states. Eventually, a numerical example is given to exhibit the results.