Reaching consensus in the sense of probability

This paper studies consensus for a team of networked agents with stochastic interactions. Specifically, consensus in the sense of probability is investigated for both fixed and switching interaction graphs that are chosen randomly from some given set. In the static case, a lower bound for the probability of consensus is calculated when each interaction graph is equally likely to be selected among set containing all possible undirected graphs. It is then shown that the (exact) probability of consensus for n agents is strictly increasing with respect to n, whenever n ≥ 3. For the case of a randomly switching directed interaction graph, the probability of consensus is equal to the probability of an event that is critical for reaching consensus under a deterministic setting. In addition, the equivalence of consensus 1) with probability 1, 2) in probability, and 3) in the r-th mean is demonstrated without requiring an i.i.d. process and linear system dynamics.