Convergence rate estimates for consensus over random graphs

Multi-agent coordination algorithms with randomized interactions have seen use in a variety of settings in the multi-agent systems literature. In some cases, these algorithms can be random by design, as in a gossip-like algorithm, and in other cases they are random due to external factors, as in the case of intermittent communications. Targeting both of these scenarios, we present novel convergence rate estimates for consensus problems solved over random graphs. Established results provide asymptotic convergence in this setting, and we provide estimates of the rate of convergence in two forms. First, we estimate decreases in a quadratic Lyapunov function over time to bound how quickly the agents' disagreement decays, and second we bound the probability of being at least a given distance from the point of agreement. Simulation results are provided to support the theoretical developments made.