Optimal tradeoff between instantaneous and delayed neighbor information in consensus algorithms

We consider a distributed consensus problem over a network, where at each time instant every node receives two pieces of information from disjoint neighboring sets: a weighted average of current states of neighbors from a primary network, and a weighted average of one-hop delayed states of neighbors from a secondary network. The proposed algorithm makes each node update its state to a weighted average of these individual averages. We show that convergence to consensus is guaranteed with non-trivial weights. We also present an explicit formula for the weights allocated to each piece of the information for the optimal rate of convergence, when the secondary network is the complement of the primary network. Finally numerical examples are given to explore the case when the neighbor sets of the agents do not cover the whole network.