Convergence Rate of Distributed Consensus with Nonuniform Delays

Motivated by broad applications in various fields of engineering, we study distributed consensus problems over a network of nodes, where the goal is to seek a consensus on the local variables at the nodes. Our main contribution is to analyze the convergence rate of distributed consensus methods in the presence of inter-node communication delays, which is one of the most fundamental issues in distributed systems. In particular, we provide an explicit formula for the convergence rate of such methods, which depends on the network topology and the delay constants. Our main technique is to utilize the discrete lime delayed version of the Cornwall-Bellman inequality, which directly takes into account the impact of delays. Finally, we provide numerical simulations to illustrate our theoretical results.