Mean Square Stability of Consensus over Fading Networks with Nonhomogeneous Communication Delays

In this paper, we propose a discrete time consensus protocol which can solve the consensus problem in the network with nonhomogeneous communication delays. We give the sufficient conditions to reach consensus and provide a closed form formula for the consensus value. Furthermore, we investigate the mean square stability (MSS) of our protocol when each link of the network can break with a given probability at each time interval. The condition for checking MSS is equivalent to checking the spectral radius of a positive matrix. To gain more insight, we further restrict our attention to spatially invariant network structure and develop a more efficient expression to check the MSS. We derive a closed form formula to determine the MSS in the limit of large delays, get useful lower and upper bounds and analyze their implications for large classes of network topologies. We find that the consensus protocol is robust to link failures in the sense the system is always mean square stable if we put restrictions on the propagation gain.