Averaging Asynchronously Using Double Linear Iterations∗

The distributed averaging problem is to devise a protocol which will enable the members of a group of n > 1 agents to asymptotically determine in a decentralized manner, the average of the initial values of their scalar agreement variables. A typical averaging protocol can be modeled by a linear iterative equation whose update matrices are doubly stochastic. Building on the ideas proposed in [1] and [2], this paper discusses an alternative approach to distributed averaging which is modeled by two simultaneously iterated linear iterative equations whose update matrices are left stochastic rather than doubly stochastic. We call such models, double linear iterations. The aim of this paper is to propose and analyze two different asynchronous double linear iteration protocols. For each protocol, each agent independently updates its variables at times determined by its own clock. It is not assumed that the agents’ clocks are synchronized or that the “event times” at which any one agent updates its variables are evenly spaced. Nor is it assumed that the communication between agents is bidirectional. By appealing to the concept of “analytic synchronization”, it is shown that both the two protocols are guaranteed to solve the asynchronous distributed averaging problem under appropriate assumptions for both fixed and time-varying directed neighbor graphs.