Design and analysis of distributed averaging with quantized communication

Consider a network whose nodes have some initial values, and it is desired to design an algorithm that builds on neighbor to neighbor interactions with the ultimate goal of convergence to the average of all initial node values or to some value close to that average. Such an algorithm is called generically 'distributed averaging', and our goal in this paper is to study the performance of a subclass of distributed averaging algorithms where the information exchange between neighboring nodes (agents) is subject to deterministic uniform quantization. With such quantization, the precise average cannot be achieved (except in exceptional cases), but some value close to it, called quantized consensus. It is shown in this paper that in finite time, the algorithm will either cause all agents to reach a quantized consensus where the consensus value is the largest integer not greater than the average of their initial values, or will lead all variables to cycle in a small neighborhood around the average, depending on initial conditions. In the latter case, tight bounds for the size of the neighborhood are given, and it is further shown that the error can be made arbitrarily small by adjusting the algorithm's parameters in a distributed manner.