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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 deterministic distributed averaging algorithms where the information exchange between neighboring nodes (agents) is subject to uniform quantization. With such quantization, convergence to the precise average cannot be achieved in general, but the convergence would be to some value close to it, called quantized consensus. Using Lyapunov stability analysis, we characterize the convergence properties of the resulting nonlinear quantized system. We show that in finite time and depending on initial conditions, the algorithm will either cause all agents to reach a quantized consensus where the consensus value is the largest quantized value not greater than the average of their initial values, or will lead all variables to cycle in a small neighborhood around the average. In the latter case, we identify tight bounds for the size of the neighborhood and we further show that the error can be made arbitrarily small by adjusting the algorithm's parameters in a distributed manner.

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Abstract-In many application scenarios sensors need to calculate the average of some local values, e.g. of local measurements. A possible solution is to rely on consensus algorithms. In this case each sensor maintains a local estimate of the global average, and keeps improving it by performing a weighted sum of the estimates of all its neighbors. The number of iterations needed to reach an accurate estimate depends on the weights used at each sensor. Speeding up the convergence rate is important also to reduce the number of messages exchanged among neighbors and then the energetic cost of these algorithms. While it is possible in principle to calculate the optimal weights, the known algorithm requires a single sensor to discover the topology of the whole network and perform the calculations. This may be unfeasible for large and dynamic sensor networks, because of sensor computational constraints and of the communication overhead due to the need to acquire the new topology after each change. In this paper we propose a new average consensus algorithm, where each sensor selects its own weights on the basis of some local information about its neighborhood. Our algorithm is tailored for networks having cluster structure, like it is common for wireless sensor networks. In realistic sensor network topologies, the algorithm shows faster convergence than other existing consensus protocols.