The effect of additive noise on consensus achievement in wireless sensor networks

Achieving consensus on common global parameters through totally decentralized algorithms is a topic that has attracted considerable attention in the last few years, in view of its potential application in sensor networks. Several algorithms, along with their convergence properties, have been studied in the literature, among which the most popular are the (weighted) average consensus based schemes. One of the most critical aspects of these algorithms is that they suffer from catastrophic noise propagation. We show that the noise affecting the system state variables has a variance that grows linearly with the time index. In addition, we prove that encoding the information on the first forward difference of the state variables rather than on the state itself improves noise resilience, since it guarantees that the asymptotic value of the consensus is affected by noise with bounded variance. The results of our in-depth analysis of the effect of additive noise on consensus algorithms are valid regardless of the noise statistics and for arbitrary network topologies, i.e., arbitrary Laplacian matrices, and contain as special cases previously known results.