Performance of consensus algorithms in large-scale distributed estimation

When consensus algorithms are used in very large networks, spreading information across the whole graph requires a long time. Hence, traditional convergence analysis, studying the essential spectral radius of the transition matrix, predicts very poor performance. However, in estimation problems, it is clear that a growing number of measurements improves the quality of the estimate, and it is natural to expect such behaviour even though the best estimate is approximated using distributed algorithms. Then, it is important to define a suitable performance metric, depending on the actual estimation or control problem in which the consensus algorithm is used. This allows to study how performance scales when both computation time and number of agents grow to infinity, for different communication graphs and choices of the algorithm.