A geometric approach towards linear consensus algorithms

In this paper, we deal with the limiting behavior of linear consensus systems in both continuous and discrete time. A geometric framework featuring the state transition matrix of the system is introduced to: (i) generalize/rediscover the existing results in the literature about convergence properties of distributed averaging algorithms, (ii) interpret, from a consensus system point of view, the Sonin´s Decomposition-Separation Theorem that has proved, as in our recent work, to be a powerful tool in asymptotic analysis of backward propagating Markov chains, and (iii) address the so-called “consensus space” of the underlying chain of a system, where by the consensus space, we mean the set of initial conditions leading to consensus.