Convergence in inhomogeneous consensus processes with positive diagonals

Consensus Algorithms and the Decomposition-Separation Theorem

Convergence properties of time inhomogeneous Markov chain based discrete and continuous time linear consensus algorithms are analyzed. Provided that a so-called infinite jet flow property is satisfied by the underlying chains, necessary conditions for both consensus and multiple consensus are established. A recenet extension by Sonin of the classical Kolmogorov-Doeblin decomposition-separation for homogeneous Markov chains to the inhomogeneous case is then employed to show that the obtained necessary conditions are also sufficient when the chain is of Class P*, as defined by Touri and Nedic. I…

度波动与共识算法的收敛时间

Degree Fluctuations and the Convergence Time of Consensus Algorithms

研究在时变图中节点度固定条件下共识算法的收敛时间，证明在B-连通图中达到ε精度所需时间为B+4n^3 B ln(2n/ε)，并给出当度恒定假设放松时收敛时间指数增长的简单证明。

Convergence to consensus in multiagent systems and the lengths of inter-communication intervals

A theorem on (partial) convergence to consensus of multiagent systems is presented. It is proven with tools studying the convergence properties of products of row stochastic matrices with positive diagonals which are infinite to the left. Thus, it can be seen as a switching linear system in discrete time. It is further shown that the result is strictly more general than results of Moreau (IEEE Transactions on Automatic Control, vol. 50, no. 2, 2005), although Moreau's results are formulated for generally nonlinear updating maps. This is shown by demonstrating the existence of an appropriate sw…

Convergence of products of stochastic matrices with positive diagonals and the opinion dynamics background

We present a convergence result for infinite products of stochastic matrices with positive diagonals. We regard infinity of the product to the left. Such a product converges partly to a fixed matrix if the minimal positive entry of each matrix does not converge too fast to zero and if either zero-entries are symmetric in each matrix or the length of subproducts which reach the maximal achievable connectivity is bounded. Variations of this result have been achieved independently in Lorenz 2005, Moreau 2005 and Hendrickx 2005. We present briefly the opinion dynamics context, discuss the relation…