Ergodicity and class-ergodicity of balanced asymmetric stochastic chains

Unconditional consensus is the property of a consensus algorithm for multiple agents, to produce consensus irrespective of the particular time or state at which the agent states are initialized. Under a weak condition, so-called balanced asymmetry, on the sequence (An) of stochastic matrices in the agents states update algorithm, it is shown that (i) the set of accumulation points of states as n grows large is finite, (ii) the asymptotic unconditional occurrence of single consensus or multiple consensuses is directly related to the property of absolute infinite flow of this sequence, as introduced by Touri and Nedić. The latter condition must be satisfied on each of the islands of the so-called unbounded interactions graph induced by (An), as defined by Hendrickx et al. The property of balanced asymmetry is satisfied by many of the well known discrete time consensus models studied in the literature.

[1]  Ιωαννησ Τσιτσικλησ,et al.  PROBLEMS IN DECENTRALIZED DECISION MAKING AND COMPUTATION , 1984 .

[2]  Behrouz Touri,et al.  Alternative characterization of ergodicity for doubly stochastic chains , 2011, IEEE Conference on Decision and Control and European Control Conference.

[3]  Sadegh Bolouki,et al.  On consensus with a general discrete time convex combination based algorithm for multi-agent systems , 2011, 2011 19th Mediterranean Conference on Control & Automation (MED).

[4]  M. Degroot.  Reaching a Consensus , 1974 .

[5]  Vicsek,et al.  Novel type of phase transition in a system of self-driven particles. , 1995, Physical review letters.

[6]  John N. Tsitsiklis,et al.  Convergence of Type-Symmetric and Cut-Balanced Consensus Seeking Systems , 2011, IEEE Transactions on Automatic Control.

[7]  Behrouz Touri,et al.  On Approximations and Ergodicity Classes in Random Chains , 2010, IEEE Transactions on Automatic Control.

[8]  Behrouz Touri,et al.  On Ergodicity, Infinite Flow, and Consensus in Random Models , 2010, IEEE Transactions on Automatic Control.

[9]  E. Seneta,et al.  Towards consensus: some convergence theorems on repeated averaging , 1977, Journal of Applied Probability.

[10]  Behrouz Touri,et al.  On existence of a quadratic comparison function for random weighted averaging dynamics and its implications , 2011, IEEE Conference on Decision and Control and European Control Conference.

[11]  John N. Tsitsiklis,et al.  Distributed Asynchronous Deterministic and Stochastic Gradient Optimization Algorithms , 1984, 1984 American Control Conference.

[12]  Jie Lin,et al.  Coordination of groups of mobile autonomous agents using nearest neighbor rules , 2003, IEEE Trans. Autom. Control..

[13]  John N. Tsitsiklis,et al.  Parallel and distributed computation , 1989 .

[14]  Sadegh Bolouki,et al.  Theorems about ergodicity and class-ergodicity of chains with applications in known consensus models , 2012, 2012 50th Annual Allerton Conference on Communication, Control, and Computing (Allerton).