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 (A_n) 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 (A_n), 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.