We study the distributed averaging problem on arbitrary connected graphs, with the additional constraint that the value at each node is an integer. This discretized distributed averaging problem models several problems of interest, such as averaging in a network with finite capacity channels and load balancing in a processor network. We describe simple randomized distributed algorithms which achieve consensus to the extent that the discrete nature of the problem permits. We give bounds on the convergence time of these algorithms for fully connected networks and linear networks.

We develop in finite case sufficient conditions for weak ergodicity of a nonstationary Markov chain r-states having transition matrices (Pn)n≥1 with Pn → P, where P has 2 irreducible and aperiodic closed classes and,possibly, transient states. AMS 2000 Subject Classification: 60J10.

In this paper we study ergodic measures on non-simple Bratteli diagrams of finite rank that are invariant with respect to the cofinal equivalence relation. We describe the structure of finite rank diagrams and prove that every ergodic invariant measure (finite or infinite) is an extension of a finite ergodic measure defined on a simple subdiagram. We find some algebraic criteria in terms of entries of incidence matrices and their norms under which such an extension remains a finite measure. Furthermore, the support of every ergodic measure is explicitly determined. We also give an algebraic condition for a diagram to be uniquely ergodic. It is proved that Vershik maps (not necessarily continuous) on finite rank Bratteli diagrams cannot be strongly mixing and always have zero entropy with respect to any finite ergodic invariant measure. A number of examples illustrating the established results is included. MSC: 37B05, 37A25, 37A20.