text: In this article, we establish orientation and connectivity based criteria for the agreement algorithm to achieve asymptotic consensus in the context of time-varying topology and communication delays. These criteria unify and extend many earlier convergence results on the agreement algorithm for deterministic and discrete-time multiagent systems.
  @cite_14: We consider continuous-time consensus seeking systems whose time-dependent interactions are cut-balanced, in the following sense: if a group of agents influences the remaining ones, the former group is also influenced by the remaining ones by at least a proportional amount. Models involving symmetric interconnections and models in which a weighted average of the agent values is conserved are special cases. We prove that such systems always converge. We give a sufficient condition on the evolving interaction topology for the limit values of two agents to be the same. Conversely, we show that if our condition is not satisfied, then these limits are generically different. These results allow treating systems where the agent interactions are a priori unknown, e.g., random or determined endogenously by the agent values. We also derive corresponding results for discrete-time systems.
  @cite_7: The paper deals with the convergence properties of the products of random (row-)stochastic matrices. The limiting behavior of such products is studied from a dynamical system point of view. In particular, by appropriately defining a dynamic associated with a given sequence of random (row-)stochastic matrices, we prove that the dynamics admits a class of time-varying Lyapunov functions, including a quadratic one. Then, we discuss a special class of stochastic matrices, a class @math , which plays a central role in this work. We then introduce balanced chains and using some geometric properties of these chains, we characterize the stability of a subclass of balanced chains. As a special consequence of this stability result, we obtain an extension of a central result in the non-negative matrix theory stating that, for any aperiodic and irreducible row-stochastic matrix @math , the limit @math exists and it is a rank one stochastic matrix. We show that a generalization of this result holds not only for sequences of stochastic matrices but also for independent random sequences of such matrices.
summary: After writing the proof of Theorem , we became aware of two recent papers both containing Theorem in the synchronous case. @cite_14 , Hendrickx and Tsitsiklis showed that the agreement algorithm achieves asymptotic consensus under assumptions A, B with @math , D1 and the so-called cut-balance condition. In light of Proposition below, the latter condition turns out to correspond to the decentralized model. Independently, Touri and Nedi 'c @cite_7 established a general convergence result for the infinite product of random stochastic matrices, and to do that, they first proved that this result holds in the deterministic case; see Lemma 5 in @cite_7 . It is easy to see that this lemma actually coincides with our Theorem in the particular case of a synchronous multiagent system. In fact, our technique for proving the lemmas in Sections and infra specialized to @math , is similar to the one used for the proof of the deterministic result in @cite_7 .