Synchronization Analysis for Stochastic Networks Through Finite Fields

Finite-field networks (FFNs) are an effective type of models to characterize the multiagent systems with limited computation, communication, and memory capabilities. In this article, we term the FFNs, which switch among a collection of modes with respect to an independently and identically distributed process, as the stochastic FFNs (SFFNs). Subsequently, the concepts of asymptotic synchronization with probability one (asymptotic SPO), finite-time SPO, and fixed-time SPO of SFFNs are, respectively, introduced. Moreover, a series of necessary and sufficient conditions is, respectively, derived for asymptotic SPO and fixed-time SPO on the basis of transition graph and network matrix. Compared with the existing results in logical networks and real-valued networks, several interesting characteristics of a synchronous SFFN are revealed. Furthermore, the relations among asymptotic SPO, fixed-time SPO, finite-time SPO, and the synchronization of each mode are established.