On Quotients of Stochastic Networks Over Finite Fields

This article studies the set stability of stochastic finite-field networks (SFFNs) via the quotient-transition-system (QTS)-based method. The QTS is constructed to preserve the complete probabilistic transition information of the original SFFN and has a comparatively smaller network scale. First, with respect to the initial partition of the state set, we obtain the smallest QTS by calculating the coarsest equivalence relation. Then, the stability relationship between SFFN and its corresponding QTS is explored. In particular, the smallest QTS corresponding to a synchronous
$n$
-node SFFN has no greater than
$n+1$
nodes. This formal simplicity gives a solid foundation for the subsequent research. Moreover, we establish a visualization interface “quotient generator” to obtain the quotients for any SFFN. After that, we explore the necessary and sufficient conditions for the set stability in distribution and the finite-time set stability with probability one of the SFFNs based on the QTS. Finally, an example concerning a 27-state SFFN is presented to demonstrate the theoretical results, indicating that its synchronization analysis can be completely characterized by the stability of a 4-node QTS. Furthermore, we analyze the relationships among the number of iterations to obtain the smallest QTS, the number of nodes in the obtained QTS, and the types of SFFNs.