On the Dominant Vertices of Constrained Edge-Labeled Graphs

Finite-valued dynamic systems (FVDSs) are a class of networked systems characterized by quantized information arising from channel communication constraints or logical alternatives, encompassing some classical models including Boolean networks and finite-field networks. By virtue of their finite-cardinal features, we can represent the dynamics of FVDSs by edge-labeled graphs. In this paper, we study the strongly/weakly dominant problems of vertices on this kind of edge-labeled graphs under both memory- and period-based constraints. By constructing two kinds of non-homogeneous Markovian stochastic graphs (NMSGs), we conclude that the identification of strongly/weakly dominant vertices on the constrained edge-labeled graphs can be equivalently converted into the almost-sure probabilistic reachability problems on NMSGs. Based on this property and the expectation expressions of these NMSGs, we derive an upper bound on the search instants, and further develop some necessary and sufficient algebraic criteria for judging strongly/weakly dominant vertices. Finally, we validate applications of the theoretical results on a$\lambda$-Switchbiological model and a typical sequential control network over finite field.