Transition Analysis of Stochastic Logical Control Networks

Transition analysis is essential for analyzing and regulating logical control networks (LCNs). The algebraic state-space representation method, which relies on the semitensor product of matrices, has been shown to allow deterministic LCNs to be represented as linear-like systems. However, due to the inherent uncertainty, it is hard to obtain the algebraic expression of an stochastic LCN. A unified paradigm for the transition analysis of LCNs with stochastic and deterministic dynamics is provided in this research. First, the algebraic expression of LCN with deterministic dynamics is reviewed. Second, the algebraic expression of LCN with stochastic dynamics is considered, where the nonequivalence between the dispersed form and the integrated form is proposed. Then, the reason for the nonequivalence is provided. After that, a consistency condition is presented to bridge the gap between the independent model and the conditionally independent model. Finally, we specifically point out that probabilistic LCN satisfies the consistency criteria, allowing one to calculate the probabilistic LCN transition matrix by using a power-reducing operator.