Asymptotical Stability of Probabilistic Boolean Networks With State Delays

This paper devotes to establishing a bridge between asymptotical stability of a probabilistic Boolean network (PBN) and a solution to its induced equations, which are induced from the PBN's transition matrix. By utilizing the semitensor product technique routinely, the dynamics of a PBN with coincident state delays can be equivalently converted into that of a higher dimensional PBN without delays. Subsequently, several novel stability criteria are derived from the standpoint of equations’ solution. The most significant finding is that a PBN is globally asymptotically stable at a predesignated one-point distribution if and only if a vector, obtained by adding 1 at the bottom of this distribution, is the unique nonnegative solution to PBN's induced equations. Moreover, the influence of coincident state delays on PBN's asymptotical stability is explicitly analyzed without consideration of the convergence rate. Consequently, such bounded state delays are verified to have no impact on PBN's stability, albeit delays are time-varying. Based on this worthwhile observation, the time computational complexity of the aforementioned approach can be reduced by removing delays directly. Furthermore, this universal procedure is summarized to reduce the time complexity of some previous results in the literature to a certain extent. Two examples are employed to demonstrate the feasibility and effectiveness of the obtained theoretical results.