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.

中文翻译：

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具有状态时滞的概率布尔网络的渐近稳定性


本文致力于在概率布尔网络 (PBN) 的渐近稳定性与其导出方程的解之间建立一座桥梁，该方程是从 PBN 的转移矩阵导出的。通过常规地利用半张量积技术，具有重合状态延迟的PBN的动力学可以等效地转换为无延迟的更高维PBN的动力学。随后，从方程解的角度导出了几个新颖的稳定性准则。最重要的发现是，当且仅当通过在该分布底部加 1 获得的向量是 PBN 导出方程的唯一非负解时，PBN 在预先指定的单点分布上是全局渐近稳定的。此外，在不考虑收敛速度的情况下，明确分析了重合状态延迟对PBN渐近稳定性的影响。因此，这种有界状态延迟被证实对 PBN 的稳定性没有影响，尽管延迟是随时间变化的。基于这一有价值的观察，上述方法的时间计算复杂度可以通过直接消除延迟来降低。此外，总结了这种通用的过程，在一定程度上降低了文献中一些先前结果的时间复杂度。通过两个算例验证了所得理论结果的可行性和有效性。