The study of logical matrix factorization provides a new insight into the matrix dimension reduction problems of biological systems. This paper develops the logical matrix factorization technique for exploring the topological structure and stability of probabilistic Boolean networks (PBNs). Firstly, the union set of distinct indices in factorized structural matrices for different modes is obtained, based on which, a size-reduced system is constructed for the original PBN. Secondly, it is proved that the topological structure of original PBN is equivalent to that of the size-reduced system. Thirdly, the equivalence of finite-time stability and stability in distribution between the original PBN and the size-reduced system is further revealed. Finally, the effectiveness of the obtained new results are verified via several Boolean models of genetic regulatory networks (GRNs).

逻辑矩阵分解的研究为生物系统的矩阵降维问题提供了新的见解。本文开发了逻辑矩阵分解技术，以探索概率布尔网络（PBN）的拓扑结构和稳定性。首先，获得了不同模式的分解结构矩阵中不同索引的并集，在此基础上，为原始PBN构造了尺寸减小的系统。其次，证明了原始PBN的拓扑结构与尺寸减小的系统相同。第三，进一步揭示了有限时间稳定性和原始PBN与尺寸减小系统之间分布的稳定性。最后，