This article studies the stability in probability of probabilistic Boolean networks and stabilization in the probability of probabilistic Boolean control networks. To simulate more realistic cellular systems, the probability of stability/stabilization is not required to be a strict one.

In this situation, the target state is indefinite to have a probability of transferring to itself. Thus, it is a challenging extension of the traditional probability-one problem, in which the self-transfer probability of the target state must be one. Some necessary and sufficient conditions are proposed via the semitensor product of matrices. Illustrative examples are also given to show the effectiveness of the derived results.

C. Huang, J. Lu, G. Zhai, J. Cao, G. Lu, M. Perc, Stability and stabilization in probability of probabilistic Boolean networks, IEEE Transactions on Neural Networks and Learning Systems 32(1) (2021) 241-251.