Stability and Set Stability in Distribution of Probabilistic Boolean Networks

We propose a new concept, stability in distribution (SD) of a probabilistic Boolean network (PBN), which determines whether the probability distribution converges to the distribution of the target state (namely, a one-point distributed random variable). In a PBN, stability with probability one, stability in the stochastic sense, and SD are equivalent. The SD is easily generalized to subset stability, i.e., to set stability in distribution (SSD). We prove that the transition probability from any state to an invariant subset (or to a fixed point) is nondecreasing in time. This monotonicity is an important property in establishing stability criteria and in calculating or estimating the transient period. We also obtain a verifiable, necessary, and sufficient condition for SD of PBNs with independently and identically distributed switching. We then show that SD problems of PBNs with Markovian switching and PBN synchronizations can be recast as SSD problems of Markov chains. After calculating the largest invariant subset of a Markov chain in a given set by the newly proposed algorithm, we propose a necessary and sufficient condition for SSDs of Markov chains. The proposed method and results are supported by examples.