Cluster Synchronization of Boolean Networks Under Probabilistic Function Perturbation

In practical gene regulatory networks, gene expression is often accompanied by stochastic gene mutation. This brief investigates the cluster synchronization problem of Boolean networks (BNs) and Boolean control networks (BCNs) under probabilistic function perturbation, which can better characterize stochastic gene mutation and explore internal synchronization mechanism of the system. By transforming BN into an equivalent stochastic logical system, the algebraic expression of BN (BCN) under probabilistic function perturbation is derived. Then, necessary and sufficient conditions are provided so as to distinguish whether systems can achieve cluster synchronization with probability one under probabilistic function perturbation. Corresponding state feedback controllers are designed as well. The effectiveness is demonstrated by numerical examples.