This paper is concerned with the local synchronization problem for the interconnected Boolean networks (BNs) without and with stochastic disturbances. For the case without stochastic disturbances, first, the limit set and the transient period of the interconnected BNs are discussed by resorting to the properties of the reachable set for the global initial states set. Second, in terms of logical submatrices of a certain Boolean vector, a compact algebraic expression is presented for the limit set of the given initial states set. Based on it, several necessary and sufficient conditions are derived assuring the local synchronization of the interconnected BNs. Subsequently, an efficient algorithm is developed to calculate the largest domain of attraction. As for the interconnected BNs with stochastic disturbances, first, mutually independent two-valued random logical variables are introduced to describe the stochastic disturbances. Then, the corresponding local synchronization criteria are also established, and the algorithm to calculate the largest domain of attraction is designed. Finally, numerical examples are employed to illustrate the effectiveness of the obtained results/ algorithms.

This paper investigates the synchronization problem for the realization-dependent probabilistic Boolean networks (PBNs) coupled unidirectionally in the drive-response configuration. The realization of the response PBN is assumed to be uniquely determined by the realization signal generated by the drive PBN at each discrete time instant. First, the drive-response PBNs are expressed in their algebraic forms based on the semitensor product method, and then, a necessary and sufficient condition is presented for the synchronization of the PBNs. Second, by resorting to a newly defined matrix operator, the reachable set from any initial state is expressed by a column vector. Consequently, an easily computable algebraic criterion is derived assuring the synchronization of the drive-response PBNs. Finally, three illustrative examples are employed to demonstrate the applicability and usefulness of the developed theoretical results.

This paper addresses the partial synchronization problem for the interconnected Boolean networks (BNs) via the semi-tensor product (STP) of matrices. First, based on an algebraic state space representation of BNs, a necessary and sufficient criterion is presented to ensure the partial synchronization of the interconnected BNs. Second, by defining an induced digraph of the partial synchronized states set, an equivalent graphical description for the partial synchronization of the interconnected BNs is established. Consequently, the second partial synchronization criterion is derived in terms of adjacency matrix of the induced digraph. Finally, two examples (including an epigenetic model) are provided to illustrate the efficiency of the obtained results.

This paper investigates the complete synchronization problem for the drive-response switched Boolean networks (SBNs) under arbitrary switching signals, where the switching signals of the response SBN follow those generated by the drive SBN at each time instant. First, the definition of complete synchronization is introduced for the drive-response SBNs under arbitrary switching signals. Second, the concept of switching reachable set starting from a given initial state set is put forward. Based on it, a necessary and sufficient condition is derived for the complete synchronization of the drive-response SBNs. Last, we give a simple algebraic expression for the switching reachable set in a given number of time steps, and two computable algebraic criteria are obtained for the complete synchronization of the SBNs. A biological example is given to demonstrate the effectiveness of the obtained main results.