A node-pinning and state-flipped approach to partial synchronization of Boolean Networks

In this paper, the partial synchronization of general Boolean Networks is studied from the perspective of nodes and states by using the semi-tensor product of matrices. First, by introducing the Hamming distances and its algebraic expression, the partial synchronization of general Boolean Networks can be transformed into an ensemble stability problem. Second, through optimizing the pinning strategy, the conditions for achieving partial synchronization of general Boolean Networks are developed, and the algorithm for finding the optimal pinning nodes is given. Subsequently, to further simplify the above condition, the related pinning node strategy is further optimized and combined by introducing state-flipped mechanism, which makes the general Boolean Networks partially synchronous under no condition, and requires the less number of nodes to be pinned compared to the previous works. Meanwhile, the corresponding algorithm for seeking the optimal pinning nodes is designed in the context of flipping the state only once. Finally, the obtained results are validated by several specific numerical examples. In summary, the current work can further enrich and improve the study of partial synchronization problems in Boolean Networks.