Lu Jianquan, Zhong Jie, Li Lulu, Ho Daniel W C, Cao Jinde
Department of Mathematics, Southeast University, Nanjing 210096, China.
School of Automation, Southeast University, Nanjing, 210096, China.
Sci Rep. 2015 Aug 28;5:13437. doi: 10.1038/srep13437.
In this paper, we analyze the synchronization problem of master-slave probabilistic Boolean networks (PBNs). The master Boolean network (BN) is a deterministic BN, while the slave BN is determined by a series of possible logical functions with certain probability at each discrete time point. In this paper, we firstly define the synchronization of master-slave PBNs with probability one, and then we investigate synchronization with probability one. By resorting to new approach called semi-tensor product (STP), the master-slave PBNs are expressed in equivalent algebraic forms. Based on the algebraic form, some necessary and sufficient criteria are derived to guarantee synchronization with probability one. Further, we study the synchronization of master-slave PBNs in probability. Synchronization in probability implies that for any initial states, the master BN can be synchronized by the slave BN with certain probability, while synchronization with probability one implies that master BN can be synchronized by the slave BN with probability one. Based on the equivalent algebraic form, some efficient conditions are derived to guarantee synchronization in probability. Finally, several numerical examples are presented to show the effectiveness of the main results.
在本文中，我们分析主从概率布尔网络（PBNs）的同步问题。主布尔网络（BN）是一个确定性BN，而从BN在每个离散时间点由一系列具有一定概率的可能逻辑函数确定。在本文中，我们首先定义了概率为1时主从PBNs的同步，然后研究概率为1时的同步。借助一种称为半张量积（STP）的新方法，将主从PBNs表示为等价的代数形式。基于该代数形式，推导了一些保证概率为1时同步的充要准则。此外，我们研究了主从PBNs的概率同步。概率同步意味着对于任何初始状态，主BN可以以一定概率被从BN同步，而概率为1时的同步意味着主BN可以以概率1被从BN同步。基于等价代数形式，推导了一些保证概率同步的有效条件。最后，给出了几个数值例子来说明主要结果的有效性。
