Synchronization of an Array of Coupled Probabilistic Boolean Networks

Two synchronization problems, synchronization with probability one and synchronization in probability, are investigated for an array of coupled probabilistic Boolean networks (CPBNs). Compared with the former one, the in-probability problem considers a more general situation, in which synchronization can be achieved with a positive probability, instead of strictly 100%. It reflects the intrinsic randomness of biological systems. For both problems, some necessary and sufficient conditions are proposed, based on which two feasible algorithms are presented for checking two kinds of synchronization, respectively. CPBNs can be seemed as a combination of coupled Boolean networks (CBNs) with assigned probabilities. There are also detailed discussions on the impact of the synchronism of CBNs on the two addressed synchronization problems. The existence of invariant synchronization subsets is studied, which deepens our understanding on the difference and difficulty of these problems. Finally, numerical simulations show the effectiveness of the theoretical results.