Stability of large-scale probabilistic Boolean networks via network aggregation

Large-scale probabilistic Boolean networks (LSPBNs) are a modeling tool used to simulate and analyze the dynamics of complex systems with uncertainty. However, due to its high computational complexity, previous research methods cannot be directly applied to study such systems. Inspired by network aggregation, this paper conducts network aggregation on LSPBNs to investigate its global stability with probability 1. It is worth mentioning that the stability conclusion proposed in this article holds for any form of network aggregation. First, the entire network is partitioned and the algebraic expressions for each subnetwork are given through the semi-tensor product of matrices. And then, a set of iterative formulas is constructed to describe and reflect the input-output coordination relationship among the subnetworks, and based on which, a sufficient condition for the global stability of LSPBNs is derived, greatly reducing computational complexity. The feasibilities of the proposed method and results are verified through examples.