A Method to Stabilize Large-Scale Boolean Networks via Pinning Control

As a discrete-time system, the dynamics of each gene node of Boolean networks (BNs) is determined by logical functions. This article aims to stabilize BNs globally through distributed pinning control. According to the existing literature, the state transition matrix-based pinning control strategy always suffers from high time complexity. The proposed approach, while likewise matrix-based, replaces the state transition matrix in favor of structure matrices based on the logical functions of each node and develops four computationally effective phases to solve the minimum set of controlled nodes and the corresponding pinning controllers, reducing the time complexity to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O(2^{\xi })$</tex-math></inline-formula>, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\xi$</tex-math></inline-formula> is a value determined by the specific network and the target state. The primary contribution is to provide a new method that eliminates time complexity as a barrier to matrix-based pinning control strategy by determining the index set of key nodes, particularly for sparse biological networks while preserving the benefits of matrix-based strategies. Finally, to demonstrate the effectiveness of the control method, gene networks with different sizes are discussed as simulations.