Asymptotic Synchronization Analysis in Drive-Response Markovian Jump Boolean Networks

Area of Science:

• Control Theory
• Network Science
• Stochastic Systems

Background:

• Boolean networks (BNs) are fundamental models in systems biology and discrete dynamical systems.
• Markovian jump Boolean networks (MJBNs) introduce stochasticity and regime switching, complicating analysis.
• Asymptotic synchronization is crucial for coordinated behavior in complex networks.

Purpose of the Study:

• To develop effective criteria for the asymptotic synchronization of drive-response MJBNs.
• To adapt synchronization techniques from deterministic BNs to the more complex MJBN framework.
• To establish a robust method for analyzing set stability in stochastic switching networks.

Main Methods:

• Employing an algebraic state space representation for MJBNs.
• Introducing an augmented variable to transform synchronization into an asymptotic set stability problem.
• Developing criteria based on stochastic processes, invariant subsets, and singular value analysis.

Main Results:

• Two novel criteria for identifying nonnegative solutions and assessing set stability in MJBNs were developed.
• The equivalence between the two proposed criteria was mathematically demonstrated.
• The criteria for MJBNs were shown to be more intricate than those for deterministic BNs.

Conclusions:

• The proposed criteria offer a rigorous framework for analyzing asymptotic synchronization in MJBNs.
• The methods provide a valuable tool for understanding and controlling complex stochastic networks.
• Numerical examples confirm the practical applicability and effectiveness of the developed theoretical concepts.