Noise Response Data Reveal Novel Controllability Gramian for Nonlinear Network Dynamics

Area of Science:

• Complex Systems
• Network Science
• Statistical Mechanics

Background:

• Controlling large-scale dynamical networks is crucial in science and engineering.
• The controllability Gramian is key for linear systems, measuring state reachability.
• Nonlinear network control, especially with scale-free topologies, remains challenging.

Purpose of the Study:

• Extend the controllability Gramian concept to nonlinear dynamical networks.
• Investigate the relationship between network controllability and statistical mechanics.
• Develop a simulation method for identifying controllable subdynamics in complex networks.

Main Methods:

• Generalized the controllability Gramian for nonlinear dynamics using Gibbs distributions.
• Analyzed networks open to environmental noise.
• Utilized Monte Carlo simulations to extract controllable subdynamics.

Main Results:

• The generalized Gramian equals the covariance matrix for noisy, uncontrolled trajectories.
• A Monte Carlo simulation approach is theoretically justified for identifying controllable dynamics.
• Established a novel connection between network controllability and statistical mechanics.

Conclusions:

• The extended Gramian provides a powerful tool for analyzing nonlinear network control.
• Environmental noise does not hinder the identification of controllable subdynamics.
• This work bridges control theory and statistical mechanics for complex systems.