Noise Response Data Reveal Novel Controllability Gramian for Nonlinear Network Dynamics

TL;DR

This paper introduces a new controllability Gramian for nonlinear networks based on noise response data, linking controllability with statistical mechanics and enabling efficient analysis of complex dynamical systems.

Contribution

It extends the controllability Gramian concept to nonlinear systems using Gibbs distribution and relates it to noise-induced covariance, facilitating practical controllability analysis.

Findings

The new Gramian equals the covariance of uncontrolled, noise-perturbed trajectories.

Monte Carlo simulations can effectively identify controllable subdynamics.

Provides a theoretical link between controllability and statistical mechanics.

Abstract

Control of nonlinear large-scale dynamical networks, e.g., collective behavior of agents interacting via a scale-free connection topology, is a central problem in many scientific and engineering fields. For the linear version of this problem, the so-called controllability Gramian has played an important role to quantify how effectively the dynamical states are reachable by a suitable driving input. In this paper, we first extend the notion of the controllability Gramian to nonlinear dynamics in terms of the Gibbs distribution. Next, we show that, when the networks are open to environmental noise, the newly defined Gramian is equal to the covariance matrix associated with randomly excited, but uncontrolled, dynamical state trajectories. This fact theoretically justifies a simple Monte Carlo simulation that can extract effectively controllable subdynamics in nonlinear complex networks. In addition, the result provides a novel insight into the relationship between controllability and statistical mechanics.