Controllability of Complex Networks with Nonlinear Dynamics

TL;DR

This paper discusses the controllability of complex networks with nonlinear dynamics, clarifying the limitations of linearization methods and reaffirming control results for linear networks with single-node control.

Contribution

It clarifies the validity of control results for linear networks with single-node control despite issues with nonlinear dynamics, and discusses the limitations of linearization.

Findings

Single-node control can fully control linear networks with 1D self-dynamics.

Linearization may fail to predict controllability in nonlinear networks.

Control results for linear networks remain valid under certain conditions.

Abstract

Sun et al. provided an insightful comment arXiv:1108.5739v1 on our manuscript entitled "Controllability of Complex Networks with Nonlinear Dynamics" on arXiv. We agree on their main point that linearization about locally desired states can be violated in general by the breakdown of local control of the linearized complex network with nonlinear state. Therefore, we withdraw our manuscript. However, other than nonlinear dynamics, our claim that a single-node-control can fully control the general bidirectional/undirected linear network with 1D self-dynamics is still valid, which is similar to (but different from) the conclusion of arXiv:1106.2573v3 that all-node-control with a single signal can fully control any direct linear network with nodal-dynamics (1D self-dynamics).