Control and controllability of nonlinear dynamical networks: a geometrical approach

TL;DR

This paper presents a practical control framework for nonlinear dynamical networks with multistability, focusing on driving the system between attractors using restricted, experimentally feasible parameter perturbations, and introduces the concept of attractor networks.

Contribution

It develops a novel geometrical control approach for nonlinear networks with multistability, incorporating restricted perturbations and the attractor network concept to assess controllability.

Findings

Noise can facilitate control due to nonlinearity.

Controllability correlates with the connectivity of the attractor network.

Framework is demonstrated on gene regulatory network models.

Abstract

In spite of the recent interest and advances in linear controllability of complex networks, controlling nonlinear network dynamics remains to be an outstanding problem. We develop an experimentally feasible control framework for nonlinear dynamical networks that exhibit multistability (multiple coexisting final states or attractors), which are representative of, e.g., gene regulatory networks (GRNs). The control objective is to apply parameter perturbation to drive the system from one attractor to another, assuming that the former is undesired and the latter is desired. To make our framework practically useful, we consider RESTRICTED parameter perturbation by imposing the following two constraints: (a) it must be experimentally realizable and (b) it is applied only temporarily. We introduce the concept of ATTRACTOR NETWORK, in which the nodes are the distinct attractors of the system, and there is a directional link from one attractor to another if the system can be driven from the former to the latter using restricted control perturbation. Introduction of the attractor network allows us to formulate a controllability framework for nonlinear dynamical networks: a network is more controllable if the underlying attractor network is more strongly connected, which can be quantified. We demonstrate our control framework using examples from various models of experimental GRNs. A finding is that, due to nonlinearity, noise can counter-intuitively facilitate control of the network dynamics.