Networkcontrology

Abstract

An increasing number of complex systems are now modeled as networks of coupled dynamical entities. Nonlinearity and high-dimensionality are hallmarks of the dynamics of such networks but have generally been regarded as obstacles to control. Here, I discuss recent advances on mathematical and computational approaches to control high-dimensional nonlinear network dynamics under general constraints on the admissible interventions. I also discuss the potential of network control to address pressing scientific problems in various disciplines.

FIG. 1.

Example of a system that is controllable but whose control trajectories are nonlocal. Adapted from J. Sun and A. E. Motter, “Controllability transition and nonlocality in network control,” Phys. Rev. Lett. 110, 208701 (2013). Copyright 2013 American Physical Society.

FIG. 2.

Illustration of a control problem in which the variables are constrained not to be increased by the intervention, which prohibits steering the trajectory directly to the desired attractor. (a) A solution that consists of bringing the system to the basin of the desired attractor. (b) The problem as it appears to the observer when the basin of attraction is not known, as in the case of networks with high-dimensional phase spaces. The dotted line indicates the future evolution of the uncontrolled system.

FIG. 3.

Schematic illustration of the control approach, for interventions based on manipulating (a) dynamical variables and (b) system parameters.

FIG. 4.

Example of associative memory network in which each letter of the word “NETWORK” is stored as an attractor. The network is 8 × 8 and each node is a phase oscillator color-coded by the phase; in stationary states each node can be in one of two states: in phase or anti-phase with respect to a reference node (marked as ON and OFF pixels, respectively). In this illustration, the control problem is to drive the network from the attractor representing a letter to the attractor presenting the next letter by only manipulating OFF-pixel nodes. The control interventions are indicated by the vertical arrows, and the subsequent evolutions toward the attractors are indicated by the oblique arrows. The gray pixels mark errors, which means that in some cases the system converged to a different, parasite attractor. The reached attractors are, nevertheless, remarkably similar to the intended ones, indicating that the method is robust even when the desired solution is not possible. Adapted from Cornelius et al., “Realistic control of network dynamics,” Nat. Commun. 4, 1942 (2013). Copyright 2013 Macmillan Publishers Limited).