Steering complex networks toward desired dynamics

Abstract

We consider networks of dynamical units that evolve in time according to different laws, and are coupled to each other in highly irregular ways. Studying how to steer the dynamics of such systems towards a desired evolution is of great practical interest in many areas of science, as well as providing insight into the interplay between network structure and dynamical behavior. We propose a pinning protocol for imposing specific dynamic evolutions compatible with the equations of motion on a networked system. The method does not impose any restrictions on the local dynamics, which may vary from node to node, nor on the interactions between nodes, which may adopt in principle any nonlinear mathematical form and be represented by weighted, directed or undirected links. We first explore our method on small synthetic networks of chaotic oscillators, which allows us to unveil a correlation between the ordered sequence of pinned nodes and their topological influence in the network. We then consider a 12-species trophic web network, which is a model of a mammalian food web. By pinning a relatively small number of species, one can make the system abandon its spontaneous evolution from its (typically uncontrolled) initial state towards a target dynamics, or periodically control it so as to make the populations evolve within stipulated bounds. The relevance of these findings for environment management and conservation is discussed.

Figure 1

Controlling the dynamics of a mixed network with uniform $k_{\text{in}}$ and $k_{\text{out}}$ distributions comprising $N=50$ nonlinearly-coupled Rössler oscillators with intra-layer coupling ${\sigma }_1=0.01$ and inter-layer coupling ${\sigma }_2=1$. (Top) Maximum Lyapunov exponent ${\lambda }_{\text{max}}$ (main panel) and synchronization error (inset) as functions of the targeting step. (Bottom) Influence index $k_{\text{out}}/k_{\text{in}}$ of the node that is pinned at each targeting step. The curves are averages of 20 different network realizations. A 4th-order Runge-Kutta method with a step of 0.01 time units has been employed for the numerical integration of the systems of $3N=150$ ordinary differential equations corresponding to each layer.

Figure 2

Controlling the dynamics of a trophic web comprising 12 species with inter-layer coupling ${\sigma }_2=0.005$. (Top) Maximum Lyapunov exponent ${\lambda }_{\text{max}}$ (main panel) as a function of the targeting step (codes indicating the species targeted at each step are described in Sect. B of the Supplementary Information). (Bottom) Synchronization error as a function of the targeting step. A 4th-order Runge–Kutta method with a step of 0.01 time units has been employed for the numerical integration of the systems of 12 ordinary differential equations corresponding to each layer.

Figure 3

Evolution of two species of the trophic web (red dotted line) pinned to a periodically repeated segment of recorded population dynamics (black discs). (Top) Population of one species of deer as a function of time in the recording and under pinning of 5 species using the recording as master dynamics. (Bottom) Population of another species of deer under the same conditions.