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.
Highlights
We consider networks of dynamical units that evolve in time according to different laws, and are coupled to each other in highly irregular ways
Controlling the dynamics of ensembles of units networking via irregular topologies is one of the foremost challenges of modern science, and, the literature of the last two decades abounds with proposals for network control
A different approach was proposed in Ref.[8], where conditions based on classical control and graph theories were given for the identification of the minimal set of nodes that, if forced to follow a prescribed time evolution, suffice to drive the entire network to the target dynamics
Summary
Ricardo Gutiérrez1*, Massimo Materassi[2], Stefano Focardi2 & Stefano Boccaletti[2,3,4,5]. The introduction of multi-layer network r epresentations[5] opened up new avenues, such as the study of complex-network targetability[6], based on considering an identical copy of the graph undergoing a desirable evolution, and gradually creating unidirectional actions from nodes of the copy to the corresponding nodes in the original network, until the latter becomes fully synchronized with the former These and other related works follow the master stability function approach[7] in assuming that dynamical units are identical, and that their coupling function at each link is the same, in order to derive analytical criteria for controllability. A different approach was proposed in Ref.[8], where conditions based on classical control and graph theories were given for the identification of the minimal set of nodes that, if forced to follow a prescribed time evolution, suffice to drive the entire network to the target dynamics This is applicable to graphs whose dynamics is unknown, and for directed and weighted connectivities, when weights may possibly be unknown too. Both aspects of a node position are combined in the influence index in the case of directed or mixed networks
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