Research On The Control Of Complex Networked Systems

Controlling collective dynamics of complex networks, which is a frontier problemof complexity science, is our ultimate goal of studying them and has a wide range ofapplications in many complex systems and dynamical processes in nature and society,such as cascading failures, epidemic and rumor spreading, synchronization and etc.However, the field of controlling complex networked systems is still in its infancy,due to the lack of general theoretical tools to address the controllability of arbitrarynetworks, despite the recent development of structural-controllability theory by Liuet.al in2011that stimulates an increasing amount of interest. In addition, there aremany important problems deserve research, such as controller setting, minimal controlenergy, nodes classification and so on.Here, we introduce an exact-controllability paradigm based on the approach ofmaximum multiplicity to determine the minimum number of controllers and identifydriver nodes required to achieve full control of arbitrary network structures and link-weight distributions. The framework reproduces the structural controllability for di-rected networks associated with structural matrices. The exact controllability of a largenumber of real and model networks is explored, leading to the general finding of diffi-culty in controlling dense networks with identical weights. The paradigm allows us todevelop an efficient and simplified tool to accurately assess the controllability of sparseand dense networks, where the former are common in real-world systems. At the sametime, we study the relation between exact-controllability and network structure.Application of our exact-controllability theory, we develop a general frameworkto study the controllability of multiplex networks with particular interest in two rep-resentative classes, i.e., multiple-relation networks and multiple-layer networks withinterlayer couplings. In the former, networks associated with different physical vari-ables share the same set of nodes and in the latter diffusion processes take place. We offer theoretical tools to reveal how the interplay among different relation networksand interconnected layers affects the system’s controllability, finding a dominant layerin the multiple-relation network and the remarkable improvement of controllability bya small fraction of interconnections in the multiple-layer network. Our framework isgenerally applicable beyond the two classes, enabling comprehensively understandingof our ability to control t he dynamics of various multiplex network systems.For control a system, the most important issue is how to select a proper inputmatrix to guarantee the controllable of this system. We conclude that an undirectedpath is controllable by driving a single node at either end of the path. For a generalcomplex network, we give a method to construct a simplest input matrix to controlits transformed network, which lead to another input matrix (generally speaking, morecomplex) for the original network. It is difficult to give a simple input matrix for theoriginal network at present. For analyzing input matrix, we divide nodes into threeparts: driver nodes, driven nodes and redundant uncontrolled nodes. We put forwarda method to ascertain each part of nodes and simulate their distribution on ER randomnetwork and BA scale-free network, respectively.Understanding exact controllability via our framework is fundamental towardsachieving actual control and exploring practical issues, such as control energy, con-trol core. We exact calculate the control energy of a controlled networks not only itsnormal bounds, which could greatly improve our ability to control complex systems.For control core, we propose a leaf remove proceeding to get a control core for sparsenetworks and calculate the density of nodes and edges of this core. We also researchthe relation between exact controllability and network structure and calculate the exactobservability of any complex networks with linear coupling.Our framework is generally applicable beyond the two classes, enabling compre-hensively understanding of our ability to control the dynamics of various multiplexnetwork systems. Our work will offer new insight into the fundamental problem ofcontrolling complex networked systems, especially for improving the exact controlla-bility framework and providing theoretical support for devising optimal control.