Robustness And Control For Dynamical Behaviors In Complex Networks

Every individual is not isolated in our society but in a network, and itself is a part of net-work. In recent years, revealing the close relationship between topology of network and network dynamics becomes a hot topic in network science. The dynamical behaviors in complex net-works are very rich, but their formation mechanisms are different from each other. In the real world, every dynamical behavior corresponds a function of the network, and uncovering the for-mation mechanism of each dynamical behavior could help us understand the behavior, as well as protect and control the function of the network. In this thesis, we will focus on the study of robustness and control for dynamical behaviors in complex networks. The factors that impact the robustness of dynamical behaviors in complex networks are found, and a new control method for synchronization-desynchronization switch is suggested.In the first chapter, the research progresses of complexity science, dynamical behaviors in complex networks, as well as the background of robustness and control in complex networks are introduced.In the second chapter, the basic knowledge of complex network and nonlinear dynamics are introduced, including network graph presentation, network expressed by matrix, basic network models, centrality of network nodes, the commonly used nonlinear oscillators, and the stability analysis of complete synchronization.In the third chapter, we systematically analysis the dynamical robustness of weighted com-plex network. We mainly study whether high-degree nodes or low-degree nodes are key nodes in impacting the dynamical robustness, actually it depends on both the degree of weighted coupling and the coupling intensity. For weakly coupled systems in weakly weighted (or unweighted) het-erogeneous networks, the low-degree nodes are more important, and the network is highly vulner-able to the failure of low-degree nodes. While for systems with other parameters, the high-degree nodes are more important, and the network is highly vulnerable to the failure of high-degree n-odes; this point is the same as in the structural robustness analysis. We also find that with random inactivation, no matter the networks are weighted or unweighted, the heterogeneous networks are always more robust than homogeneous network except for one special parameter where the robustness of these two kinds of network are the same.In the fourth chapter, a new method to find the backbone network of a full network is sug-gested. Many biology networks are robustness for its function, and for each network there is a minimal subnetwork which can maintain the main function of the full network, it is called the backbone network. To find this backbone network, we suggest a new algorithm, tinker algorithm. With this algorithm, we can accurately and efficiently find the backbone network in various mod-els, such as Boolean network model, stochastic model, and ordinary differential equations model. We also compare the topology of all backbone networks obtained from different models, and find very little difference between them.In the fifth chapter, we investigate dynamical behaviors of coupled chaotic Rossler oscillators on complex network, and find two different types of periodic windows with the variation of cou-pling strength. Under a moderate coupling, the periodic window is intermittent, and the attractors within the window extremely sensitively depend on the initial conditions, coupling strength and topology of the network. Therefore, after every slight perturbation on initial condition, coupling strength or network, the periodic attractor can be destroyed and substituted by a chaotic one, or vice versa. In contrast, under an extremely weak coupling, another types of periodic window ap-pears. This periodic window is sustained and unchanged for different types of network structure. The attractors within the window insensitively depend on the initial conditions, coupling strength and topology of the network. We also find that the attractors is general splay states, where all oscillators take the same orbit but with distinct phases. The phase differences of all oscillators are almost discrete and randomly distributed except that directly linked oscillators are more likely to have different phases.In the sixth chapter, we suggest an unified precise control method, pull-push control method, for a synchronization-desynchronization switch. With this method, synchronization can be achieve when the original systems are desynchronous by pulling (or protecting) one node or a certain subset of nodes, whereas desynchronization can be accomplished when the systems are already synchronous by pushing one node or a certain subset of nodes. Then, we test the efficiency for several different centralities according to which the controlled nodes are chosen, including degree centrality, closeness centrality, betweenness centrality and eigenvector centrality strategy. It turns out that the generalized eigenvector centrality of critical synchronization mode of the Laplacian matrix is the most efficient.We summarize the whole thesis in the last chapter.