Robustness Of Networked Oscillators System

Recently, the study on networked complex systems has attracted great attention. For such systems, one of the most important issues is the network robustness, i.e., the ability to maintain normal functions or structures when suffering some kinds of de-liberate attacks or malfunctions. For instance, it is crucial for a power-grid whether it could work or not when some parts in the grids have been destroyed. Previously, many works have been done for this problem in the framework of percolation and network connectivity [15,16]. However, most of them only focus on the network topology and ignore the dynamics of nodes in the systems. In2004, Daido et. al. studied the aging transition in a system composed of active and inactive elements [24]. By introducing node dynamics, this model is more practical and suitable to investigate the dynamical robustness of networked systems. However, their theoretical treatment only involved the globally coupled networks-the very special case of networks.In this thesis, we further study the aging transition by extending the network topology to general cases. Our particular interest focuses on the dynamical robust-ness of the networked system. By employing the mean-field approximation and linear stability analysis, we analytically give the critical points of aging transitions in dif-ferent network topologies, which have been verified by numerical experiments based on Stuart-Landau oscillators and Rossler oscillators. Mainly, our study obtains the following results:1. For homogeneous system, we find that the robustness is related to the mean degree of the network. With the increase of the mean degree, the robustness generally decreases. This result implies that adding links may be helpful to the connectivity of the system, but harmful to the dynamical robustness of such systems.2. The robustness of heterogeneous system depends on the mean degrees of both active and inactive oscillators. It is more robust when the mean degree of inactive oscillators is larger than the mean degree of active ones.3. When the mean degree of the network is fixed, it is in principle uncertain whether the homogeneous system or the heterogeneous system is more robust. How-ever, the probability is larger with which the heterogeneous system is more robust than the homogeneous one. 4. There exist fluctuations for the critical points of the aging transitions when randomly flipping active oscillators into inactive ones. Specially, the fluctuations are more obvious in the heterogenous systems. These phenomena can be successfully explained by plotting the phase diagram in parameter plane.5. We investigate the robustness of the system with active and inactive oscillators by comparing three typical strategies of flipping oscillators. Contrary to our intuition, it is shown that the most effective method to destroy the system is to start the flipping from the node with minimal degree. This implies that the hub’nodes with large de-grees are not necessarily the most important in all situations. This result can also be explained by our theory.This work is helpful to understand the behaviors of networked dynamical system-s, and shed light on certain practical applications as well.