This paper investigates the synchronization phenomena in scale-free dynamical networks, focusing on the robustness and fragility of synchronization. Scale-free networks, characterized by a power-law distribution of node degrees, are found to be robust against random node removal but fragile to the removal of highly connected nodes. The authors use a scale-free network model based on the Barabási-Albert model and analyze the synchronization stability of the network through linearization and Lyapunov function methods. They show that the second-largest eigenvalue of the coupling matrix, which characterizes the synchronizability, remains almost unchanged when a small fraction of randomly chosen nodes are removed. However, when a small fraction of the most connected nodes are removed, the second-largest eigenvalue decreases significantly, leading to a loss of synchronization. The paper concludes that scale-free networks exhibit a "robust yet fragile" property, where they can tolerate random errors but are vulnerable to targeted attacks.This paper investigates the synchronization phenomena in scale-free dynamical networks, focusing on the robustness and fragility of synchronization. Scale-free networks, characterized by a power-law distribution of node degrees, are found to be robust against random node removal but fragile to the removal of highly connected nodes. The authors use a scale-free network model based on the Barabási-Albert model and analyze the synchronization stability of the network through linearization and Lyapunov function methods. They show that the second-largest eigenvalue of the coupling matrix, which characterizes the synchronizability, remains almost unchanged when a small fraction of randomly chosen nodes are removed. However, when a small fraction of the most connected nodes are removed, the second-largest eigenvalue decreases significantly, leading to a loss of synchronization. The paper concludes that scale-free networks exhibit a "robust yet fragile" property, where they can tolerate random errors but are vulnerable to targeted attacks.