This paper investigates the synchronization properties of scale-free dynamical networks, which are characterized by power-law connectivity distributions. It shows that such networks are robust against random node removal but fragile to targeted removal of highly connected nodes. Scale-free networks are inhomogeneous, with most nodes having few connections and a small number of highly connected nodes. This inhomogeneity makes them error-tolerant but vulnerable to attacks. The synchronization stability of a scale-free network depends on the second-largest eigenvalue of its coupling matrix. Numerical results show that the second-largest eigenvalue remains nearly unchanged when a small fraction of nodes is randomly removed, indicating robustness. However, when highly connected nodes are removed, the second-largest eigenvalue increases significantly, leading to a loss of synchronization. The paper also compares scale-free networks with locally regular networks, showing that the latter have lower synchronizability due to their uniform connectivity. The analysis is supported by simulations using a Chua's oscillator model, demonstrating that scale-free networks can achieve chaotic synchronization with appropriate coupling strength. The study concludes that scale-free networks are robust to random failures but vulnerable to targeted attacks, highlighting their 'robust yet fragile' nature.This paper investigates the synchronization properties of scale-free dynamical networks, which are characterized by power-law connectivity distributions. It shows that such networks are robust against random node removal but fragile to targeted removal of highly connected nodes. Scale-free networks are inhomogeneous, with most nodes having few connections and a small number of highly connected nodes. This inhomogeneity makes them error-tolerant but vulnerable to attacks. The synchronization stability of a scale-free network depends on the second-largest eigenvalue of its coupling matrix. Numerical results show that the second-largest eigenvalue remains nearly unchanged when a small fraction of nodes is randomly removed, indicating robustness. However, when highly connected nodes are removed, the second-largest eigenvalue increases significantly, leading to a loss of synchronization. The paper also compares scale-free networks with locally regular networks, showing that the latter have lower synchronizability due to their uniform connectivity. The analysis is supported by simulations using a Chua's oscillator model, demonstrating that scale-free networks can achieve chaotic synchronization with appropriate coupling strength. The study concludes that scale-free networks are robust to random failures but vulnerable to targeted attacks, highlighting their 'robust yet fragile' nature.