<p>As an efficient approach to controlling the collective dynamics of networked oscillators, pinning synchronization has been broadly studied by researchers from different fields. For complex networks of weakly coupled chaotic oscillators, a simple way to achieve network synchronization is pinning all oscillators by an externally added controller, namely the strategy of global pinning. Yet global pinning is not reliable in many practical scenarios, as sporadic and occasional failures of the pinning couplings are inevitable, due to either random failures or intentional attacks. From the perspective of system management and protection, an issue of paramount importance is evaluating accurately how the synchronization degree of a complex network is deteriorated when some oscillators are disconnected from the controller, i.e., the problem of sporadic control failure. Here, by complex networks of coupled chaotic oscillators under global pinning, we investigate how the synchronizability of the networks is affected when a few oscillators are unpinned. We find that, instead of the large-degree oscillators, the network synchronizability is more deteriorated when the small-degree oscillators are unpinned. We demonstrate numerically this intriguing phenomenon in different network models, and the numerical results are analyzed by the methods of eigenvector-based analysis and matrix perturbation theory.</p>