Distributed software-defined networks SDN, consisting of multiple inter-connected network domains, each managed by one SDN controller, is an emerging networking architecture that offers balanced centralized control and distributed operations. Under such a networking paradigm, most existing works focus on designing sophisticated controller-synchronization strategies to improve joint controller-decision-making for inter-domain routing. However, there is still a lack of fundamental understanding of how the performance of distributed SDN is related to network attributes, thus it is impossible to justify the necessity of complicated strategies. In this regard, we analyze and quantify the performance enhancement of distributed SDN architectures, which is influenced by intra-/inter-domain synchronization levels and network structural properties. Based on a generic network model, we establish analytical methods for performance estimation under four canonical inter-domain synchronization scenarios. Specifically, we first derive an asymptotic expression to quantify how dominating structural and synchronization-related parameters affect the performance metric. We then provide performance analytics for an important family of networks, where all links are of equal preference for path constructions. Finally, we establish fine-grained performance metric expressions for networks with dynamically adjusted link preferences. Our theoretical results reveal how network performance is related to synchronization levels and intra-/inter-domain connections, the accuracy of which is confirmed by simulations based on both real and synthetic networks. To the best of our knowledge, this is the first work quantifying the performance of distributed SDN in terms of network structural properties and synchronization levels.