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. In such 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 impossible to justify the necessity of complicated strategies. In this regard, we analyse and quantify how the performance enhancement of distributed SDN architectures is influenced by inter-domain synchronization levels, in terms of the resulting number of abstracted routing clusters, and network structural properties. Based on a generic network model incorporating link preference for path constructions, we establish analytical lower bounds for quantifying the routing performance under any arbitrarily given network synchronization status. The significance of these performance bounds is that they can be used to quantify the contribution of controller synchronization levels in improving the network performance under different network parameters, which therefore serves as a fundamental guidance for future SDN performance analysis and protocol designs.