Due to specific characteristics of electric distribution systems and their high vulnerability against natural disasters, providing appropriate methods for resiliency analysis and identifying system weaknesses is of great importance. Increasing penetration of distributed generation (DG) resources and developing microgrid (MG) technology provide modularity to distribution networks. Since the system capability in splitting into multi-independent sub-sections benefits its performance in severe circumstances, the level of modularization can be construed as a measure for resiliency analysis. In this study, a novel framework based on the modularity idea is proposed for quantifying the resiliency level of electric distribution systems. Utilizing graph related theories, a new path-based approach is presented to extract the possible set of formable MGs and their servicing areas. Further, electrical and topological features, switching limitations, and reconfiguration options are also included in the MGs exploration process. Then, by introducing new dependency-based indices and considering various uncertainties, the efficiency of the formed MGs and the impact of disasters on the survivability level of the sections are evaluated. In addition, an efficient approximate approach is developed for implementing the method on large-scale distribution networks. The proposed algorithms have been simulated on several case studies with the results confirming their effectiveness and computability.