Abstract
Identifying critical edges is a fundamental problem in network science, with direct implications for understanding robustness, vulnerability, and functional organization in complex systems. Existing approaches typically rely on single-scale structural properties—either local or global—and thus fail to capture the inherently multi-scale nature and weighted interactions of real-world networks, which are often organized into communities. This paper introduces a multi-scale framework for quantifying edge importance by integrating local community structure with global connectivity patterns. Communities are first detected using the Leiden algorithm, after which edges are classified into intra-community and inter-community categories. The importance of intra-community edges, which sustain local cohesion, is measured by their frequency across all possible Minimum Spanning Trees (MSTs) within each community, weighted by a community factor that accounts for size, internal strength, and density. The importance of inter-community edges, which act as global bridges, is assessed through a composite metric combining local bridgeness with their necessity for preserving connectivity in the condensed community network via super-edge MST analysis. A unified ranking is then established, prioritizing inter-community edges due to their central role in maintaining global connectivity. Extensive experiments on diverse real-world networks demonstrate that attack strategies guided by the proposed metric cause the fastest degradation of global connectivity and the steepest decline in local efficiency. These results confirm the effectiveness of the framework in identifying edges that are structurally vital and functionally indispensable across multiple scales.