Many complex networks commonly exhibit community structure in their underlying organizations, i.e., they contain multiple groups of nodes having more connections inside a group and less interactions among groups. This special structure not only offers key insights into understanding the network organization principles but also plays a vital role in maintaining the normal function of the whole system. As a result, any significant change to the network communities, due to elementwise failures,

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... potentially redefine their organizational structures and consequently lead to the malfunction or undesirable corruption of the entire system. Therefore, identifying network elements that are essential to its community structure is a fundamental and important problem. However, to the best of our knowledge, this research direction has not received much been attention in the literature. In this paper, we study the structural vulnerability of overlapping complex network communities to identify nodes that are important in maintaining the complex structure organization. Specifically, given a network and a budget of k nodes, we want to identify k critical nodes whose their exclusions transforms the current network community structure. To effectively analyze this vulnerability on overlapping communities, we propose the concept of generating edges and provide an optimal algorithm for detecting the Minimal Generating Edge Set (MGES) in a network community. We suggest genEdge, an effective solution based on this MGES. Empirical results on both synthesized networks with known community structures, and real data including Reality cellular data, Foursquare and Facebook social traces confirm the efficacy of our approach.