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Social networks are attracting significant interest from researchers in different domains, especially with the advent of social networking systems which enable large-scale collection of network information. However, as much as analysis of such social networks can benefit researchers, it raises serious privacy concerns for the people involved in them. To address such privacy concerns, several techniques, such as k-anonymity-based approaches, have been proposed in the literature to provide user anonymity in published social networks. However, these methods usually introduce a large amount of distortion to the original social network graphs, thus, raising serious questions about their utility for useful social network analysis. Consequently, these techniques may never be applied in practice. We propose two methods to enhance edge-perturbing anonymization methods based on the concepts of structural roles and edge betweenness in social network theory. We experimentally show significant improvements in preserving structural properties in an anonymized social network achieved by our approach compared to the original algorithms over several data sets.