Privacy risks are an important issue to consider during the release of network data to protect personal information from potential attacks. Network data anonymization is a successful procedure used by researchers to prevent an adversary from revealing the user's identity. Such an attack is called a re-identification attack. However, this is a tricky task where the primary graph structure should be maintained as much as feasible within the anonymization process. Most existing solutions used edge-perturbation methods directly without any concern regarding the structural information of the graph. While that preserving graph structure during the anonymization process requires keeping the most important knowledge edges in the graph without any modifications. This paper introduces a high utility K-degree anonymization method that could utilize edge betweenness centrality ( ) as a measure to map the edges that have a central role in the graph. Experimental results showed that preserving these edges during the modification process will lead the anonymization algorithm to better preservation for the most important structural properties of the graph. This method also proved its efficiency for preserving community structure as a trade-off between graph utility and privacy.