The problem of anonymization in large networks and the utility of released data are considered in this paper. Although there are some anonymization methods for networks, most of them cannot be applied in large networks because of their complexity. In this paper, we devise a simple and efficient algorithm for k-degree anonymity in large networks. Our algorithm constructs a k-degree anonymous network by the minimum number of edge modifications. We compare our algorithm with other well-known k-degree anonymous algorithms and demonstrate that information loss in real networks is lowered. Moreover, we consider the edge relevance in order to improve the data utility on anonymized networks. By considering the neighbourhood centrality score of each edge, we preserve the most important edges of the network, reducing the information loss and increasing the data utility. An evaluation of clustering processes is performed on our algorithm, proving that edge neighbourhood centrality increases data utility. Lastly, we apply our algorithm to different large real datasets and demonstrate their efficiency and practical utility.