Enormous amounts of data collected from social networks or other online platforms are being published for the sake of statistics, marketing, and research, among other objectives. The consequent privacy and data security concerns have motivated the work on degree-based data anonymization. We propose and study a new multi-objective anonymization approach that generalizes the known degree anonymization problem and attempts at improving it as a more realistic model for data security/privacy. Our suggested model guarantees a convenient privacy level based on modifying the degrees in a way that respects some given local restrictions, per node, such that the total modifications at the global level (in the whole graph/network) are bounded by some given value. The corresponding multi-objective graph realization approach is solved using Integer Linear Programming to obtain the best possible solutions. Our experimental studies provide empirical evidence of the effectiveness of the new approach; by specifically showing that the introduced anonymization algorithm has a negligible effect on the way nodes are clustered, thereby preserving valuable network information while significantly improving data privacy.