Following the trend of preserving privacy in online-social-network publishing, various anonymization mechanisms have been designed and applied. Differential privacy is an approach that guarantees the privacy level. Many existing mechanisms claim that they can also preserve the utility very well during anonymization. However, their utility analysis is always based on some specifically chosen metrics. While the existing metrics only partially present the graph utility, this paper aims to find a novel approach that describes the network in multiple scales. Persistent homology is a high-level metric, in that it reveals the parameterized topological features with various scales, and it is applicable for real-world applications. In this paper, four differential privacy mechanisms with different abstraction models are analyzed with traditional graph metrics and with persistent homology. The evaluation results demonstrate that all algorithms can partially or conditionally preserve certain graph utilities, but none of them are suitable for all metrics. Furthermore, none of the existing mechanisms fully preserves persistent homology, especially in high dimensions, which implies that the true graph utility is lost
