The risks of publishing privacy-sensitive data have received considerable attention recently. Several de-anonymization attacks have been proposed to re-identify individuals even if data anonymization techniques were applied. However, there is no theoretical quantification for relating the data utility that is preserved by the anonymization techniques and the data vulnerability against de-anonymization attacks. In this paper, we theoretically analyze the de-anonymization attacks and provide conditions on the utility of the anonymized data (denoted by anonymized utility) to achieve successful de-anonymization. To the best of our knowledge, this is the first work on quantifying the relationships between anonymized utility and de-anonymization capability. Unlike previous work, our quantification analysis requires no assumptions about the graph model, thus providing a general theoretical guide for developing practical de-anonymization/anonymization techniques. Furthermore, we evaluate state-of-the-art de-anonymization attacks on a real-world Facebook dataset to show the limitations of previous work. By comparing these experimental results and the theoretically achievable de-anonymization capability derived in our analysis, we further demonstrate the ineffectiveness of previous de-anonymization attacks and the potential of more powerful de-anonymization attacks in the future.