Social network based trust relationships present a critical foundation for designing trustworthy systems, such as Sybil defenses, secure routing, and anonymous/censorshipresilient communications. A key issue in the design of such systems, is the revelation of users' trusted social contacts to an adversary-information that is considered sensitive in today's society. In this work, we focus on the challenge of preserving the privacy of users' social contacts, while still enabling the design of social trust based applications. First, we propose LinkMirage, a community detection based algorithm for anonymizing links in social network topologies; LinkMirage preserves community structures in the social topology while anonymizing links within the communities. LinkMirage considers the evolution of the social network topologies, and minimizes privacy leakage due to temporal dynamics of the system. Second, we define metrics for quantifying the privacy and utility of a time series of social topologies with anonymized links. We analyze the privacy and utility provided by LinkMirage both theoretically, as well as using real world social network topologies: a Facebook dataset with 870K links and a large-scale Google+ dataset with 940M links. We find that our approach significantly outperforms the existing state-of-art. Finally, we demonstrate the applicability of LinkMirage in real-world applications such as Sybil defenses, reputation systems, anonymity systems and vertex anonymity. We also prototype LinkMirage as a Facebook application such that real world systems can bootstrap privacy-preserving trust relationships without the cooperation of the OSN operators.