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Open-access distributed systems such as peer-to-peer systems are particularly vulnerable to sybil attacks, where a malicious user creates multiple fake identities (called sybil nodes). Without a trusted central authority that can tie identities to real human beings, defending against sybil attacks is quite challenging. Among the small number of decentralized approaches, our recent SybilGuard protocol leverages a key insight on social networks to bound the number of sybil nodes accepted. Despite its promising direction, SybilGuard can allow a large number of sybil nodes to be accepted. Furthermore, SybilGuard assumes that social networks are fast-mixing, which has never been confirmed in the real world. This paper presents the novel SybilLimit protocol that leverages the same insight as SybilGuard, but offers dramatically improved and near-optimal guarantees. The number of sybil nodes accepted is reduced by a factor of Θ(√n), or around 200 times in our experiments for a million-node system. We further prove that SybilLimit's guarantee is at most a logn factor away from optimal when considering approaches based on fast-mixing social networks. Finally, based on three large-scale real-world social networks, we provide the first evidence that real-world social networks are indeed fast-mixing. This validates the fundamental assumption behind SybilLimit's and SybilGuard's approach.