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This paper considers the problem of how to construct and maintain an overlay structured P2P network based on the small world paradigm. Two main attractive properties of a small world network are (1) low average hop distance between any two randomly chosen nodes, and (2) high clustering coefficient of nodes. Having a low average hop distance implies a low latency for object lookup, while having a high clustering coefficient implies the underlying network can effectively provide object lookup even under heavy demands (for example, in a flash crowd scenario). We present a small world overlay protocol (SWOP) for constructing a small world overlay P2P network. We compare the performance of our system with that of other structured P2P networks such as Chord. We show that the SWOP protocol can achieve improved object lookup performance over the existing protocols. We also exploit the high clustering coefficient of a SWOP network to design an object replication algorithm that can effectively handle heavy object lookup traffic. As a result, a SWOP network can quickly and efficiently deliver popular and dynamic objects to a large number of requesting nodes. To the best of our knowledge, ours is the first piece of work that addresses how to handle dynamic flash crowds in a structured P2P network environment.