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The graph structure of a peer-to-peer network defines the neighboring relations between peer nodes. The nodes can issue queries only with the assistance of the graph edges. For that reason it is essential to find the most valuable edges, that is, the edges that provide the most answers for the queries of a given node. However, the basic peer-to-peer protocols construct this graph on a random manner and they do not take the content of the stored or searched documents into account. Our solution improves the performance of a peer-to-peer network by constructing semantic profiles based on document metadata followed by a comparison of the semantic similarities on generalized topics, where these topic generalizations are made with an adaptive algorithm that utilize a well-tried full-language taxonomy. We expect that as time goes on our protocol, the SemPeer, transforms the graph so that neighboring nodes can answer each others queries with greater probability than with the previous graph structures.