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In this paper, we propose a generalization of the basic flooding search strategy for decentralized unstructured peer-to-peer (P2P) networks. In our algorithm a peer forwards a query to one of its neighbors using a probability that is a function of the number of connections in the overlay network of both. Moreover, this probability may also depend on the distance from the query originator. To analyze the performance of the proposed search strategy in heterogeneous decentralized unstructured P2P networks, we develop a generalized random graph (GRG)-based model that takes into account the high variability in the number of application level connections that each peer establishes, and the nonuniform distribution of resources among peers. Furthermore, the model includes an analysis of peer availability, i.e., the capability of relaying queries of other peers, as a function of the query generation rate of each peer. Validation of the proposed model is carried out comparing the model predictions with simulations conducted on real overlay topologies obtained from crawling the popular file sharing application Gnutella.