We introduce a scalable searching algorithm for finding nodes and contents in random networks with Power-Law (PL) and heavy-tailed degree distributions. The network is searched using a probabilistic broadcast algorithm, where a query message is relayed on each edge with probability just above the bond percolation threshold of the network. We show that if each node caches its directory via a short random walk, then the total number of accessible contents exhibits a first-order phase transition,

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... nsuring very high hit rates just above the percolation threshold. In any random PL network of size, N, and exponent, 2 ≤τ < 3, the total traffic per query scales sub-linearly, while the search time scales as O( N). In a PL network with exponent, τ≈ 2, any content or node can be located in the network with probability approaching one in time O( N), while generating traffic that scales as O(^2 N), if the maximum degree, k_max, is unconstrained, and as O(N^1/2+ϵ) (for any ϵ>0) if k_max=O(√(N)). Extensive large-scale simulations show these scaling laws to be precise. We discuss how this percolation search algorithm can be directly adapted to solve the well-known scaling problem in unstructured Peer-to-Peer (P2P) networks. Simulations of the protocol on sample large-scale subnetworks of existing P2P services show that overall traffic can be reduced by almost two-orders of magnitude, without any significant loss in search performance.