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Network structure construction and global state maintenance are expensive in large-scale dynamic peer-to-peer (P2P) networks. With inherent topology independence and low state maintenance overhead, random walk is an excellent tool in such network environments. However, the current uses are limited to unguided or heuristic random walks with no guarantee on their converged node visitation probability distribution. Such a convergence guarantee is essential for strong analytical properties and high performance of many P2P applications. In this paper, we investigate an approach for random walks to converge to application-desired node visitation probability distributions while only requiring information about the direct neighbors of each peer. Our approach is guided by the Metropolis-Hastings algorithm for Monte Carlo Markov Chain sampling. Our contributions are threefold. First, we analyze the convergence time of the random walk node visitation probability distribution on common P2P network topologies. Second, we analyze the fault tolerance of our random walks in dynamic networks with potential walker losses. Third, we present the effectiveness of random walks in assisting three realistic network applications: random membership subset management, search, and load balancing. Both search and load balancing desire random walks with biased node visitation distributions to achieve application-specific goals. Our analysis, simulations, and Internet experiment demonstrate the advantage of our random walks compared with alternative topology-independent index-free approaches.