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Random scale-free overlay topologies provide a number of properties like for example high resilience against failures of random nodes, small (average) diameter as well as good expansion and congestion characteristics that make them interesting for the use in large-scale distributed systems. A number of these properties have been shown to be influenced by the exponent of their power law degree distribution. In this article, we present a distributed rewiring scheme that is suitable to effectuate random scale-free overlay topologies with an adjustable degree distribution exponent. The scheme uses a biased random walk strategy to sample new endpoints of edges being rewired and relies on an equilibrium model for scale-free networks. The bias of the random walk strategy can be tuned to produce random scale-free networks with arbitrary degree distribution exponents greater than two. We argue that the rewiring strategy can be implemented in a distributed fashion based on a node's information about its immediate neighbors. We present both analytical arguments as well as results that have been obtained in simulations of the proposed protocol.