The growth of peer-to-peer applications on the Internet motivates interest in general purpose overlay networks. The construction of overlays connecting a large population of transient nodes poses several challenges. First, connections in the overlay should reflect the underlying network topology, in order to avoid overloading the network and to allow good application performance. Second, connectivity among active nodes of the overlay should be maintained, even in the presence of high failure rates or when a large proportion of nodes is not active. Finally, the cost of using the overlay should be spread evenly among peer nodes for fairness reasons as well as for the sake of application performance. To preserve scalability, we seek solutions to these issues that can be implemented in a fully decentralized manner and rely only on local knowledge from each node. In this paper, we propose an algorithm called the Localiser.