— Unstructured p2p and overlay network applications often require that a random graph be constructed, and that some form of random node selection take place over that graph. A key and difficult requirement of many such applications is heterogeneity: peers have different node degrees in the random graph based on their capacity. Using simulations, this paper compares a number of techniques—some novel and some variations on known approaches—for heterogeneous graph construction and random node selection on top of such graphs. Our focus is on practical criteria that can lead to a genuinely deployable toolkit that supports a wide range of applications. These criteria include simplicity of operation, support for node heterogeneity, quality of random selection, efficiency and scalability, load balance, and robustness. We show that all these criteria can more-or-less be met by all the approaches. Our novel approach, however, stands out as the best from a practical perspective because of...