Topology management schemes conserve energy in wireless ad hoc networks by identifying redundant nodes that may turn off their radios or other components while maintaining connectivity. We present Naps, a randomized topology management scheme that does not rely on geographic location information, provides exibility in the target density of waking nodes, and sends only a periodic heartbeat message between waking neighbors; thus it is implementable even on modest hardware. We formally analyze the connectivity of the waking graphs produced by Naps, showing that these graphs have nearly complete connectivity even at relatively low densities. We examine simulation results for a wide range of initial deployment densities and for heterogeneous and mobile deployments.