A resistor-network picture of transitions is appropriate for the study of energy absorption by weakly chaotic or weakly interacting driven systems.Such ”sparse” systems reach a novel non- equilibrium steady state (NESS) once coupled to a bath. In the stochastic case there is an analogy to the physics of percolating glassy systems, and an extension of the fluctuation-dissipation phenomenology is proposed. In the mesoscopic case the quantum NESS might differ enormously from the stochastic NESS, with saturation temperature determined by the sparsity. A toy model where the sparsity of the system is modeled using a log-normal random ensemble is analyzed. We also develop a method for calculating the current around a ring in a NESS.