  We study the steady state of a multiply-connected system that is driven out
of equilibrium by a sparse perturbation. The prototype example is an $N$-site
ring coupled to a thermal bath, driven by a stationary source that induces
transitions with log-wide distributed rates. An induced current arises, which
is controlled by the strength of the driving, and an associated topological
term appears in the expression for the energy absorption rate. Due to the
sparsity, the crossover from linear response to saturation is mediated by an
intermediate regime, where the current is exponentially small in $\sqrt{N}$,
which is related to the work of Sinai on "random walk in a random environment".
