We randomly perturb a large graph G in a natural manner: independently and identically distributed at every vertex x of G, we draw a random family of rooted graphs and connect x to the root of each graph in the family. We study the density of states (DOS) of the resulting sprinkled graph in the limit of large G. By relating our model to both site percolation as well as the Anderson model on G, we analyze the atoms and the support of the DOS. We derive an expansion of the DOS in the limit of sparse sprinkling. Based on joint work with Antti Knowles.