Bootstrap percolation is a classical model for the spread of information in a network. In the round-based version, nodes of an undirected graph become active once at least r neighbors were active in the previous round. We propose the perturbed percolation process: a superposition of two percolation processes on the same node set. One process acts on a local graph with activation threshold 1, the other acts on a global graph with threshold rrepresenting local and global edges, respectively. We consider grid-like local graphs and expanders as global graphs on n nodes. For the extreme case r = 1, all nodes are active after O(log n) rounds, while the process spreads only polynomially fast for the other extreme case r ≥ n. For a range of suitable values of r, we prove that the process exhibits both phases of the above extremes: It starts with a polynomial growth and eventually transitions from at most cn to n active nodes, for some constant c ∈ (0, 1), in O(log n) rounds. We observe this behavior also empirically, considering additional global-graph models.