Exhaustive percolation in binary avalanches

We introduce the concept of binary avalanches as a generalization of commonly investigated site or bond percolation processes. Binary avalanches are capable of displaying a behavior that we call exhaustive percolation, where the fraction of nodes that are not affected by an avalanche approaches zero in the large system limit. We present a numerical example of exhaustive percolation on a directed lattice and analytical results for directed random networks. For random networks, the transition to exhaustive percolation is second order with scaling properties different from ordinary percolation. Our numerical calculations indicate that the exhaustive percolation transition on the directed lattice is also second order. Our analytic results for random networks improve the understanding of dynamical properties in random Boolean networks.