Bootstrap percolation on the random graph $G_{n,p}

Bootstrap percolation on the random graph G... is a process of spread of "activation" on a given realization of the graph with a given number of initially active nodes. At each step those vertices which have not been active but have at least r≥2 active neighbors become active as well. We study the size A... of the final active set. The parameters of the model are, besides r (fixed) and n (tending to ...), the size a=a(n) of the initially active set and the probability p=p(n) of the edges in the graph. We show that the model exhibits a sharp phase transition: depending on the parameters of the model, the final size of activation with a high probability is either n-o(n) or it is o(n). We provide a complete description of the phase diagram on the space of the parameters of the model. In particular, we find the phase transition and compute the asymptotics (in probability) for A...; we also prove a central limit theorem for A... in some ranges. Furthermore, we provide the asymptotics for the number of steps until the process stops. (ProQuest: ... denotes formulae/symbols omitted.)