AbstractWe consider diffusion in random graphs with given vertex degrees. Our diffusionmodel can be viewed as a variant of a cellular automaton growth process: assumethat each node can be in one of the two possible states, inactive or active. Theparameters of the model are two given functions θ : N → N and α : N → [0, 1].At the beginning of the process, each node v of degree dv becomes active withprobability α(dv ) independently of the other vertices. Presence of the active verticestriggers a percolation process: if a node v is active, it remains active forever. Andif it is inactive, it will become active when at least θ(dv) of its neighbors are active.In the case where α(d) = α and θ(d) = θ, for each d ∈ N, our diffusion model isequivalent to what is called bootstrap percolation. The main result of this paper isa theorem which enables us to find the final proportion of the active vertices...