Bootstrap Percolation and Diffusion in Random Graphs with Given Vertex Degrees

Abstract

We consider diffusion in random graphs with given vertex degrees. Our diffusion model can be viewed as a variant of a cellular automaton growth process: assume that each node can be in one of the two possible states, inactive or active. The parameters of the model are two given functions $\theta: {\Bbb N} \rightarrow {\Bbb N}$ and $\alpha:{\Bbb N} \rightarrow [0,1]$. At the beginning of the process, each node $v$ of degree $d_v$ becomes active with probability $\alpha(d_v)$ independently of the other vertices. Presence of the active vertices triggers a percolation process: if a node $v$ is active, it remains active forever. And if it is inactive, it will become active when at least $\theta(d_v)$ of its neighbors are active. In the case where $\alpha(d) =\alpha$ and $\theta(d) =\theta$, for each $d \in {\Bbb N}$, our diffusion model is equivalent to what is called bootstrap percolation. The main result of this paper is a theorem which enables us to find the final proportion of the active vertices in the asymptotic case, i.e., when $n \rightarrow \infty$. This is done via analysis of the process on the multigraph counterpart of the graph model.