BOOTSTRAP PERCOLATION ON THE RANDOM GRAPH Gn,p

作者：Janson Svante^*; Luczak Tomasz; Turova Tatyana; Vallier Thomas

来源：Annals of Applied Probability, 2012, 22(5): 1989-2047.

DOI：10.1214/11-AAP822

• Bootstrap percolation
• random graph
• sharp threshold

Bootstrap percolation on the random graph C-n,C-p is a process of spread of %26quot;activation%26quot; 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 %26gt;= 2 active neighbors become active as well. %26lt;br%26gt;We study the size A* of the final active set. The parameters of the model are, besides r (fixed) and n (tending to infinity), 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.

• 出版日期2012-10