Bootstrap percolation and the geometry of complex networks

On a geometric model for complex networks (introduced by Krioukov et al.) we investigate the bootstrap percolation process. This model consists of random geometric graphs on the hyperbolic plane having $N$ vertices, a dependent version of the Chung-Lu model.The process starts with infection rate p=p(N). Each uninfected vertex with at least r infected neighbors becomes infected, remaining so forever.We identify a function p_c(N)=o(1) such that a.a.s. when p >> p_c(N) the infection spreads to a positive fraction of vertices, whereas when p << p_c(N) the process cannot evolve. Moreover, this behavior is ``robust'' under random deletions of edges.