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 NN vertices, a dependent version of the Chung–Lu model. The process starts with infection rate p=p(N)p=p(N). Each uninfected vertex with at least r≥1 infected neighbors becomes infected, remaining so forever. We identify a function pc(N)=o(1)pc(N)=o(1) such that a.a.s. when p≫pc(N)p≫pc(N) the infection spreads to a positive fraction of vertices, whereas when p≪pc(N)p≪pc(N) the process cannot evolve. Moreover, this behavior is “robust” under random deletions of edges.