BOOTSTRAP PERCOLATION ON RANDOM GEOMETRIC GRAPHS

Abstract

Bootstrap percolation (BP) has been used effectively to model phenomena as diverse as emergence of magnetism in materials, spread of infection, diffusion of software viruses in computer networks, adoption of new technologies, and emergence of collective action and cultural fads in human societies. It is defined on an (arbitrary) network of interacting agents whose state is determined by the state of their neighbors according to a threshold rule. In a typical setting, BP starts by random and independent “activation” of nodes with a fixed probabilityp, followed by a deterministic process for additional activations based on the density of active nodes in each neighborhood (θ activated nodes). Here, we study BP on random geometric graphs (RGGs) in the regime when the latter are (almost surely) connected. Random geometric graphs provide an appropriate model in settings where the neighborhood structure of each node is determined by geographical distance, as in wirelessad hocand sensor networks as well as in contagion. We derive boundspc′,pc″ on the critical thresholds such that for allp > p″cfull percolation takes place, whereas forp < p′cit does not. We conclude with simulations that compare numerical thresholds with those obtained analytically.