Bootstrap percolation with individual thresholds

Bootstrap percolation is a random process that models the spread of activation on a linked structure. Starting with an initial set of active vertices, in each round all inactive vertices with at least r active neighbors become and remain active until there are no inactive vertices with enough active neighbors left. Due to its numerous applications in modeling of physical processes, bootstrap percolation has been intensively studied on various (deterministic and random) graph models. The thereby recurringly encountered sharp threshold phenomenon states that there exists a percolation threshold such that we typically either have percolation or no percolation, depending on whether the active starting set lies above or below this value. That is, with high probability either the activation spreads to almost the whole structure, or the process ceases with almost no additionally activated vertices. The main goal of this thesis is to extend the current results for r-neighbor bootstrap percolation to a model where each vertex i draws its individual activation threshold ri from a given distribution whose support is assumed to be independent from the number of vertices. This enhancement is mainly motivated by real-world applications, as for example neural networks and viral marketing. We prove the existence of a sharp threshold for an underlying ErdősRényi random graph model and provide estimates for the final set size and the time until percolation. Moreover, we investigate the bootstrap percolation process on a directed random graph model with arbitrary degree distribution. Under certain assumptions about the distribution of the out-degrees we can show a similar sharp threshold result. Namely, starting below a certain percolation threshold results with high probability in no percolation, whereas starting above typically leads to a final active set of linear size in the number of vertices.