# Bootstrap Percolation in Random Geometric Graphs

#### Abstract

Bootstrap Percolation, sometimes used to model the spread of disease, is a dynamic process on a graph in which a vertex becomes infected if it has too many edges to infected vertices. For bootstrap percolation in a random geometric graph, the size of the initially infected set can determine whether or not the system will percolate. If we choose the initially infected set to be too small, the system will not percolate. Likewise, if we choose our initially infected set to be large enough, then the system will percolate with high probability. We explore just how large our initially infected set should be in order for it to percolate. Furthermore, we explore the local resilience of the graph with respect to bootstrap percolation and with respect to connectivity. We discover that we have a series of “bad configurations” that we want to avoid, or else our system will percolate, and we test how many edges we can remove from our graph in order to avoid these bad configurations. For the local resilience of the connectivity property, we looked into how many edges we could delete from the graph while keeping the graph connected.

*This paper has been withdrawn.*

Bootstrap Percolation in Random Geometric Graphs

Bootstrap Percolation, sometimes used to model the spread of disease, is a dynamic process on a graph in which a vertex becomes infected if it has too many edges to infected vertices. For bootstrap percolation in a random geometric graph, the size of the initially infected set can determine whether or not the system will percolate. If we choose the initially infected set to be too small, the system will not percolate. Likewise, if we choose our initially infected set to be large enough, then the system will percolate with high probability. We explore just how large our initially infected set should be in order for it to percolate. Furthermore, we explore the local resilience of the graph with respect to bootstrap percolation and with respect to connectivity. We discover that we have a series of “bad configurations” that we want to avoid, or else our system will percolate, and we test how many edges we can remove from our graph in order to avoid these bad configurations. For the local resilience of the connectivity property, we looked into how many edges we could delete from the graph while keeping the graph connected.