#### Title of Abstract

Bootstrap Percolation in Random Geometric Graphs

#### Faculty Mentor

One WU mentor: Arran Hamm, Ph.D. hamma@winthrop.edu

#### College

College of Arts and Sciences

#### Department

Mathematics

#### Faculty Mentor

Arran Hamm, Ph.D.

#### 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.

#### Additional Fields About Your Abstract

Please check this if you understand.

#### Grant Support

This project is supported by an Institutional DevelopmentAward (IDeA) from the National Institute of General MedicalSciences (2 P20 GM103499) from the National Institutes of Health.

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.