Mathematical Models and Methods for Behavior in Social Networks: Urban Crime, Self-Exciting Interactions, and Information Spread

We present three applications of mathematics to problems in the study of social networks. The problems are different enough that each requires its own model and set of methods. We investigate these models through theory and simulation. The first application is to urban crime and police response. We prove some stability results for an agent-based model and present an efficient algorithm for numerical simulation. The second application is to communication patterns in social networks. Adopting a point-process model to reflect observed temporal bursts in communication, we discuss parameter estimation for such models. We then consider the problem of filling in missing data in records of communication, formulating it as a variational problem and developing a numerical method to solve it. The third application is to the spread of information within social networks. We introduce a simple model, provide some theoretical results, and discuss results of simulations.