Random Processes and Large Networks

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

This research project is a foundational theoretical study of mathematical models of interactions within large networks. There has been extensive study of many different models of, for instance, human interaction via social networks, which involves modeling both the network structure and the interaction rules. Studying the mathematical underpinnings of such probability models is intended to facilitate the dissemination of techniques among different research communities and provide additional tools for the design, analysis, and empirical investigation of large networks. Separately, by emphasizing novel mathematical research problems arising from readily available real-world data, it is hoped to stimulate interest amongst undergraduate and graduate students in pursuing novel research paths. One technical aspect concerns the compactification methodology for studying limits of finite discrete random structures. This is well studied in the case of graphons as limits of dense graphs; this project continues the study of limits of finite reversible Markov chains and initiates study of limits of first passage percolation on arbitrary finite weighted graphs. Another theoretical aspect concerns further study of scale-invariant random spatial networks and their construction via dynamic proximity graphs. Novel research problems arising from readily available real-world data concern spatial queues (exemplified by airport security lines), the topology of road networks on sampled points, and Nash equilibria for an online game one can observe in real time.