Probabilistic and Extremal Combinatorics

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Random geometric graphs can be used to model many large scale networks, such as the Internet and social networks. Such graphs are particularly suited to the modeling of large-scale sensor and transceiver networks, which are becoming more common as electronic devices become smaller and cheaper and are interconnected in very large networks. Modeling the behavior of these networks is becoming more and more important, and the analysis of the behavior of these networks when they become extremely large is becoming increasingly relevant in practical applications. This award supports research on the properties of these large-scale networks, as well as training of graduate students and early-career researchers in the mathematics of random graphs. The study of random geometric graphs originated with questions about the way fluids seep through porous media. More recently, the study of large-scale electronic and communication networks has prompted many questions about random geometric graphs. The basic model of random geometric graphs was proposed by Gilbert over fifty years ago: take points randomly in the plane according to a Poisson point process of unit intensity, and join two whenever they are within a prescribed distance of each other. The central question concerning this model is: for what values of the prescribed distance do we obtain an infinite connected component? Surprisingly, even after fifty years, only rough upper and lower bounds are known for the critical value of the prescribed distance. Some properties of this Gilbert model are known, but many other questions still remain unanswered. This research project addresses some of these questions, as well as other questions about related models of random graph inspired by both percolation theory and large-scale communication networks.