Random Processes and Large Networks

本研究项目是对大型网络中相互作用的数学模型的基础理论研究。人们已经广泛研究了许多不同的模型，例如，通过社会网络进行的人类交互，这涉及到对网络结构和交互规则进行建模。研究这种概率模型的数学基础旨在促进技术在不同研究群体中的传播，并为大型网络的设计、分析和实证调查提供额外的工具。另外，通过强调从现成的真实世界数据中产生的新的数学研究问题，希望能激发本科生和研究生追求新的研究途径的兴趣。其中一个技术方面涉及研究有限离散随机结构极限的紧凑化方法。在图作为稠密图的极限的情况下，这一点得到了很好的研究；本项目继续研究有限可逆马氏链的极限，并开始研究任意有限赋权图的首次通过渗流的极限。另一个理论方面涉及进一步研究尺度不变的随机空间网络及其通过动态邻近图的构造。从现成的真实世界数据中产生的新的研究问题涉及空间队列(以机场安检线为例)、采样点上的道路网络拓扑以及人们可以实时观察的在线游戏的纳什均衡。

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.