Probabilistic and Extremal Combinatorics

随机几何图可以用于对许多大规模网络进行建模，例如互联网和社交网络。 这样的图特别适合于大规模传感器和收发器网络的建模，随着电子设备变得更小、更便宜并且在非常大的网络中互连，这种网络变得越来越普遍。对这些网络的行为进行建模变得越来越重要，并且当这些网络变得非常大时，对这些网络的行为的分析在实际应用中变得越来越重要。该奖项支持对这些大规模网络特性的研究，以及对研究生和早期职业研究人员进行随机图数学方面的培训。随机几何图形的研究起源于流体在多孔介质中渗流的问题。最近，对大规模电子和通信网络的研究引发了许多关于随机几何图的问题。随机几何图的基本模型是由吉尔伯特在50多年前提出的：根据单位强度的泊松点过程在平面上随机取点，只要两个点之间的距离在规定的范围内，就将它们连接起来。关于这个模型的中心问题是：对于什么样的规定距离值，我们得到一个无限连通分量？令人惊讶的是，即使在50年后，对于规定距离的临界值，也只有粗略的上限和下限。这个吉尔伯特模型的一些性质是已知的，但许多其他问题仍然没有答案。本研究项目解决了其中的一些问题，以及受逾渗理论和大规模通信网络启发的随机图相关模型的其他问题。

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.