Random Geometric Graphs

这个项目的目的是为了进一步研究随机几何图。这门学科是从应用问题发展而来的，最初的问题是关于流体如何渗入多孔介质。最近，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Gilbert在五十多年前提出的：根据强度为1的Poisson点过程在平面上随机取点，当两点之间的距离在r以内时，将两个点连接起来。吉尔伯特问的主要问题是：对于r的什么值，我们可以得到无限连通分量？令人惊讶的是，即使在50年后，关于r的临界值，只有相当糟糕的上下界是已知的。这个Gilbert模型的一些性质是已知的，但许多其他问题仍然没有答案。在渗流理论和大规模通信网络的启发下，随机几何图可以用来对现实生活中存在的许多大规模网络进行建模，特别适合于大规模传感器网络和收发网络的建模。随着电子设备变得更小、更便宜，现在拥有非常大的互联设备网络并不少见。对这些网络的行为进行建模变得越来越重要，当这些网络变得非常庞大时，对其行为的分析在现实世界的应用中变得越来越重要。所提出的研究将有助于理解这些大规模网络的性质。

The aim of this project is to further the study of random geometric graphs. This subject has grown out of applied problems, with the first questions about the way fluids seep through porous media. More recently, many questions about random geometric graphs have been prompted by the study of large scale electronic and communication networks. 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 intensity 1, and join two whenever they are within a distance r of each other. The main question Gilbert asked is the following: for what values of r do we obtain an infinite connected component? Surprisingly, even after fifty years, only rather poor upper and lower bounds are known for the critical value of r. Some properties of this Gilbert model are known, but many other questions still remain unanswered. The proposed project considers some of these questions, and many other questions about related models of random graph inspired by both percolation theory and large scale communication networks.Random geometric graphs can be used to model many large scale networks that exist in real life, such as the internet, social networks, and are particularly suited to the modeling of large scale sensor and transceiver networks. As electronic devices become smaller and cheaper, it is now not uncommon to have very large networks of interconnected devices. 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 real world applications. The research proposed will aid the understanding of the properties of these large scale networks.