Geometric graphs to model a world of connections

万维网的出现打开了我们的视野，让我们看到了信息的互联本质。我们不再期望在一个专业管理的图书馆中找到我们的事实，这个图书馆是以分级的方式组织的。取而代之的是，我们访问由许多独立参与者的集体行动建立的复杂信息网络。这类网络的一个有趣的方面是，它们的链接结构本身包含信息。要提取这些信息并使用它在网络中导航，我们需要了解链接是如何形成的。图论是数学的一个分支，它可以帮助研究、刻画和建模链接结构。 在数学术语中，图是一个网络，是顶点(或节点)的集合，其中一些顶点对通过一条边(或链接)连接在一起。图形是非常通用的对象，因此它们可以在各种应用程序中使用。传统上，顶点是匿名的、可互换的，而图论家感兴趣的是链接的结构。但在信息网络中，这种假设是有问题的。例如，在Web中，顶点与包含文本、图像和其他内容的网页相关联，并且在社交网络中，用户具有爱好、兴趣、地理位置和其他个人特征。我们可以通过在特征空间中嵌入顶点来适应图的概念，特征空间代表了关于顶点的底层信息。顶点位于空间中的图称为几何图。 我的研究将集中在几何图的研究以及它们在提高我们对信息网络的理解方面的应用。

The advent of the World Wide Web has opened our eyes to the connected nature of information. No longer do we expect to find our facts in a professionally managed library, organized in a hierarchical manner. Instead, we access complex information networks, built by the collective actions of many independent participants. An interesting aspect of such networks is that their link structure itself contains information. To extract this information and use it to navigate the network, we need to understand how the links are formed. Graph theory is a branch of mathematics that can help to study, characterize and model the link structure. In mathematical terms, a graph is a network, a collection of vertices (or nodes), where some pairs of vertices are joined by an edge (or link). Graphs are very general objects, and as a result they can be used in a variety of applications. Traditionally, vertices are anonymous, interchangeable, and what graph theorists are interested in is the structure of the links. But in information networks, this assumption is problematic. In the Web, for example, a vertex is associated with a Web page, which contains text, images and other content, and in social networks, a user has hobbies, interests, a geographical location, and other personal characteristics. We can adapt the concept of a graph by embedding the vertices in a feature space, which represents the underlying information about the vertices. Graphs where the vertices are located in a space are called geometric graphs. My research will focus on the study of geometric graphs and their use in improving our understanding of information networks.