Geometric graphs to model a world of connections

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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.