Graphs and complex networks

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The last decade has witnessed a great deal of research on modelling real-world complex networks such as the link structure of the web. Graphs, consisting of vertices and edges between them, make natural models which simulate and predict properties of complex networks. My goals are to devise and analyze new realistic models for complex networks, using tools from geometry, infinite graphs, and probability theory. On-line social networks (where nodes represent people and edges correspond to social interactions) such as Facebook and Twitter, are an important type of complex network. Such networks contain many dense subgraphs representing tight-knit communities. I will propose new evolutionary models for on-line social networks, as an approach to classifying their community structure. Applications of graph models may lead, in the case of the web, to more efficient search engines; in on-line social networks, they may help map the spread of social influence.