In my talk, I will describe some recent work on a model of complex networks that was proposed recently by Krioukov et al.
Random geometric graphs are obtained by taking a random set of points in the plane (usually either generated by a Poisson process or sampled i.i.d.~uniformly from a large disk or square), and then joining any pair of points by an edge whose distance is less than some parameter r>0.
The subject essentially goes back to the work of E.N.~Gilbert, who defined a very similar model in 1961. They have since become the focus of considerable research effort and very precise answers are now known to many of the natural questions for this model.

A natural question is what happens if we define this model not on the ordinary euclidean plane but instead on the hyperbolic plane. Perhaps quite surprisingly, it turns out that for some choices of the parameters hyperbolic random geometric graphs display features that are usually associated with so-called complex networks. Famously, around the turn of the century it was noted that many ``real'' networks stemming from diverse fields (chemical reaction networks, the internet and social networks) share certain characteristics.
Ordinary, euclidean random geometric graphs are very far from showing these characteristics, but recently Krioukov-Papadopoulos-Kitsak-Vahdat-Boguna observed that there is a way to define random geometric graphs on the hyperbolic plane such that they display these characteristics.
In two joint works with Michel Bode and Nikolaos Fountoulakis we studied the largest component and the transition from a disconnected to a connected graph in this model, and in a recent work with my master student Staps we studied the graph diameter.
Our results are very different from the results for all other models in the literature as far as we are aware.