In this talk we will discuss two different approaches to proving that a sequence of (random) graphs converges to some limit object.
Firstly, we will introduce a class of functions known as graphons, which can be thought of as uncountable graphs. We will discuss how to define a sensible probabilitistic notion of graph convergence, and how to utilize the link between graphs and graphons to translate this notion to the somewhat “nicer” space of graphons in order to prove that a sequence of graphs converges. Secondly, we will point towards limitations in the above approach and discuss how the Gromov-Hausdorff-Prokhorov metric can be a useful alternative, limiting the discussion to sequences of trees. If time permits we will end the talk by introducing some concrete examples of convergent sequences of random graphs. The talk is meant to be introductory and will not rely on any previous knowledge of graphs or graph convergence.