There are two main categories of the cross-section of networks and extremes. The first is extreme values observed at a network on say a 2-d Euclidean space. THe second is the extremes of certain characteristics of a network e.g. the degree distribution or other centrality measures. This presentation is on the latter.
The premise is like this. Simulate from a relatively simple preferential attachment (PA) model, and observe the bivariate extremal dependence between the in-degrees and out-degrees. You will observe very strong asymptotic dependence (or even complete dependence), which is different to that in empirical networks. This is an indication that the PA model is not adequate enough for real data. Some theoretical results are being explained in simpler terms in the presentation.