Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erd\H{o}s-R\'enyi random graph

Understand

A wide array of random graph models have been postulated to understand properties of observed networks.

• Typically these models have a parameter $t$ and a critical time $t_c$ when a giant component emerges.
• It is conjectured that for a large class of models, the nature of this emergence is similar to that of the Erd\H{o}s-R\'enyi random graph, in the sense that (a) the sizes of the maximal components in the critical regime scale like $n^{2/3}$, and (b) the structure of the maximal components at criticality (rescaled by $n^{-1/3}$) converges to random fractals.
• To date, (a) has been proven for a number of models using different techniques.