Scaling limits of random graph models at criticality: Universality and   the basin of attraction of the Erd\H{o}s-R\'enyi random graph
  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. This paper develops a general program for proving (b) that requires three ingredients: (i) in the critical scaling window, components merge approximately like the multiplicative coalescent, (ii) scaling exponents of susceptibility functions are the same as that of the Erd\H{o}s-R\'enyi random graph, and (iii) macroscopic averaging of distances between vertices in the barely subcritical regime. We show that these apply to two fundamental random graph models: the configuration model and inhomogeneous random graphs with a finite ground space. For these models, we also obtain new results for component sizes at criticality and structural properties in the barely subcritical regime. 
Shankar Bhamidi, Nicolas Broutin, Sanchayan Sen, Xuan Wang
math.PR math.CO