Title: Universality of scaling limits of critical random graphs

Abstract

Over the last few years a wide array of random graph models have been postulated to understand properties of empirically observed networks. Most of these models come with a parameter $t$ (usually related to edge density) and a (model dependent) critical time $t_c$ which specifies when a giant component emerges. There is evidence to support that for a wide class of models, under moment conditions, the nature of this emergence is universal and looks like the classical Erdős-Rényi random graph, in that the extent of the range of the parameter in which phase transition occurs, but also of the sizes and structures of the large connected components within this range (when equipped with the graph distance) converge to random fractals related to Aldous' celebrated continuum random tree.

Our aim is to present and discuss a general program for proving such results. The results apply to a number of fundamental random graph models including the configuration model, inhomogeneous random graphs modulated via a finite kernel and bounded size rules.