Critical random graphs and the differential equations technique

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 that 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 the sense of the critical scaling window and (a) the sizes of the components in this window (all maximal component sizes scaling like n2/3) and (b) the structure of components (rescaled by n−1/3) converge to random fractals related to the continuum random tree. The aim of this note is to give a non-technical overview of recent breakthroughs in this area, emphasizing a particular tool in proving such results called the differential equations technique first developed and used extensively in probabilistic combinatorics in the work of Wormald [52, 53] and developed in the context of critical random graphs by the authors and their collaborators in [10–12].