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

TL;DR

This paper establishes a general framework for proving the universality of the critical component structure in various random graph models, extending known results from Erdős-Rényi graphs to more complex models.

Contribution

It develops a unified approach to prove the convergence of critical components to fractal structures in broad classes of random graphs, including configuration and inhomogeneous models.

Findings

Proved component size scaling as n^{2/3} in new models.

Established structural convergence to fractals at criticality.

Derived new critical and subcritical regime properties for specific models.

Abstract

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.