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# Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erdős-Rényi random graph

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 the sense of the critical scaling window and (a) the sizes of the components in this window (all maximal component sizes scaling like $n^{2/3}$) and (b) the structure of components (rescaled by $n^{−1/3}$) converge to random fractals related to the continuum random tree. Till date, (a) has been proven for a number of models using different techniques while (b) has been proven for only two models, the classical Erdős-Rényi random graph and the rank-1 inhomogeneous random graph. The aim of this paper is to develop a general program for proving such results. The program requires three main ingredients: (i) in the critical scaling window, components merge approximately like the multiplicative coalescent (ii) scaling exponents of susceptibility functions are the same as the Erdős-Rényi random graph and (iii) macroscopic averaging of expected distances between random points in the same component in the barely subcritical regime. We show that these 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. Thus these models all belong to the domain of attraction of the classical Erdős-Rényi random graph. As a by product we also get results for component sizes at criticality for a general class of inhomogeneous random graphs.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.inria.fr/hal-01092563
Contributor : Nicolas Broutin Connect in order to contact the contributor
Submitted on : Tuesday, December 9, 2014 - 7:28:53 AM
Last modification on : Friday, November 18, 2022 - 9:27:42 AM

### Identifiers

• HAL Id : hal-01092563, version 1
• ARXIV : 1411.3417

### Citation

Shankar Bhamidi, Nicolas Broutin, Sanchayan Sen, Xuan Wang. Scaling limits of random graph models at criticality: Universality and the basin of attraction of the Erdős-Rényi random graph. {date}. ⟨hal-01092563⟩

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