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

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 Erdos-Renyi 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 \erdos 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 in the barely subcritical regime are the same as the Erdos-Renyi 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 Erdos-Renyi random graph. As a by product we also get the first known results for component sizes at criticality for a general class of inhomogeneous random graphs. This is joint work with Xuan Wang, Sanchayan Sen and Nicolas Broutin.

**Category**: Probability**Duration**: 01:34:51**Date**: January 15, 2015 at 4:25 PM**Views**: 103-
**Tags:**seminar, Probability Seminar

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