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A wide array of random graph models have been postulated to understandproperties of observed networks. Typically these models have a parameter $t$and a critical time $t_c$ when a giant component emerges. It is conjecturedthat for a large class of models, the nature of this emergence is similar tothat of the Erd\H{o}s-R\'enyi random graph, in the sense that (a) the sizes ofthe maximal components in the critical regime scale like $n^{2/3}$, and (b) thestructure of the maximal components at criticality (rescaled by $n^{-1/3}$)converges to random fractals. To date, (a) has been proven for a number ofmodels using different techniques. This paper develops a general program forproving (b) that requires three ingredients: (i) in the critical scalingwindow, components merge approximately like the multiplicative coalescent, (ii)scaling exponents of susceptibility functions are the same as that of theErd\H{o}s-R\'enyi random graph, and (iii) macroscopic averaging of distancesbetween vertices in the barely subcritical regime. We show that these apply totwo fundamental random graph models: the configuration model and inhomogeneousrandom graphs with a finite ground space. For these models, we also obtain newresults for component sizes at criticality and structural properties in thebarely subcritical regime.

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