Universality for critical heavy-tailed network models: metric structure of maximal components

The aim of this paper is to understand general universality principles for random network models whose component sizes in the critical regime lie in the multiplicative coalescent universality class but with heavy tails resulting in hubs. For the multiplicative coalescent in this regime, limit (random) metric spaces via appropriate tilts of inhomogeneous continuum random trees were derived by Bhamidi et al. (2015). In this paper we derive sufficient uniform asymptotic negligibility conditions for general network models to satisfy in the barely subcritical regime such that, if the model can be appropriately coupled to a multiplicative coalescent as one transitions from the barely subcritical regime through the critical scaling window, then the maximal components belong to the same universality class as in Bhamidi et al. (2015). As a canonical example, we study critical percolation on configuration models with heavy-tailed degrees. Of independent interest, we derive refined asymptotics for various susceptibility functions and maximal diameter in the barely subcritical regime. These estimates, coupled with the universality result, allow us to derive the asymptotic metric structure of the large components through the critical scaling window for percolation.