We study limits of the largest connected components (viewed as metric spaces) obtained by critical percolation on uniformly chosen graphs and configuration models with heavy-tailed degrees. For rank-one inhomogeneous random graphs, such results were derived by Bhamidi, van der Hofstad, Sen (2018) . We develop general principles under which the identical scaling limits as in  can be obtained. Of independent interest, we derive refined asymptotics for various susceptibility functions and the maximal diameter in the barely subcritical regime.
"Universality for critical heavy-tailed network models: Metric structure of maximal components." Electron. J. Probab. 25 1 - 57, 2020. https://doi.org/10.1214/19-EJP408