Critical percolation on scale-free random graphs I: New universality class for the configuration model

Souvik Dhara, Remco van der Hofstad, Johan S. H. van Leeuwaarden

Published 2019-09-12Version 1

In this paper, we study the critical behavior of percolation on a configuration model with degree distribution satisfying an infinite second-moment condition, which includes power-law degrees with exponent $\tau \in (2,3)$. It is well known that, in this regime, many canonical random graph models, such as the configuration model, are \emph{robust} in the sense that the giant component is not destroyed when the percolation probability stays bounded away from zero. Thus, the critical behavior is observed when the percolation probability tends to zero with the network size, despite of the fact that the average degree remains bounded. In this paper, we initiate the study of critical random graphs in the infinite second-moment regime by identifying the critical window for the configuration model. We prove scaling limits for component sizes and surplus edges, and show that the diameter of the largest critical components is tight, which contrasts with the previous universality classes arising in the literature. This introduces a third and novel universality class for the critical behavior of percolation on random networks, that is not covered by the multiplicative coalescent framework due to Aldous and Limic \cite{AL98}. We also prove concentration of the component sizes outside the critical window, and that a unique, complex \emph{giant component} emerges after the critical window. This completes the picture for the percolation phase transition on the configuration model.