Barely Supercritical Percolation on Poissonian Scale-free Networks

We study the giant component problem slightly above the critical regime for percolation on Poissonian random graphs in the scale -free regime, where the vertex weights and degrees have a diverging second moment. Critical percolation on scale -free random graphs has been observed to have incredibly subtle features that are markedly different compared to those in random graphs with a converging second moment. In particular, the critical window for percolation depends sensitively on whether we consider single- or multi -edge versions of the Poissonian random graph. In this paper, and together with our companion paper [3], we build a bridge between these two cases. Our results characterize the part of the barely supercritical regime where the size of the giant components are approximately same for the single- and multi -edge settings. The methods for establishing concentration of giant for the single- and multi -edge versions are quite different. While the analysis in the multi -edge case is based on scaling limits of exploration processes, the single -edge setting requires identification of a core structure inside certain high -degree vertices that forms the giant component.