Barely supercritical percolation on Poissonian scale-free networks

TL;DR

This paper investigates the behavior of the giant component in barely supercritical percolation on Poissonian scale-free networks, revealing nuanced differences between single- and multi-edge models and establishing conditions for their similarity.

Contribution

It characterizes the barely supercritical regime where giant component sizes are similar in single- and multi-edge Poissonian scale-free networks, bridging the gap between these models.

Findings

Giant component sizes are approximately the same in certain supercritical regimes for both models.

Different analytical methods are required for single- and multi-edge cases.

The study highlights the sensitivity of critical percolation features to network edge configurations.

Abstract

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 have been observed to have incredibly subtle features that are markedly different compared to those in random graphs with 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 with Bhamidi, 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.