Barely Supercritical Percolation on Poissonian Scale-free Networks

Souvik Dhara, R. van der Hofstad

2024, v.30, Issue 1, 27-55

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 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.

doi:10.61102/1024-2953-mprf.2024.30.1.001

Keywords: percolation, giant component, scale-free, inhomogeneous random graphs

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