Structures in supercritical scale-free percolation

TL;DR

This paper investigates the properties of supercritical scale-free percolation on integer lattices, providing bounds on graph distances, characterizing recurrence and transience, and revealing hierarchical structures in infinite variance regimes.

Contribution

It offers new bounds on graph distances, characterizes recurrence and transience in low dimensions, and demonstrates hierarchical structures in infinite variance degree regimes.

Findings

Bounds for graph distance in infinite degree regime

Complete characterization of recurrence vs. transience in dimensions 1 and 2

Existence of hierarchical structures with infinite variance degrees

Abstract

Scale-free percolation is a percolation model on $\mathbb{Z}^d$ which can be used to model real-world networks. We prove bounds for the graph distance in the regime where vertices have infinite degrees. We fully characterize transience vs. recurrence for dimension 1 and 2 and give sufficient conditions for transience in dimension 3 and higher. Finally, we show the existence of a hierarchical structure for parameters where vertices have degrees with infinite variance and obtain bounds on the cluster density.