Improved bounds for polylogarithmic graph distances in scale-free percolation and related models

TL;DR

This paper improves the understanding of graph distances in scale-free percolation and related models by providing new lower bounds that match upper bounds in certain regimes, simplifying proofs and extending to first passage percolation.

Contribution

It offers the first matching lower bounds for a larger parameter range, improves previous bounds, and introduces a simpler proof technique that extends to other models.

Findings

Matching bounds for a larger parameter range

Tail bounds on short path probabilities

Shape theorems and tight bounds for neighborhood sizes

Abstract

In this paper, we study graph distances in the geometric random graph models scale-free percolation SFP, geometric inhomogeneous random graphs GIRG, and hyperbolic random graphs HRG. Despite the wide success of the models, the parameter regime in which graph distances are polylogarithmic is poorly understood. We provide new and improved lower bounds. In a certain portion of the parameter regime, those match the known upper bounds. Compared to the best previous lower bounds by Hao and Heydenreich, our result has several advantages: it gives matching bounds for a larger range of parameters, thus settling the question for a larger portion of the parameter space. It strictly improves the lower bounds by Hao and Heydenreich for all parameters settings in which those bounds were not tight. It gives tail bounds on the probability of having short paths, which imply shape theorems for the $k$-neighbourhood of a vertex whenever our lower bounds are tight, and tight bounds for the size of this $k$-neighbourhood. And last but not least, our proof is much simpler and not much longer than two pages, and we demonstrate that it generalizes well by showing that the same technique also works for first passage percolation.