Graph distances in scale-free percolation: the logarithmic case

TL;DR

This paper investigates the behavior of graph distances in a scale-free percolation model, focusing on the logarithmic regime and providing improved bounds and precise exponents for the relationship between graph and Euclidean distances.

Contribution

It offers new bounds on logarithmic exponents and identifies the correct exponent in the light tail regime for scale-free percolation graph distances.

Findings

Graph distances are (poly-)logarithmic in Euclidean distance in certain regimes.

Improved bounds on logarithmic exponents are established.

The exact exponent is identified in the light tail regime.

Abstract

Scale-free percolation is a stochastic model for complex networks. In this spatial random graph model, vertices $x,y\in\mathbb{Z}^d$ are linked by an edge with probability depending on i.i.d.\ vertex weights and the Euclidean distance $|x-y|$. Depending on the various parameters involved, we get a rich phase diagram. We study graph distances and compare it to the Euclidean distance of the vertices. Our main attention is on a regime where graph distances are (poly-)logarithmic in the Euclidean distance. We obtain improved bounds on the logarithmic exponents. In the light tail regime, the correct exponent is identified.