Logarithmic typical distances in preferential attachment models

TL;DR

This paper establishes that typical distances in a preferential attachment network grow logarithmically with the network size, with precise asymptotics derived using advanced probabilistic and spectral methods.

Contribution

It provides a rigorous proof of the logarithmic typical distances in preferential attachment models with positive fitness, introducing new spectral convergence techniques.

Findings

Typical distances are close to _ u(n)

Distance growth rate depends on the exponential growth parameter 

Uses novel spectral radius convergence proof

Abstract

We prove that the typical distances in a preferential attachment model with out-degree $m\geq 2$ and strictly positive fitness parameter are close to $\log_\nu{n}$, where $\nu$ is the exponential growth parameter of the local limit of the preferential attachment model. The proof relies on a path-counting technique, the first- and second-moment methods, as well as a novel proof of the convergence of the spectral radius of the offspring operator under a certain truncation.