# Weighted distances in scale-free preferential attachment models

## TL;DR

This paper investigates weighted distances in scale-free preferential attachment networks with infinite variance degree distributions, identifying two universality classes affecting typical weighted distances and hopcounts.

## Contribution

It introduces a classification of weight distributions into explosive and conservative classes, with explicit asymptotic behaviors for weighted distances and hopcounts.

## Key findings

- In the explosive class, weighted distances converge to a sum of two i.i.d. variables.
- In the conservative class, weighted distances tend to infinity with explicit growth rates.
- The paper characterizes fluctuations around the main growth term under mild assumptions.

## Abstract

We study three preferential attachment models where the parameters are such that the asymptotic degree distribution has infinite variance. Every edge is equipped with a non-negative i.i.d. weight. We study the weighted distance between two vertices chosen uniformly at random, the typical weighted distance, and the number of edges on this path, the typical hopcount. We prove that there are precisely two universality classes of weight distributions, called the explosive and conservative class. In the explosive class, we show that the typical weighted distance converges in distribution to the sum of two i.i.d. finite random variables. In the conservative class, we prove that the typical weighted distance tends to infinity, and we give an explicit expression for the main growth term, as well as for the hopcount. Under a mild assumption on the weight distribution the fluctuations around the main term are tight.