# Weighted distances in scale-free preferential attachment models

arXiv: https://arxiv.org/abs/1907.01907 · HTML: https://mlsanity.com/paper/1907.01907/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 3 Jul 2019.

## Built on by (most cited first, 1 of 1)

- [A network community detection method with integration of data from multiple layers and node attributes](https://mlsanity.com/paper/2305.13012/) (2023)

## Abstract (arXiv)

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.