Geometric preferential attachment in non-uniform metric spaces

TL;DR

This paper studies how degree sequences in geometric preferential attachment graphs behave in various metric spaces, showing convergence to power-law distributions influenced by the space's properties.

Contribution

It extends preferential attachment models to general metric spaces, analyzing degree distribution convergence and power-law behavior in finite and infinite settings.

Findings

Degree sequences in finite spaces converge to power-law distributions.

In infinite spaces, degree distributions in certain subsets also exhibit power-law tails.

Behavior depends on the attractiveness function and the metric space's properties.

Abstract

We investigate the degree sequences of geometric preferential attachment graphs in general compact metric spaces. We show that, under certain conditions on the attractiveness function, the behaviour of the degree sequence is similar to that of the preferential attachment with multiplicative fitness models investigated by Borgs et al. When the metric space is finite, the degree distribution at each point of the space converges to a degree distribution which is an asymptotic power law whose index depends on the chosen point. For infinite metric spaces, we can show that for vertices in a Borel subset of $S$ of positive measure the degree distribution converges to a distribution whose tail is close to that of a power law whose index again depends on the set.