Tauberian Theory for Multivariate Regularly Varying Distributions with Application to Preferential Attachment Networks

TL;DR

This paper develops a multivariate Tauberian theorem for regularly varying distributions and applies it to analyze the joint degree distribution in directed preferential attachment networks, revealing their heavy-tailed behavior.

Contribution

It introduces a multivariate Abel-Tauberian theorem for non-standard regularly varying measures and applies it to network degree distributions.

Findings

Joint in- and out-degree distribution exhibits regularly varying tails.

Provides a theoretical foundation for analyzing heavy-tailed network data.

Establishes a link between distribution tails and their transforms in multivariate settings.

Abstract

Abel-Tauberian theorems relate power law behavior of distributions and their transforms. We formulate and prove a multivariate version for non-standard regularly varying measures on $\mathbb{R}_+^p$ and then apply it to prove that the joint distribution of in- and out-degree in a directed edge preferential attachement model has jointly regularly varying tails.