Multivariate Regular Variation of Preferential Attachment Models

TL;DR

This paper applies multivariate regular variation to analyze the extremal behavior of preferential attachment models, revealing dependence structures and convergence properties of node degrees in heavy-tailed networks.

Contribution

It introduces a novel framework combining regular variation, martingale methods, and Pólya urn models to study extremal dependence in preferential attachment networks.

Findings

Edge counts are multivariate regularly varying.

Explicit dependence structure via Dirichlet distributions.

New convergence results for node degrees as network grows.

Abstract

We use the framework of multivariate regular variation to analyse the extremal behaviour of preferential attachment models. To this end, we follow a directed linear preferential attachment model for a random, heavy-tailed number of steps in time and treat the incoming edge count of all existing nodes as a random vector of random length. By combining martingale properties, moment bounds and a Breiman type theorem we show that the resulting quantity is multivariate regularly varying, both as a vector of fixed length formed by the edge counts of a finite number of oldest nodes, and also as a vector of random length viewed in sequence space. A P\'{o}lya urn representation allows us to explicitly describe the extremal dependence between the degrees with the help of Dirichlet distributions. As a by-product of our analysis we establish new results for almost sure convergence of the edge counts in sequence space as the number of nodes goes to infinity.