Condensation phenomena in preferential attachment trees with neighbourhood influence

TL;DR

This paper introduces a new preferential attachment tree model where vertex evolution depends on individual and neighbor weights, revealing a condensation phenomenon and power-law degree distribution, generalizing previous models.

Contribution

It presents a novel model incorporating neighborhood influence in preferential attachment trees, extending existing results and analyzing condensation and degree distribution behaviors.

Findings

Condensation phenomenon occurs under certain conditions.

Degree distribution converges to a power-law-like distribution.

Almost-sure convergence of key quantities in absence of condensation.

Abstract

We introduce a model of evolving preferential attachment trees where vertices are assigned weights, and the evolution of a vertex depends not only on its own weight, but also on the weights of its neighbours. We study the distribution of edges with endpoints having certain weights, and the distribution of degrees of vertices having a given weight. We show that the former exhibits a condensation phenomenon under a certain critical condition, whereas the latter converges almost surely to a distribution that resembles a power law distribution. Moreover, in the absence of condensation, we prove almost-sure setwise convergence of the related quantities. This generalises existing results on the Bianconi-Barab\'{a}si tree as well as on an evolving tree model introduced by the second author.