High degree vertices in the Power of Choice model combined with Preferential Attachment

TL;DR

This paper analyzes the asymptotic behavior of the top degrees in a preferential attachment model with local choice, revealing a sublinear power-law decay for the top k degrees, contrasting with standard models.

Contribution

It provides the first asymptotic analysis of the top degrees in a combined local choice and preferential attachment model, showing a sublinear power-law behavior.

Findings

Top k degrees exhibit sublinear power-law decay.

The model has a persistent hub with linear maximum degree.

Contrasts with standard preferential attachment where all top degrees are similarly sublinear.

Abstract

We find assimpotics for the first $k$ highest degrees of the degree distribution in an evolving tree model combining the local choice and the preferential attachment. In the considered model, the random graph is constructd in the following way. At each step, a new vertex is introduced. Then, we connect it with one (the vertex with the largest degree is chosen) of $d$ ($d>2$) possible neighbors, which are sampled from the set of the existing vertices with the probability proportional to their degrees. It is known that the maximum of the degree distribution in this model has linear behavior. We prove that $k$-th highest dergee has a sublinear behavior with a power depends on $d$. This contrasts sharply with what is seen in the preferential attachment model without choice, where all highest degrees in the degree distribution has the same sublinear order. The proof is based on showing that the considered tree has a persistent hub by comparison with the standard preferential attachment model, along with martingale arguments.