Deletion of oldest edges in a preferential attachment graph

TL;DR

This paper studies a modified preferential attachment graph where oldest edges are periodically removed, revealing how degree distributions shift between power law and exponential tails depending on parameters.

Contribution

It introduces a new model with edge deletion in preferential attachment graphs and analyzes the resulting degree distribution and connectivity properties.

Findings

Degree of vertices is either zero or geometrically distributed.

Power law degree distribution emerges when p exceeds a threshold.

A unique giant component exists if and only if m ≥ 2.

Abstract

We consider a variation on the Barab\'asi-Albert random graph process with fixed parameters $m\in \mathbb{N}$ and $1/2 < p < 1$. With probability $p$ a vertex is added along with $m$ edges, randomly chosen proportional to vertex degrees. With probability $1 - p$, the oldest vertex still holding its original $m$ edges loses those edges. It is shown that the degree of any vertex either is zero or follows a geometric distribution. If $p$ is above a certain threshold, this leads to a power law for the degree sequence, while a smaller $p$ gives exponential tails. It is also shown that the graph contains a unique giant component whp if and only if $m\geq 2$.