A counter-example to persistence in generalised preferential attachment trees

TL;DR

This paper presents a counter-example to a conjecture about persistent hubs in generalized preferential attachment trees, showing that the conjecture does not hold under certain conditions.

Contribution

It provides a specific counter-example disproving a previous conjecture about the existence of persistent hubs in the model.

Findings

Counter-example disproves the conjecture about persistent hubs.

The counter-example is a modification of a known counter-example by Galganov and Ilienko.

The result clarifies limitations of the conjecture under the given conditions.

Abstract

Consider a generalised preferential attachment tree with attachment function $f$, that is a random tree, where at each time-step a node connects to an existing node $v$ with probability proportional to $f(\mathrm{deg}(v))$, where $\mathrm{deg}(v)$ denotes the degree of the node in the existing tree. We provide a counter-example to a conjecture of the author asserting that under the assumption $\sum_{j=1}^{\infty} \frac{1}{f(j)^2} < \infty$ there is a persistent hub in the model, that is, a single node that has the maximal degree for all but finitely many time-steps. The counter-example is a minor modification of a related counter-example due to Galganov and Ilienko.