Non-convergence of proportions of types in a preferential attachment graph with three co-existing types

TL;DR

This paper presents a counterexample in a multi-type preferential attachment model showing non-convergence of type proportions, and also identifies conditions where convergence does occur, challenging previous conjectures.

Contribution

It provides the first known counterexample to the conjecture of convergence in a three-type preferential attachment model, and explores conditions for convergence.

Findings

Counterexample with three types showing non-convergence

Conditions under which convergence of proportions holds

Challenging previous conjectures on type proportions

Abstract

We consider the preferential attachment model with multiple vertex types introduced by Antunovi\'c, Mossel and R\'acz. We give an example with three types, based on the game of rock-paper-scissors, where the proportions of vertices of the different types almost surely do not converge to a limit, giving a counterexample to a conjecture of Antunovi\'c, Mossel and R\'acz. We also consider another family of examples where we show that the conjecture does hold.