Fragility of non-convergence in preferential attachment graphs with three types

TL;DR

This paper studies the behavior of type proportions in preferential attachment graphs with three types, revealing that certain models with similar rules to non-converging cases actually do converge, contrary to expectations.

Contribution

It introduces two new models with three types in preferential attachment graphs that, despite similarities to non-converging models, demonstrate convergence of type proportions.

Findings

Two new models with three types converge over time.

Small random perturbations do not prevent convergence.

Contrasts with previous non-converging rock-paper-scissors model.

Abstract

Preferential attachment networks are a type of random network where new nodes are connected to existing ones at random, and are more likely to connect to those that already have many connections. We investigate further a family of models introduced by Antunovi\'{c}, Mossel and R\'{a}cz where each vertex in a preferential attachment graph is assigned a type, based on the types of its neighbours. Instances of this type of process where the proportions of each type present do not converge over time seem to be rare. Previous work found that a "rock-paper-scissors" setup where each new node's type was determined by a rock-paper-scissors contest between its two neighbours does not converge. Here, two cases similar to that are considered, one which is like the above but with an arbitrarily small chance of picking a random type and one where there are four neighbours which perform a knockout tournament to determine the new type. These two new setups, despite seeming very similar to the rock-paper-scissors model, do in fact converge, perhaps surprisingly.