Competing types in preferential attachment graphs with community structure

TL;DR

This paper extends a two-type preferential attachment model to include community structure, analyzing how community interactions influence the synchronization of type proportions across communities.

Contribution

It introduces a community-structured extension of the preferential attachment model and identifies conditions for synchronization of type proportions between communities.

Findings

Synchronization depends on community structure and assignment rules.

Communities may fail to synchronize under weak interaction.

Conditions for guaranteed synchronization are provided.

Abstract

We extend the two-type preferential attachment model of Antunovi\'c, Mossel and R\'acz, where each new vertex takes its type according to a defined rule based on the types of its neighbours, to incorporate community structure, and investigate whether the proportions of vertices of each type synchronise between communities. The behaviour depends both on the choice of community structure and on the type assignment rule. For essentially all cases where the single community model has more than one possible limit, communities may fail to synchronise for weakly interacting communities. Even when the single community model almost surely converges to a deterministic limit, synchronisation is not guaranteed. However, we give natural conditions on the assignment rule and, for two communities, on the structure, either of which will imply synchronisation to this limit, and each of which is essentially best possible. We also give an example where the proportions of types almost surely do not converge, which is impossible in the single community model.