Competing types in preferential attachment graphs with community structure

TL;DR

The paper generalizes the two-type AMR model to geopref9 graphs with a finite community structure and analyzes whether community-level red-proportions synchronize. By linking the dynamics to the fixed points of the rule-generated polynomial $R$, it uncovers regimes of non-synchronisation under weak coupling when multiple linearly stable fixed points exist, and provides sharp conditions guaranteeing convergence to a common limit when a unique fixed point dominates. The results leverage stochastic approximation and urn-coupling techniques to prove synchronization in the linear model with sufficient inter-community interaction, while also presenting examples where non-convergence or different limits across communities occur. The findings enhance understanding of reinforcement processes on multi-community networks and reveal how interaction strength and local rules shape global convergence behavior with potential implications for models of contagion, opinion dynamics, and network growth.

Abstract

We extend the two-type preferential attachment model of Antunović, Mossel and Rácz, 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.

Figures (7)

• Figure 1: The values of $Z^{(t)}_{1},Z^{(t)}_{2},Z^{(t)}_{3}$ in the framework of Proposition \ref{['threecycle']} with $m=7$ plotted over time (log scale), showing apparent non-convergence.
• Figure 2: An example of $R(z)$ satisfying the conditions of Lemma \ref{['lem:shift-up']} with $z_1=0.4$ and $z_2=0.6$. The curve must lie outside the highlighted region. For some suitable $\varepsilon>0$ the value of $R(z)$ is above $z_1+4\mu_i^{-1}\varepsilon$ for $z\in(z_1-2\varepsilon,z_2+2\varepsilon)$, as shown by the dotted lines.
• Figure 3: An example where neither condition in Lemma \ref{['rcond']} is satisfied; the solid black line shows $R(z)$. The values $z'=0.2$ and $z"=0.8$ violate \ref{['two-point-condition']}, and $R(R(z))$ (dashed red line) has linearly stable fixed points close to $0.1$ and $0.9$.
• Figure 4: Plots of $R(z)$ for the examples given in Sections \ref{['majwins']}, \ref{['randvis']}, \ref{['minority']} and \ref{['tp-rule']}.
• Figure 5: Examples of simulations with two weakly linked communities and $m=3, p_0=p_1=0,p_2=p_3=1$. In the left simulation different types are dominating in the two communities, with proportions of red vertices in the two communities being $0.877$ and $0.077$. In the right simulation blue appears to be dominating in both communities, with proportions of red vertices in the two communities being $0.057$ and $0.021$.

Theorems & Definitions (36)

• Theorem 3.1
• Remark 3.2
• Proposition 3.3
• Theorem 3.4
• Remark 3.5
• Theorem 3.6
• Theorem 3.7
• Theorem 3.8
• Proposition 3.9
• Theorem 3.10