The voter model on random regular graphs with random rewiring

Abstract: We consider the voter model with binary opinions on a random regular graph with $n$ vertices of degree $d \geq 3$, subject to a rewiring dynamics in which pairs of edges are rewired, i.e., broken into four half-edges and subsequently reconnected at random. A parameter $\nu \in (0,\infty)$ regulates the frequency at which the rewirings take place, in such a way that any given edge is rewired exponentially at a rate $\nu$ in the limit as $n\to\infty$. We show that, under the joint law of the random rewiring dynamics and the random opinion dynamics, the fraction of vertices with either one of the two opinions converges on time scale $n$ to the Fisher-Wright diffusion with an explicit diffusion constant $\vartheta_{d,\nu}$ in the limit as $n\to\infty$. In particular, we identify $\vartheta_{d,\nu}$ in terms of a continued-fraction expansion and analyse its dependence on $d$ and $\nu$. A key role in our analysis is played by the set of discordant edges, which constitutes the boundary between the sets of vertices carrying the two opinions.

• The paper presents a convergence finding where opinion densities on dynamic regular graphs evolve toward a Fisher-Wright diffusion process.
• It derives a key diffusion constant using continued-fraction expansions that quantitatively links rewiring rates with stochastic opinion updates.
• The analysis leverages duality with coalescing random walks to rigorously establish exponential tail behavior for consensus times in the model.

Overview of the Voter Model on Random Regular Graphs with Random Rewiring

This paper investigates the voter model with binary opinions on dynamic random regular graphs. Specifically, it explores a scenario where each vertex within a graph represents an agent holding one of two possible opinions, and these graphs undergo continuous rewiring—altering the connections between nodes based on a Poisson-driven dynamic process.

Key Aspects of the Model

• Graph Structure and Rewiring Dynamics: The model operates on random regular graphs, where each node is of degree $d$ ($d \geq 3$). The rewiring process allows edges to be broken and randomly reassigned at different rates controlled by a parameter $\nu$, indicating the frequency of these dynamic changes.
• Voter Model Mechanism: In this graph, each agent randomly selects a neighbor and adopts their opinion at each tick of a Poisson process. The interplay between these opinion changes and the rewiring of the graph constitutes the main focus of the study.

Theoretical Contributions

1. Convergence to Wright-Fisher Diffusion: A primary result is the convergence of opinion densities—proportion of vertices with a particular opinion—to a Fisher-Wright diffusion over an appropriate timescale. This result elucidates how the interaction between local opinion exchanges and the rewiring of networks influences macroscopic opinion dynamics.
2. Identification of Diffusion Constant $\vartheta_{d,\nu}$: Central to this convergence is the diffusion constant $\vartheta_{d,\nu}$, mathematically characterized through continued-fraction expansions. This novel expression for $\vartheta_{d,\nu}$ not only makes the analysis tractable but also ties the dynamic properties of the network to the stochastic processes governing opinion dynamics.
3. Duality with Coalescing Random Walks: The paper leverages a duality between the voter model and a system of coalescing random walks, making the convergence proofs more rigorous and tractable.
4. Exponential Tail of Meeting Times: The work establishes precise asymptotics for the meeting time of two stationary random walks on these graphs, which is essential to understanding consensus times in the model.

Practical and Theoretical Implications

• Social and Biological Systems: By understanding how information disperses and reaches consensus in dynamically rewired networks, insights can be gleaned into how beliefs or behaviors spread across social settings or genetic traits proliferate within biological populations.
• Network Science and Dynamic Graphs: The findings contribute to the broader understanding of processes on dynamic networks—networks where the topology is not static but evolves with time, reflecting more realistic scenarios compared to static models.

Future Directions

The study highlights several avenues for future research, including:

• Exploring different types of rewiring mechanisms and their implications for diffusion constants.
• Extending the analysis to networks with non-uniform node degree distributions or even weighted edges, which may mimic more complex realistic systems.
• Considering additional dynamics, such as compounding rewiring with growth or decay mechanisms within the network topology.

In summary, the paper presents a comprehensive analysis of the voter model on dynamic random regular graphs, offering significant insights into the convergence behavior and the role of network dynamics in opinion formation processes. By addressing both theoretical underpinnings and potential real-world applications, it contributes valuably to the understanding of dynamic interactions on evolving network structures.