The incremental voter model: mean-field analysis and convergence to equilibrium

Large language models (LLMs) increasingly interact with one another in multi-agent systems, from simulations of human discourse to influence operations and fully LLM-driven social platforms. These interactions give rise to new regimes of opinion propagation that are not yet well understood. We investigate whether classical opinion dynamics models, which have long been used to explain how interactions shape collective beliefs in human societies, can capture the behavior of LLM networks. We find that, while naive averaging-style models fail to track LLMs' opinion dynamics, simple modifications yield substantial gains in modeling fidelity. In particular, bias, an innate opinion toward which agents regress, emerges as a significant driver of LLM opinion dynamics, with its inclusion reducing cumulative estimated mean opinion error by up to 88%. We additionally find that these conclusions generalize across model families, discussion topics, and networks.

In a multi-agent system, there are often stubborn followers of specific opinions or beliefs. Motivated by this observation, in this paper, we aim to understand how stubborn agents affect the distribution of opinions in a network where both stubborn and non-stubborn agents interact with each other. To do so, we assume that all agents have an opinion in the set $\{0,1\}$ and each non-stubborn agent updates its opinion according to the $2k$\textit{-choices rule}, where the agent samples $2k$ neighbours (including both stubborn and non-stubborn neighbours) uniformly at random and adopts the majority opinion among the sampled group of neighbours and itself. We assume that a proportion of agents, $γ_i$, are stubborn followers of opinion $i\in \{0,1\}$. It is natural to expect that the steady-state distribution of the opinions in the network will be dominated by the opinion with the larger proportion of stubborn followers. We show that while this is true, the time to reach steady-state depends heavily on the values of the parameters $γ_0$ and $γ_1$. When the individual values of these parameters, as well as their difference, are small, it can take an exponentially long time (in the network size) to reach the steady-state. In sharp contrast, when at least one of the parameters $γ_0$ and $γ_1$ is large, the network reaches the steady-state in a time that is only logarithmic in the network size. Hence, there exists a sharp phase transition in the network dynamics based on the proportions of stubborn agents. We also characterise the behaviour of the system when the parameters $γ_0$ and $γ_1$ lie on the boundary of the phase transition. In this boundary region, we show using Stein's method that the dynamics are driven by a diffusion process which takes polynomial time to mix.

Understanding social polarization requires integrating insights from psychology, sociology, and complex systems science. Agent-based modeling provides a natural framework to combine perspectives from different fields and explore how individual cognition shapes collective outcomes. This study introduces a novel agent-based model that integrates two cognitive and social mechanisms: the desire to be unique within a group (optimal distinctiveness theory) and the tendency to simplify complex information (cognitive compression). In the model, virtual agents interact in pairs and decide whether to adopt each other's opinions by balancing two opposing drives: maximizing opinion diversity within their local social group while simplifying the overall opinion landscape, with both evaluated using Shannon entropy. We show that the combination of these mechanisms can reproduce real-world patterns, such as the emergence of distinct heterogeneous opinion clusters. Moreover, unlike many existing models where opinions become fixed once opinion groups form, individuals in our model continue to adjust their opinions after clusters emerge, leading to ongoing variation within and between opinion groups. Computational experiments reveal that polarization emerges when local group sizes are moderate (consistent with Dunbar's number), while smaller groups cause fragmentation and larger ones hinder distinct cluster formation. Higher cognitive compression increases unpredictability, while lower compression produces more consistent group structures. These results demonstrate how simple psychological rules can generate complex, realistic social behavior and advance understanding of polarization in human societies.