Opinion Consensus Formation Among Networked Large Language Models

TL;DR

The paper tests whether DeGroot-style consensus can predict group behavior in networked LLMs by simulating multi-round interactions on directed graphs and mapping messages to sentiment-based opinions. It finds that consensus emerges with disagreement decaying exponentially, but the limiting opinion deviates from the DeGroot weighted average and is biased by topic and pretraining. Convergence rates match spectral graph theory, scaling with the second-largest eigenvalue magnitude $|\lambda_2|$, with a halving-time relation $t_{1/2} = \ln 2 / (- \ln |\lambda_2|)$. An open dataset of 764 experiments across 8 topics and prompting strategies is released to support future research and guide resource-efficient multi-agent LLM deployments.

Abstract

Can classical consensus models predict the group behavior of large language models (LLMs)? We examine multi-round interactions among LLM agents through the DeGroot framework, where agents exchange text-based messages over diverse communication graphs. To track opinion evolution, we map each message to an opinion score via sentiment analysis. We find that agents typically reach consensus and the disagreement between the agents decays exponentially. However, the limiting opinion departs from DeGroot's network-centrality-weighted forecast. The consensus between LLM agents turns out to be largely insensitive to initial conditions and instead depends strongly on the discussion subject and inherent biases. Nevertheless, transient dynamics align with classical graph theory and the convergence rate of opinions is closely related to the second-largest eigenvalue of the graph's combination matrix. Together, these findings can be useful for LLM-driven social-network simulations and the design of resource-efficient multi-agent LLM applications.

Figures (7)

• Figure 1: The average standard deviation of agents' opinions with respect to the number of iterations across $50$ bitcoin-related experiments. The shaded region indicates the standard error of the mean (SEM) across $50$ simulations.
• Figure 2: Left: Initial opinion distributions, Right: Final opinion distributions. Each row belongs to a set of experiments with a different topic and initial opinion distribution. For example, the first row denotes a set of experiments where the initial opinion distribution is highly skewed towards "for" on bitcoin sentiment, while for the second row, the initial majority is "against". The error bars denote the SEM with respect to a total of $150$ experiments.
• Figure 3: Average standard deviation of agents’ opinions over interaction rounds for different values of Erdős--Rényi $p$. Each curve corresponds to a group of simulations within the indicated $p$ range, with $n$ denoting the number of experiments. Larger $p$ values result in faster convergence to the consensus with less disagreement.
• Figure 4: Halving time of disagreement between LLM agents, changing with respect to the second eigenvalue of the combination matrix. The eigenvalues are arranged into $30$ discrete bins, and the total number of experiments is $110$. The halving time shown on the y-axis is the mean across all experiments. SEM is denoted with bars around the mean. The dashed red curve shows the theoretical halving time.
• Figure 5: Organization of the Social-LLM-Networks repository.