⚡ electrical engineering

Opinion Consensus Formation Among Networked Large Language Models

This paper investigates how networked large language models reach consensus through multi-round text exchanges, finding that while their convergence dynamics align with classical DeGroot models regarding decay rates and graph topology, their final opinions are driven more by inherent biases and discussion subjects than by initial conditions or network centrality.

Original authors: Iris Yazici, Mert Kayaalp, Stefan Taga, Ali H. Sayed

Published 2026-01-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Iris Yazici, Mert Kayaalp, Stefan Taga, Ali H. Sayed

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a group of 20 friends sitting in a circle, each holding a different opinion about a controversial topic like "Is Bitcoin a good investment?" or "Should we be vegan?" In the real world, if these friends talked to each other for a while, they might eventually agree on something.

This paper asks a simple question: If we replace those human friends with Artificial Intelligence (AI) chatbots, will they behave the same way? Specifically, do they follow the same mathematical rules that scientists have used for decades to predict how human groups reach agreement?

Here is the breakdown of what the researchers found, using simple analogies:

1. The Old Rulebook (The DeGroot Model)

For a long time, mathematicians have used a rulebook called the DeGroot model to predict group opinions. Think of it like a weighted average.

The Theory: If you trust your neighbor 50% and yourself 50%, your new opinion is the average of yours and theirs. If you are a "hub" in the network (connected to many people), your opinion should carry more weight in the final group decision.
The Prediction: The final group opinion should be a perfect mathematical blend of everyone's starting opinions, weighted by how central they are in the network.

2. The Experiment: AI Friends in a Chat Room

The researchers set up a digital playground with 20 AI agents.

The Setup: They connected these AIs in different shapes (some in a tight circle, some in a loose web).
The Rules: They told the AIs, "You are either 'self-confident' (trust yourself more) or 'open-minded' (trust others more)." They gave them a topic (like "Ghosting" or "Gene Editing") and let them chat back and forth for 80 rounds.
The Measurement: After every chat, a separate AI acted as a judge, scoring the tone of the conversation from "Strongly Against" (-3) to "Strongly For" (+3).

3. The Big Surprise: The "Echo Chamber" Effect

The researchers found two main things:

A. They do agree, but not for the reasons we thought.
Just like the old rulebook predicted, the AI agents eventually stopped disagreeing. Their opinions converged, and the "noise" between them died down exponentially (like a echo fading away).

However: The final opinion they reached did not match the mathematical prediction.
The Analogy: Imagine a group of people trying to guess the weight of a pumpkin. The old math says the final guess should be the average of everyone's starting guesses. But in this experiment, the AIs ignored their starting guesses almost entirely. Instead, they all ended up agreeing on a specific number that seemed to come from inside the AI itself, not from the group discussion.

B. The "Inherent Bias" is the real boss.
Why did they ignore the math? Because the AIs have built-in biases from their training (the data they were fed before the experiment started).

The Analogy: Imagine the AIs are like people who grew up in a specific culture. Even if you put them in a room with people from all over the world, they might all secretly lean toward a specific cultural viewpoint.
The Finding: If the topic was "Bitcoin," the AIs tended to lean negative, regardless of whether they started out positive or negative. If the topic was "Veganism," they leaned positive. The topic and the AI's internal training mattered far more than who was talking to whom or who started with what opinion.

4. What Did Follow the Old Math?

While the final answer was wrong according to the old math, the speed at which they agreed was spot on.

The Analogy: Think of a crowded room where everyone is shouting. The old math predicts exactly how long it takes for the room to go quiet.
The Finding: The researchers found that the time it took for the AIs to stop disagreeing matched the mathematical predictions perfectly. If the network was more connected (more friends talking to more friends), they agreed faster. If the network was sparse, it took longer. The "speed limit" of the agreement was determined by the shape of the network, just as the old math said.

5. The "Self-Confidence" Factor

The researchers also tested what happens if they explicitly tell the AIs, "Trust yourself 80% and your friends 20%."

The Result: When the AIs were given these specific "trust weights," they reached a consensus much more easily. When the researchers removed these instructions, the AIs were more chaotic and disagreed more at the end. It seems that giving AI agents clear rules on how much to trust themselves helps them settle down faster.

Summary

Do AI agents reach a consensus? Yes, they stop arguing and agree on a point.
Does the old math predict what they agree on? No. The final opinion is driven by the AI's hidden biases and the topic, not by the starting opinions of the group.
Does the old math predict how fast they agree? Yes. The speed of agreement depends entirely on how the network is connected, just like the old math predicted.

The Takeaway: If you want to simulate a social network with AI, you can trust the math to tell you how fast they will agree, but you cannot trust the math to tell you what they will agree on. The AI's internal "personality" (bias) overrides the group dynamics.

Technical Summary: Opinion Consensus Formation Among Networked Large Language Models

Problem Formulation

This work investigates whether classical mathematical models of opinion formation, specifically the DeGroot consensus framework, can accurately predict the group behavior of Large Language Models (LLMs) acting as agents in a network. While the DeGroot model is well-established in signal processing and control theory for describing how agents reach consensus through repeated averaging of neighbors' opinions, its empirical validation in real-world social settings is often limited by the cost and complexity of human data collection. The authors posit that LLMs offer a controllable, low-cost testbed to model these dynamics. The core problem is twofold:

Determine if LLM agents interacting over communication graphs exhibit consensus behavior similar to human social networks.
Analyze whether the specific mechanics of LLM consensus (limiting opinions and convergence rates) align with DeGroot's theoretical predictions, particularly regarding the influence of initial conditions, network topology, and inherent model biases.

Methodology

The authors constructed a multi-agent simulation environment using the AutoGen framework, deploying Google's Gemini 2.0 Flash as the underlying LLM. The experimental setup involved the following components:

Network Topology: A network of $K=20$ agents was simulated over directed, weighted graphs. The primary topology used was the Erdős–Rényi random graph with varying connectivity parameters ( $p$ ), alongside fully connected and circular (ring) topologies to test connectivity extremes.
Agent Interaction: Agents exchanged text-based messages over multiple rounds (80 rounds per experiment). The interaction weights (trust levels assigned to neighbors and self-weights) were enforced via system prompts. Two agent character types were defined: "self-confident" (higher self-weight) and "open-minded" (lower self-weight).
Opinion Quantification: To map text responses to numerical beliefs, a separate, independent LLM (gpt-5-nano) performed sentiment analysis on all responses. Opinions were scored on a scale of $[-3, 3]$ (from "against" to "for") and normalized to $[0, 1]$ .
Experimental Variables: The study varied discussion topics (8 distinct topics including Bitcoin, Euthanasia, Veganism, etc.), initial opinion distributions (for, neutral, against), and graph connectivity.
Dataset: The authors compiled a dataset of 764 experiments containing over 1.2 million LLM responses, covering 8 topics and various network configurations.

Key Results

1. Emergence of Consensus and Decay of Disagreement

The experiments demonstrated that LLM agents typically reach a consensus. The standard deviation (STD) of agents' sentiment scores decreased exponentially over interaction rounds, approaching a near-zero steady state. This decay rate was found to be closely related to the second-largest modulus eigenvalue ( $|\lambda_2|$ ) of the graph's combination matrix, aligning with spectral graph theory predictions for convergence speed.

2. Divergence from DeGroot Predictions

Despite the emergence of consensus, the limiting opinion (the final consensus value) significantly deviated from the DeGroot model's prediction.

Insensitivity to Initial Conditions: Contrary to the DeGroot theorem, which states the final consensus should be a network-centrality-weighted average of initial opinions, the LLM consensus was largely insensitive to initial conditions.
Topic-Dependent Bias: The final consensus point was strongly dependent on the discussion subject and the inherent biases of the LLMs (likely stemming from pretraining and alignment phases). For instance, on the topic of Bitcoin, LLMs exhibited a negative bias regardless of initial "for" or "against" distributions, whereas on Veganism, a positive bias emerged.
Model Accuracy: When comparing final LLM opinions to DeGroot predictions, the Root Mean Squared Error (RMSE) was 0.32. Classification accuracy based on the DeGroot model was approximately 32% for ternary classification and 60% for binary classification, both comparable to or only marginally better than random guessing.

3. Impact of System Prompts and Weights

The study found that explicitly enforcing combination matrix weights via system prompts significantly increased the likelihood of reaching consensus. "Weighted" experiments (where self-weights were explicitly defined) resulted in a final disagreement STD of 0.083, whereas "weightless" experiments (where specific weights were removed from prompts) resulted in a significantly higher final diversity of 0.165.

4. Convergence Rate and Graph Theory

While the value of the consensus did not match DeGroot theory, the rate of convergence did. The "halving time" of disagreement (the number of rounds required for opinion variance to reduce by half) was empirically shown to be proportional to $\frac{\ln 2}{-\ln |\lambda_2|}$ . This confirms that the transient dynamics of LLM networks follow established graph-theoretical results, even if the steady-state equilibrium does not.

Significance and Claims

The paper claims that while LLMs can serve as effective, low-cost testbeds for simulating social network dynamics, their consensus behavior differs fundamentally from classical models in terms of steady-state outcomes.

Bias Dominance: The authors highlight that in LLM networks, inherent model biases and topic sensitivity override the influence of initial conditions and network centrality, which are central to the DeGroot model.
Convergence Dynamics: The study validates that the speed of consensus in LLM networks is governed by the spectral properties of the communication graph (specifically $|\lambda_2|$ ), suggesting that classical graph theory remains useful for designing resource-efficient multi-agent LLM applications where the timing of consensus is critical.
Resource Efficiency: Understanding these convergence rates allows for better estimation of the interaction rounds required to achieve consensus under cost constraints in multi-agent systems.

The authors conclude that these findings assist in clarifying the potential of LLMs for social-simulation research and guide the design of multi-agent systems, while noting that further work is needed to explore strategic agents and other LLM architectures.