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The Contagion Tensor: A Framework for Measuring Output-Distribution Coupling in Multi-Agent LLM Systems -- and Auditing the Claims It Enables

This paper introduces the Contagion Tensor and the Coupling Amplification Factor (CAF) as a quantitative framework for measuring and falsifying output-distribution coupling across agents, modalities, and time in multi-agent LLM systems, validated through simulations and real-API experiments that distinguish genuine coupling effects from design artifacts.

Original authors: Zewen Liu

Published 2026-06-30

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Zewen Liu

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 room full of people (AI agents) talking to each other. Sometimes, they start thinking alike. Sometimes, they start saying the same things. Sometimes, if you show them a picture instead of just text, they might react in a totally different, more chaotic way.

For a long time, scientists studying these AI groups had a problem: they could see that things were happening, but they couldn't measure exactly how much the agents were "catching" each other's behavior. It was like trying to describe the weather by saying "it feels stormy" without a thermometer.

This paper introduces a new "thermometer" called the Contagion Tensor and a specific reading called the Coupling Amplification Factor (CAF). Here is how it works, broken down simply:

1. The "Contagion Tensor" (The 3D Map)

Think of the Contagion Tensor as a giant, 3D spreadsheet.

One side tracks different types of input (Text vs. Images).
Another side tracks how many agents are in the room.
The third side tracks time (how long they've been talking).

Every single cell in this spreadsheet measures how much an AI's answer has drifted away from being "random" or "neutral." If an AI starts leaning heavily toward one type of answer, that cell lights up. This map lets researchers see exactly where and when the "contagion" is happening.

2. The "CAF" (The Thermometer Reading)

From that 3D map, the authors created a simple number called CAF (Coupling Amplification Factor). Think of this as a "Chaos Score."

CAF = 1.0: The agents are acting independently. They are like strangers in a library; they aren't influencing each other.
CAF > 1.0: The agents are amplifying each other's weirdness. They are like a group of friends at a party who start laughing louder and louder together.
CAF < 1.0: The agents are calming down and becoming identical. They are like a choir that has practiced so much they all sound exactly the same.

3. The Big Discovery: "The Magic Trick"

The authors ran a massive experiment with 8 different scenarios. At first, the results looked amazing:

When the agents talked via Text, they calmed down (CAF < 1).
When the agents looked at Images, they went wild and amplified each other (CAF > 1).

It looked like images were a "super-connector" that made AI agents influence each other much more strongly than text did.

But then, they pulled the plug.

The authors realized their computer simulation had a tiny "glitch" (a specific code module) that treated images differently than text. They turned this module OFF and ran the experiment again.

The Result: The "Image Super-Connector" effect vanished! The image results dropped from "wild" (1.40) to "calm" (0.87), exactly the same as the text results.

The Lesson: The "magic" wasn't real. It was a design flaw in their own simulation. This proves the value of their new tool: it allowed them to catch a fake discovery before anyone else did.

4. Testing with Real AI (The Reality Check)

To make sure their tool wasn't just good at finding fake glitches, they tested it on real AI models (DeepSeek and GPT-4o-mini) using real APIs.

Text Results: When real AI agents talked via text, they stayed independent (CAF ≈ 1.0). If they had diverse personalities, they actually became more similar (CAF < 1), like a choir finding harmony.
Image Results: When they used a real AI with actual vision capabilities (looking at real JPEG images), the "super-connector" effect reappeared. The CAF jumped to 1.72.

This confirmed that while the simulation had a glitch, the idea was real: Real images do make AI agents influence each other more strongly than text does.

5. Why This Matters (The "Audit" Protocol)

The most important part of this paper isn't just the numbers; it's the method.

The authors propose a new rule for all AI research: The Ablation Protocol.
Before you claim that a new AI behavior is "emergent" (a magical property of the group), you must:

Identify the specific code part that might be causing it.
Turn that part OFF.
Run the test again.
If the effect disappears, admit it was just a design artifact, not magic.

Summary

This paper gives the field of AI a ruler to measure how much AI agents influence each other. It used that ruler to find a fake discovery in their own simulation (proving the ruler works) and then used it to find a real discovery: Real images make AI agents "catch" each other's behavior much faster than text does.

It's a call for scientists to stop guessing and start measuring, and to always check if their own tools are creating the illusions they see.

Technical Summary: The Contagion Tensor and Coupling Amplification Factor

Problem Statement
As multi-agent Large Language Model (LLM) systems become prevalent in chatbot ensembles, retrieval-augmented systems, and social simulations, their output distributions evolve through repeated exchange. A critical gap exists in the field: while single-model bias probes (e.g., BBQ, StereoSet) and qualitative case studies exist, there is no principled, reusable measurement framework to quantify how output distributions couple across modalities, agents, and time. Specifically, the community lacks a metric to determine if adding a second modality amplifies coupling, if memory increases temporal persistence, or if calibration on a subset of conditions guarantees validity on others. Without such a framework, claims regarding emergent behaviors or bias propagation remain qualitatively descriptive and quantitatively un-falsifiable.

Methodology: The Contagion Tensor and CAF Family
The paper introduces the Contagion Tensor ( $T \in \mathbb{R}^{M \times N \times T}$ ), a measurement framework where each cell $T[m, n, t]$ represents the Jensen-Shannon Divergence (JSD) between an agent's output distribution ( $w_{n,t}^m$ ) and a uniform reference distribution ( $w_0$ ). This tensor captures distributional drift across modalities ( $M$ ), agents ( $N$ ), and time steps ( $T$ ).

From this tensor, the authors derive the Coupling Amplification Factor (CAF), a family of ratio-based metrics defined as:
$CAF = \frac{E[T_{condition}]}{E[T_{baseline}]}$
where $E[T]$ is the expectation of the tensor entries. CAF is a unitless, baseline-referenced ratio where:

$CAF > 1$: Indicates amplified coupling.
$CAF = 1$: Indicates independent superposition.
$CAF < 1$: Indicates output convergence (homogenization).

The framework unifies four previously conflicting metrics (denoted by $\Gamma$ symbols in prior work) into a single notation with four variants:

$CAF_{base}$ : Full-axis reference (modality $\times$ agent count $\times$ time).
$CAF_{net}$ : Agent-level coupling (replacing spectral radius claims).
$CAF_{cross}$ : Cross-modal coupling.
$CAF_{temp}$ : Temporal persistence.

To ensure rigor, the paper employs a modality-ablation protocol: identifying code modules that could produce effects by design, disabling them while holding all seeds and parameters fixed, and re-measuring. This distinguishes genuine coupling from design artifacts.

Key Results
The framework was instantiated in a $2 \times 2 \times 2$ full-factorial simulation design and validated against real API experiments (DeepSeek-Chat and GPT-4o-mini).

Simulation Artifact Detection: Under default parameters, image conditions appeared to exhibit super-linear coupling ( $CAF \approx 1.40$ ) while text conditions were sub-linear ( $CAF \approx 0.84$ ). However, disabling the single module responsible for image-specific noise and structural shifts caused all image conditions to collapse to sub-linear values ( $CAF \approx 0.87$ ), with a mean shift of $\Delta = -0.53$ . Text conditions remained unchanged. This demonstrated that the apparent "modality bifurcation" was an artifact of the simulation's perturbation module, not an inherent property of the agents.
Real-API Validation:
- Text Conditions: Under uniform personas, text-only BOUNDARY SYNC communication produced $CAF \approx 1.0$ in both DeepSeek-Chat and GPT-4o-mini, indicating no measurable coupling relative to isolation.
- Persona Diversity: When agents were assigned diverse personas, text conditions converged ($CAF = 0.88$), indicating output homogenization rather than divergence.
- Real Vision: A within-model comparison on GPT-4o-mini revealed that real vision inputs (C5) produced a statistically significant super-linear effect ($CAF = 1.72$, 95% CI [1.700, 1.733]), contrasting with text-only conditions ($CAF = 1.02$). This validates the simulation's qualitative prediction that image modalities drive coupling, though the magnitude differs due to the simulation's artificial injection.
Metric Comparison: The paper shows that simpler metrics (entropy ratios, variance ratios) and pairwise mutual information fail to distinguish between text and image conditions, whereas CAF successfully captures the category-level structural differences.

Significance and Claims
The paper makes two distinct contributions, modestly framed as providing tools for measurement and auditing rather than claiming the discovery of a specific emergent phenomenon:

Measurement-Instrument Level: The CAF family provides the first quantitative, falsifiable metric for output-distribution coupling. By using a baseline-referenced ratio, it allows researchers to determine if observed effects are genuine or artifacts of the measurement instrument itself.
Methodology Level: The paper proposes a transferable ablation protocol as a standard quality gate for multi-agent LLM simulations. This protocol requires researchers to identify, disable, and re-measure specific modules to verify that claimed emergent behaviors are not design artifacts.

The authors argue that the field's urgent need is not another metric, but a discipline for verifying that observed coupling effects are genuine. The framework's value is demonstrated by its ability to reveal that previous "discoveries" (such as the modality bifurcation) were artifacts, and by its successful application in detecting a genuine super-linear coupling effect in real vision-capable models. The paper concludes that making a phenomenon measurable—and providing the tools to audit those measurements—is a scientific contribution in itself, even when the initial measurements reveal more about the instruments than the phenomenon.