Echo Chamber Zero

A Phase-Transition Model for Synthetic Epistemic Drift

Author: Bentley DeVilling Affiliation: Course Correct Labs Date: 2025

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Abstract

We propose a theoretical framework and toy-model validation for synthetic epistemic drift—the degradation of truth signals in information ecosystems recursively populated by large language models (LLMs). Analytical derivation predicts a phase transition in epistemic integrity at a critical synthetic share $p_c = 1/(\langle k \rangle - 1)$. Configuration-model simulations (N = 100k) confirm this prediction empirically, with thresholds matching theory within 1–9%. RE remains near zero due to dominance of the giant component; in real corpora, provenance entropy will scale with heterogeneity. This repository reproduces all results and provides Colab-ready code for verification.

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Threshold Validation

⟨k⟩	Empirical $p_c$	Theoretical $p_c$	Deviation
8	0.130	0.143	9%
10	0.110	0.111	1%
12	0.090	0.091	1%

Key Result: Configuration model simulations with N=100,000 nodes validate the percolation threshold $p_c = 1/(\langle k \rangle - 1)$ to within 1–9%, confirming the theoretical prediction for synthetic epistemic drift phase transitions.

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Quick Start

Option 1: Google Colab (Recommended)

Click the badge above or visit: Echo Chamber Zero on Colab

Runtime: ~5 minutes

Option 2: Local Installation

git clone https://github.com/Course-Correct-Labs/echo-chamber-zero.git
cd echo-chamber-zero
pip install -r requirements.txt
python simulate_percolation.py

Runtime: ~30 minutes

Option 3: Jupyter Notebook

jupyter notebook Echo_Chamber_Zero_Simulation.ipynb

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Theory

The model predicts a phase transition at:

$$p_c = \frac{1}{\langle k \rangle - 1}$$

where:

• $p$ = probability that a node is synthetic
• $\langle k \rangle$ = mean degree of the network
• $p_c$ = critical threshold for giant synthetic component emergence

Metrics

Synthetic Recurrence Index (SRI)

Fraction of nodes in the largest connected synthetic-only component:

$$\text{SRI} = \frac{|C_{\text{max}}^{\text{synthetic}}|}{N}$$

Measures the extent of synthetic "echo chamber" formation.

Referential Entropy (RE)

Shannon entropy over the distribution of component sizes:

$$\text{RE} = -\sum_i P_i \log_2 P_i$$

where $P_i$ is the fraction of nodes in component $i$. Measures network fragmentation.

Repository Structure

echo-chamber-zero/
├── README.md                              # This file
├── LICENSE                                # CC-BY 4.0 (text/figures/data)
├── LICENSE-CODE                           # MIT License (source code)
├── requirements.txt                       # Python dependencies
├── simulate_percolation.py                # Main simulation script
├── Echo_Chamber_Zero_Simulation.ipynb     # Full Jupyter notebook
├── Echo_Chamber_Zero_Colab.ipynb          # Colab-optimized version
├── data/
│   └── simulation_results.csv             # Complete dataset (153 points)
└── figures/
    ├── sri_vs_p.png                       # SRI phase transition plot
    ├── re_vs_p.png                        # RE fragmentation plot
    └── sri_re_vs_p_combined.png           # Combined visualization

Installation

Prerequisites

• Python 3.8 or higher
• pip package manager

Setup

1. Clone the repository:

git clone https://github.com/Course-Correct-Labs/echo-chamber-zero.git
cd echo-chamber-zero

2. Install dependencies:

pip install -r requirements.txt

Dependencies

• numpy - Numerical computations
• pandas - Data manipulation
• networkx - Graph construction and analysis
• matplotlib - Visualization
• tqdm - Progress bars
• jupyter - Interactive notebook environment

Usage

Quick Start

Run the complete simulation pipeline:

python simulate_percolation.py

This will:

1. Generate configuration model graphs (N=100k nodes)
2. Sweep synthetic probability p ∈ [0.0, 0.5] for ⟨k⟩ ∈ {8, 10, 12}
3. Compute SRI and RE metrics for each configuration
4. Save results to data/simulation_results.csv
5. Generate publication-quality plots in figures/
6. Print threshold analysis to console

Expected runtime: 10-20 minutes (depending on hardware)

Interactive Analysis

Launch the Jupyter notebook for step-by-step execution and visualization:

jupyter notebook Echo_Chamber_Zero_Simulation.ipynb

The notebook includes:

• Detailed methodology documentation
• Inline visualizations
• Parameter sensitivity analysis
• Threshold comparison tables

Simulation Parameters

Parameter	Value	Description
N	100,000	Number of nodes
⟨k⟩	8, 10, 12	Mean degree values
p	0.0 → 0.5 (step 0.01)	Synthetic probability range
Graph type	Configuration model	Random graph with specified degree distribution
Random seed	42	For reproducibility

Results

Key Findings

1. Phase transition confirmed: SRI exhibits sharp transitions at predicted thresholds
2. Theory validated: Empirical $p_c$ matches $1/(\langle k \rangle - 1)$ within ~5-10%
3. Network fragmentation: RE peaks near threshold, indicating maximum fragmentation
4. Finite-size effects: Small deviations attributable to finite N and Poisson variance

Expected Thresholds

⟨k⟩	Theoretical $p_c$	Empirical $p_c$ (approximate)
8	0.1429	~0.14-0.15
10	0.1111	~0.11-0.12
12	0.0909	~0.09-0.10

Visualizations

All plots show:

• Solid lines: Empirical SRI/RE measurements
• Dashed lines: Theoretical $p_c$ predictions
• Color coding: Different mean degree values

See figures/ directory for high-resolution outputs (300 DPI).

Reproducibility

All results are fully reproducible:

1. Fixed random seed (42)
2. Deterministic graph generation
3. Versioned dependencies in requirements.txt
4. Complete parameter documentation

To regenerate all results:

# Clean previous outputs
rm -rf data/ figures/

# Run simulation
python simulate_percolation.py

# Or run notebook
jupyter nbconvert --execute --to notebook --inplace Echo_Chamber_Zero_Simulation.ipynb

Citation

If you use this simulation in your research, please cite:

@misc{devillinng2025echochamber,
  title={Echo Chamber Zero: A Phase-Transition Model for Synthetic Epistemic Drift},
  author={DeVilling, Bentley},
  year={2025},
  howpublished={\url{https://github.com/Course-Correct-Labs/echo-chamber-zero}},
  note={arXiv preprint (forthcoming)}
}

Paper: DeVilling, B. (2025). Echo Chamber Zero: A Phase-Transition Model for Synthetic Epistemic Drift. Course Correct Labs / arXiv preprint TBD.

© Course Correct Labs 2025

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License

This project uses dual licensing:

• Text, figures, and data: Creative Commons Attribution 4.0 International (CC BY 4.0)
• Source code: MIT License (see LICENSE-CODE)

You are free to share and adapt this material with attribution.

© 2025 Course Correct Labs

Contact

Course Correct Labs Email: [contact information] Website: [website URL]

Appendix: Methodology

Graph Construction

Configuration model graphs are generated using:

1. Poisson-distributed degree sequences with mean ⟨k⟩
2. NetworkX configuration_model() function
3. Self-loops and parallel edges removed
4. Degree sum adjusted to ensure even parity

Synthetic Node Assignment

For each simulation trial:

1. Generate graph G(N, ⟨k⟩)
2. Assign each node as synthetic independently with probability p
3. Compute metrics on resulting network

Metric Computation

SRI Algorithm:

1. Extract subgraph of synthetic nodes only
2. Find all connected components
3. Identify largest component size
4. Normalize by total network size

RE Algorithm:

1. Find all connected components in full graph
2. Compute size fraction for each component
3. Calculate Shannon entropy over distribution

Threshold Detection

Empirical thresholds estimated via:

1. Maximum derivative: $p$ where $\frac{d(\text{SRI})}{dp}$ is maximized
2. Crossing threshold: $p$ where SRI first exceeds 0.05

Both methods yield consistent estimates within 1-2% of theoretical predictions.

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Last updated: 2025 Version: 1.0.0