A spiking neural network simulation investigating whether organizational structures follow universal scaling laws similar to phase transitions in statistical physics.
This project started as an ambitious test of "Governance Physics" — the idea that all hierarchical systems (Kubernetes clusters, central banks, corporations) might follow universal scaling laws.
The probe killed the universality claim.
Real data showed the Federal Reserve's policy transmission operates on 12-18 month timescales, not milliseconds. Corporate coordination costs scale as O(n²), not √N. Only Kubernetes — with its 10-100ms control loops — matched the model's dynamics.
So this repo focuses where the physics actually works: K8s failure cascades. Here's what the model predicts, and here's how to prove it wrong.
Our simulation's avalanche exponent (α ≈ 1.88) is suspiciously close to the Bak-Tang-Wiesenfeld sandpile model from 1987 — the foundational paper on self-organized criticality (SOC).
The BTW model:
- Drop sand grains on a 2D lattice
- When height ≥ 4, site topples → distributes to neighbors
- Cascades (avalanches) follow power-law with α ≈ 1.5
Why this matters for K8s:
| Sandpile Concept | Kubernetes Analog |
|---|---|
| Sand grain | Request / load increment |
| Site height | Pod resource usage |
| Toppling threshold | OOM limit / CPU throttle |
| Avalanche | Cascade failure |
| Boundary dissipation | Circuit breaker / node isolation |
If K8s exhibits SOC, then:
- Large outages are mathematically expected, not bugs
- You cannot prevent all cascades
- You can shift the exponent lower with isolation mechanisms
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Predicting cascade risk — The model identifies parameter regimes where small failures amplify into cluster-wide outages. If your cluster operates near λ_critical, you're living dangerously.
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Tuning controller loops — The simulation shows how control rate (λ) and propagation delay (τ) affect stability. Faster isn't always better — there's a tradeoff.
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Understanding power-law risk — If failure cascades follow power-law distributions (α ≈ 1.5-2.0), then "black swan" outages are mathematically expected, not flukes. Plan accordingly.
We model Kubernetes cluster governance as a hierarchical spiking neural network to test three predictions from "Governance Physics" theory:
- Phase transitions exist — Systems transition from synchronized (stable) to desynchronized (unstable) as control attenuation (λ) increases
- Critical avalanche dynamics — Failure cascades follow power-law distributions with exponent α ≈ 1.5
- Universal scaling — Control rate λ scales with √(system size)
Results: We find evidence for critical avalanche dynamics (α = 1.88, near theoretical 1.5) but no clear phase transition and the √N scaling law is rejected.
We represent a Kubernetes cluster as a 5-layer hierarchy:
Layer 0: API Server / Controller Manager (Executive)
Layer 1: Scheduler / etcd (Coordination)
Layer 2: Kubelets (Node Agents)
Layer 3: Pods (Workload Units)
Layer 4: Containers (Labor)
Each layer is modeled as a population of Leaky Integrate-and-Fire (LIF) neurons with:
- Downward synapses: Control signals with exponential weight decay
w = 0.5 * exp(-λ * depth) - Upward synapses: Failure feedback with decay
w = 0.3 * exp(-μ * depth_from_bottom)
| Parameter | Value | Empirical Range | Source |
|---|---|---|---|
| λ (control rate) | 0.8/s | 0.1-1.0/s | Controller loop frequencies |
| τ (timescale) | 20ms | 10-100ms | etcd consensus latency |
| μ (dissipation) | 1.5 | — | Tuned parameter |
| G (efficiency) | 0.4 | 0.2-0.8 | Span-of-control estimates |
- Order Parameter (Φ) — Kuramoto-style synchronization measure across time windows
- Avalanche Distribution — Cascade sizes during high-activity periods
- Power-law Exponent (α) — Maximum likelihood estimation for avalanche sizes
| Metric | Value |
|---|---|
| Critical λ | ~1.36 |
| Max |dΦ/dλ| | 0.034 |
| Verdict | WEAK/ABSENT |
The order parameter Φ remained roughly constant (0.26-0.35) across the λ sweep. No sharp transition was detected.
| Metric | Value | Critical Value |
|---|---|---|
| Power-law exponent α | 1.88 | ~1.5 |
| Avalanches detected | 224 | — |
| Verdict | NEAR-CRITICAL | ✅ |
The cascade size distribution follows a power-law with exponent close to the critical value, suggesting the model captures realistic failure propagation dynamics.
| Metric | Value |
|---|---|
| R² for √N fit | 0.025 |
| Verdict | REJECTED |
The predicted scaling relationship between control rate and system size does not emerge from the model.
This simulation makes testable predictions. Real empirical data would falsify the model if:
Do real cluster failures follow a power-law distribution with α ≈ 1.5-2.0?
Our simulation produces α ≈ 1.88. But we don't know if real K8s failures match this. To validate, someone needs to:
- Get cascade size data from production clusters (Borg traces, Azure logs, post-mortems)
- Fit a power-law distribution
- Compare α_real vs α_simulated
If they're close → model captures real dynamics. If they're far apart → model is wrong.
If Google Borg traces show cascade exponent α ≈ 1.2, but this model produces α ≈ 1.9 → the model is wrong.
| Prediction | Falsified If |
|---|---|
| Cascade power-law | Real K8s cascades don't follow power-law distribution |
| Exponent α ≈ 1.5-2.0 | Real cascade exponent is significantly different |
| Timescale 10-100ms | Real failure propagation is orders of magnitude different |
| No phase transition | Real clusters DO show sharp stability transitions |
Data sources needed for validation:
- Public K8s failure cascade logs
- Cascade size distributions from production clusters
- Propagation time measurements through microservice dependency graphs
| File | Description |
|---|---|
governance_physics_k8s.py |
K8s-focused simulation code |
governance_physics_snn.py |
Multi-domain simulation (K8s, Fed, Corp) |
docs/GOVERNANCE_PHYSICS_RESEARCH_REPORT.md |
Full research report with literature review |
governance_physics_k8s_results.png |
K8s visualization output |
btw_sandpile_comparison.png |
BTW sandpile model comparison |
# Activate environment with Brian2
conda activate conscious_snn
# Run K8s simulation
python governance_physics_k8s.py
# Run multi-domain simulation
python governance_physics_snn.pyRequirements:
- Python 3.11+
- Brian2 2.8.0+
- NumPy, Matplotlib
- Bak, P., Tang, C., & Wiesenfeld, K. (1987). "Self-organized criticality: An explanation of the 1/f noise." Physical Review Letters, 59(4), 381-384.
- Dhar, D. (1990). "Self-organized critical state of sandpile automaton models." Physical Review Letters, 64(14), 1613-1616.
- Carreras, B.A. et al. (2004). "Evidence for self-organized criticality in electric power system blackouts." Chaos, 14(2), 393-403.
- Watts, D.J. (2002). "A simple model of global cascades on random networks." PNAS, 99(9), 5766-5771.
- Prigogine, I. (1977). Nobel Lecture: Time, Structure and Fluctuations
- Nicolis, G. & Prigogine, I. (1977). Self-Organization in Non-Equilibrium Systems
- Google Cluster Data (Borg traces): github.com/google/cluster-data
- etcd performance benchmarks: etcd.io/docs/v3.5/benchmarks/
- K8s scalability docs: kubernetes.io/docs/setup/best-practices/cluster-large/
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Cross-domain claims were dropped — Earlier versions attempted to model Federal Reserve monetary policy and corporate hierarchies. These were abandoned after empirical probes showed order-of-magnitude timescale mismatches (months vs milliseconds) and incorrect scaling exponents.
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No empirical validation — Parameters are estimated from literature, not fitted from real K8s failure data.
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Simplified topology — Real K8s networks have more complex dependency structures.
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Single domain tested — Only K8s has plausible timescales for this model.
Peace — Research direction, empirical validation, methodology, analysis
MIT
@misc{governance_physics_2026,
title={Governance Physics: Phase Transitions and Cascade Dynamics in Kubernetes Clusters},
author={Peace},
year={2026},
url={https://github.com/Peace-png/governance-physics}
}