Compounding Vulnerability — Reproducible Code Package

Paper: Compounding Vulnerability: Hub Removal Can Trigger Cascade Phase Transitions While Degrading Percolation Robustness in Barabási-Albert Networks
Author: Federico Cachero (federicohernancachero@gmail.com)
arXiv: 2603.04838
Target: Physical Review E
Date: March 2026

---

Overview

This repository contains simulation code, derived results (RESULTS_*.json), and figure-generation scripts to reproduce the main findings:

• Targeted hub removal (top degree-ranked fraction) raises the bond-percolation threshold (reduced robustness to random edge failure).
• The same intervention can expand the Watts cascade window and trigger a cascade phase transition (subcritical → supercritical) at intermediate thresholds.
• A controlled "hub vulnerability" experiment isolates the firewall mechanism as primarily dynamical (threshold-mediated), not purely topological.
• The cascade branching factor z₁(φ, α, m) predicts threshold crossing: pre-removal z₁ = 0.850 (subcritical) → post-removal z₁ = 1.195 (supercritical).

429,000+ simulation trials back the claims in the paper.

What's New (v4 — March 7, 2026)

• Title updated: "Can Trigger" (hedged claim) + "Barabási-Albert" (specific scope)
• Abstract shortened: 1 concise paragraph (was 5)
• Real-world table improved: Absolute cascade sizes (Cmax pre/post) replace misleading percentages
• Claims softened: No-Pareto observation labeled "empirical" (not proven impossibility)
• References added: Cohen et al. (2000), Brummitt et al. (2012)
• Propositions relabeled: Prop 1→Definition 1, Prop 2→Observation 1 for honesty
• z₁ independently verified: z1_audit.py confirms all Table V values within ±0.0003

What's New (v2 — March 6, 2026)

• Newman-Ziff susceptibility-peak percolation replaces S=0.5 as primary method (+347% ΔF)
• Configuration-model validation: Effect is STRONGER on config model (38% vs 23%), validating z₁ framework
• Finite-size scaling: Effect INCREASES with N (+352% at N=2K → +412% at N=10K)
• Heterogeneous thresholds: Effect persists under all distributions (+349% to +1,955%)
• 4 real-world SNAP networks: Cascade amplification universal (+2,082% to +60,272%)
• Lean 4 proofs updated: 5 claims, no sorry

Requirements

python >= 3.10
pip install -r requirements.txt

Repository Structure

Core Simulations

• simulate_production.py — core synthetic-network simulations (finite-size scaling)
• phi_sweep_large.py — φ-sweep at large N (transition sharpening)
• hub_vulnerability_test.py — controlled experiment (hub thresholds vs removal)
• z1_analytical.py — analytical checks / z₁ calculations
• verify_quick.py — quick sanity checks
• simulate_empirical.py — empirical network simulations

V2 Validation (fixes/)

• newman_ziff_percolation.py — susceptibility-peak percolation method
• newman_ziff_multiscale.py — finite-size scaling at N=2K/5K/10K
• config_model_validation.py — configuration-model comparison
• heterogeneous_thresholds.py — uniform/beta/truncated-normal threshold tests
• increased_trials_cascade.py — 1000-trial cascade sweep with CIs
• additional_real_networks.py — 4 SNAP real-world networks

Real-World Networks (realworld/)

• realworld_validation.py — Bitcoin OTC + AS-733 validation
• realworld_validation_results.json — results
• INTERPRETATION.md — analysis of real-world findings

Figure Generation

• generate_figures.py, generate_paper_figures.py, generate_phi_figure_v4.py

Results

• RESULTS_*.json — stored outputs from all simulations
• fixes/*.json — v2 validation results

Paper

• papers/paper-a/arxiv/ — arXiv v1 source
• papers/paper-a/journal-v2/ — journal submission (v2, PRE format)

Formal Verification

• proofs/P2Complete.lean — Lean 4 formalization (5 claims, no sorry)

Reproducing Results

# Core simulations
python verify_quick.py
python simulate_production.py
python hub_vulnerability_test.py
python phi_sweep_large.py

# V2 validation
cd fixes/
python newman_ziff_percolation.py
python config_model_validation.py
python heterogeneous_thresholds.py
python newman_ziff_multiscale.py
python additional_real_networks.py

# Empirical
python simulate_empirical.py

# Figures
python generate_figures.py
python generate_phi_figure_v4.py

Key Results

Metric	Pre-removal	Post-removal	Change
Bond percolation p_c (susceptibility peak, N=2,000)	0.174 ± 0.009	0.776 ± 0.036	+347%
Cascade size at φ=0.22 (1,000 trials)	0.86% [CI: 0.43-1.30]	23.1% [CI: 21.3-24.9]	Phase transition
z₁ branching factor at φ=0.22	0.850	1.195	Subcritical → supercritical
Hub vulnerability experiment (fixed topology)	A: 0.28%	B: 95.0% (vuln) / C: 18.7% (removed)	Dynamical > topological

License

• Code: MIT (see LICENSE)
• Paper/manuscript assets: CC BY 4.0 (see LICENSE-PAPER)