🌌 Complex Quantum Network Geometries

This C++ project simulates the evolution of complex quantum networks following the model described by Ginestra Bianconi in Complex Quantum Network Geometries: Evolution and Phase Transition (2015).

📘 Theoretical Background

The simulation grows simplicial 2-complexes (i.e. networks built from triangles) that evolve under:

• Fermi-Dirac statistics: each link can host at most one extra triangle (m = 2).
• Bose-Einstein statistics: each link can host an unlimited number of triangles (m = ∞).

Each link has an energy ε determined by node energies ( \omega_i, \omega_j ), e.g.:

• Linear: ( \varepsilon_{ij} = \omega_i + \omega_j )
• Quadratic: ( \varepsilon_{ij} = J(J + 1), ; J = \frac{\omega_i + \omega_j}{2} )

Growth is controlled by an inverse temperature parameter ( \beta ). At high β, networks undergo structural phase transitions.

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🧠 Key Features

• Fermi-Dirac & Bose-Einstein quantum network evolution
• Poisson-distributed node energies (configurable)
• Dynamic link selection weighted by energy + triangle count
• Automatic export of:
  • 📏 Max shortest path distance
  • 📶 Max degree ( k_{\text{max}} )
  • 🧠 Entropy rate ( H^{[1]} )
  • 🧩 Modularity (Louvain method)
  • 🔗 Clustering coefficient ( C )
  • 🌀 Node curvature ( R_i )
• Python scripts for:
  • Metric plots vs. ( \beta )
  • Metric evolution over time
  • Error bars over trials
  • Degree, clustering, and curvature distributions
  • Automatic detection of phase transition ( \beta_c )

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🔧 Compilation

mkdir build
cd build
cmake ..
make

🚀 Run Simulations (with multiple seeds for error bars)

Edit main.cpp to select:

• Number of trials (default: 3)
• Beta values and network sizes

#define ERROR_BAR_TRIALS 3
std::vector<double> betas = {0.01, 0.2, 1.0, 5.0};
std::vector<int> Ns = {5000};

Then run:

./quantum_net

This will export CSVs into build/raw_csv/.

📊 Visualize Results

python3 visualize_errorbars.py
python3 plot_network_metrics.py

This will generate plots in:

build/graphs/

📁 Project Folder Structure

├── CMakeLists.txt
├── main.cpp
├── node.cpp/hpp, link.cpp/hpp, triangle.cpp/hpp
├── network.cpp/hpp
├── growth_engine.cpp/hpp
├── metrics.cpp/hpp
├── plot_network_metrics.py
├── visualize_errorbars.py
├── README.md
└── build/
    ├── raw_csv/     # Exported data
    ├── graphs/      # Final plots
    └── metrics/     # Degree, curvature, clustering

📘 Reference

Bianconi, G. (2015). Complex Quantum Network Geometries: Evolution and Phase Transition. arXiv:1503.04739