Hyperbolic Geometry of Semantic Networks

Cross-Linguistic Evidence from Word Association Data

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Overview

This repository contains research on network geometry using Ollivier-Ricci curvature analysis across multiple implementations:

• Julia: Reference implementation (validated)
• Rust: Performance-focused implementation
• Sounio: Type-safe implementation with epistemic computing

Key result: Curvature sign change in random regular graphs at density η = ⟨k⟩²/N, with finite-size scaling η_c(N) = 3.75 − 14.62/√N (R² = 0.995). Semantic network geometry follows from two parameters: η (density) and C (clustering).

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Key Findings

Curvature Sign Change (Feb 2026 — updated)

• Density parameter: η = ⟨k⟩²/N determines the sign of mean ORC in random k-regular graphs

  • η < η_c(N) → Negative curvature (hyperbolic regime)
  • η > η_c(N) → Positive curvature (spherical regime)
  • η_c(N) = 3.75 − 14.62/√N (fitted on N ∈ {50, 100, 200, 500, 1000}, R² = 0.995)
  • η_c(100) ≈ 2.29, η_c(200) ≈ 2.72, η_c(500) ≈ 3.10, η_c^∞ ≈ 3.75

• Validated on 11 real semantic networks (SWOW ES/EN/ZH/NL, ConceptNet EN/PT, WordNet, BabelNet, depression)

  • Dutch SWOW (η = 7.56 >> η_c) is spherical (κ̄ = +0.10): confirms prediction
  • Clustering coefficient C modulates curvature within sub-critical regime
  • Two-parameter model (η + C) classifies all 11 networks (post-hoc; validation pending)

Original Findings

• 11 semantic networks analyzed across 7 languages (SWOW, ConceptNet, WordNet, BabelNet, depression)
• Three geometric regimes: Hyperbolic (C > 0.10, η < η_c), Euclidean (C < 0.02), Spherical (η > η_c)
• Metric dependence: All networks flip to spherical under sphere-embedded ORC (Cayley-Dickson tower)
• Cross-linguistic consistency across language families

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Repository Structure

hyperbolic-semantic-networks/
├── README.md                    # This file
├── CHANGELOG.md                 # Version history
│
├── julia/                       # Julia implementation
│   ├── src/                     # Core modules
│   ├── experiments/             # Phase transition experiments
│   └── phase_transition_pure_julia.jl  # Validated experiment
│
├── rust/                        # Rust implementation
│   ├── curvature/               # Curvature computation
│   └── null_models/             # Random graph generation
│
├── experiments/                 # Sounio-powered experiments
│   ├── 01_epistemic_uncertainty/   # Phase transition sweep
│   ├── 02_null_model/              # Configuration null model ensemble
│   ├── 03_forman_ricci/            # Forman vs Ollivier comparison
│   ├── 04_uncertainty_scaling/     # Uncertainty at phase transition
│   ├── 05_hypercomplex/           # Hypercomplex curvature embedding
│   ├── 06_spectral_geometry/     # Spectral gap phase transition
│   └── 07_scale_n500/            # N=100 all methods + N=200 spectral
│
├── results/                     # Computed results
│   ├── experiments/             # Phase transition data
│   ├── curvature/               # Curvature metrics
│   └── swow_clustering_coefficients.json
│
├── archive/                     # Historical docs and session artifacts
│
├── manuscript/                  # Main manuscript
│   ├── main.md                  # Complete manuscript
│   └── figures/                 # Publication figures
│
├── code/                        # Python analysis scripts
│   ├── analysis/                # Analysis pipeline
│   └── figures/                 # Figure generation
│
└── data/                        # Data
    ├── raw/                     # Original SWOW data
    └── processed/               # Processed networks

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Quick Start

Requirements

Julia:

julia --project -e 'using Pkg; Pkg.instantiate()'

Rust:

cd rust && cargo build --release

Sounio (for new experiments):

cd path/to/sounio/compiler
cargo build --release
export PATH=$PATH:$(pwd)/target/release

Run Phase Transition Experiment

# Julia (validated reference)
julia phase_transition_pure_julia.jl

# Results in: results/experiments/phase_transition_pure_julia.json

Reproduce Analysis

# Complete analysis pipeline
cd code/analysis
python run_analysis_pipeline.py

# Generate figures
cd ../figures
python generate_all_figures.py

CPC 2026 Extension

The repository now includes a dedicated CPC 2026 paper pipeline for:

Entropic Curvature in Hyperbolic Semantic Manifolds Indexes Psychopathology-Like Transitions

Artifacts and code live in:

• code/cpc2026/
• results/cpc2026/
• figures/cpc2026/
• manuscript/cpc2026_paper.md

Run the full extension from the repository root with:

make cpc2026

This CPC pipeline reuses the validated SWOW-EN exact-LP curvature artifact, adds node-level entropic curvature and valence annotations, simulates regime-specific semantic trajectories, computes trajectory statistics, and generates a CPC-specific figure set.

CPC 2026 O-SSM Extension

The CPC lane now also includes an octonionic state-space extension that bridges this repository to the canonical Sounio checkout at:

• github.com/sounio-lang/sounio

What this extension adds:

• code/cpc2026/ossm_bridge/
  • builds 8D SWOW node vectors and exports compact Sounio input bundles
• code/cpc2026/ossm_reference_simulator.py
  • generates the full paper-scale O-SSM artifacts in Python
• code/cpc2026/ossm_analysis.py
  • computes O-SSM-specific metrics and the Markov-vs-O-SSM comparison table
• code/cpc2026/generate_ossm_figures.py
  • generates the O-SSM figure set
• results/cpc2026/sounio_parity/
  • stores the bounded parity artifacts emitted by the canonical Sounio runner

Run the full cross-repo O-SSM lane with:

make cpc2026-ossm

Operational note:

• The canonical Sounio repo provides the executable parity lane under examples/cognitive_ossm/.
• The full 10,000 x 500 O-SSM result artifacts are currently generated by the Python reference mirror in this repo, because that is the reproducible paper-scale path available today.
• The versioned snapshot stores results/cpc2026/ossm_trajectories_{regime}.csv.gz and data/cpc2026/trajectories_{regime}_input.npz; the raw .csv and .npy counterparts remain local-only because they exceed GitHub's file-size limits.
• results/cpc2026/ossm_release_manifest.json records the frozen archive inventory and SHA-256 checksums for the versioned O-SSM snapshot.

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Sounio Experiments

All experiments are self-contained .sio programs demonstrating the phase transition with Sounio's effect system (with IO, Mut, Div, Panic) and type-safe fixed-size arrays.

Experiment 01: Phase Transition Sweep

Ollivier-Ricci curvature across k-regular graphs (N=20, k=2..18). Demonstrates the universal transition from hyperbolic to spherical geometry.

bash experiments/01_epistemic_uncertainty/run.sh
# → results/sounio/phase_transition_sounio.csv

Experiment 02: Configuration Null Model Ensemble

5 independent realizations per k-value from the configuration model C(N,k). Tests whether curvature is a structural invariant of the degree sequence.

bash experiments/02_null_model/run.sh
# → results/sounio/configuration_null.csv

Experiment 03: Forman-Ricci vs Ollivier-Ricci

Compares two discrete Ricci curvature notions on the same graphs:

• Forman: combinatorial O(deg²) per edge, no optimal transport
• Ollivier: optimal transport O(n² × sinkhorn_iter) per edge

bash experiments/03_forman_ricci/run.sh
# → results/sounio/forman_comparison.csv

Experiment 04: Uncertainty Scaling at Phase Transition

Multi-seed ensemble analysis with Shannon entropy of geometry classification. Shows that epistemic uncertainty peaks at the phase transition (k²/N ≈ 2.5).

bash experiments/04_uncertainty_scaling/run.sh
# → results/sounio/uncertainty_scaling.csv

Experiment 05: Hypercomplex Curvature Embedding

Embeds graph nodes into hypercomplex hyperspheres — S³ (quaternion), S⁷ (octonion), S¹⁵ (sedenion) — via landmark BFS distances, then computes Ollivier-Ricci curvature using geodesic distances instead of integer hop-counts. Showcases Hamilton product (associative) and Cayley-Dickson product (non-associative).

• Phase A (N=20): Validates embeddings reproduce the known phase transition
• Phase B (N=50): Breaks the N=20 barrier using landmark-based embedding

bash experiments/05_hypercomplex/run.sh
# → results/sounio/hypercomplex_curvature.csv

Experiment 06: Spectral Geometry of Phase Transition

Independent validation via eigenvalues of the adjacency matrix. Computes the second eigenvalue λ₂ using power iteration on the shifted matrix (A+kI) with deflation against the known trivial eigenvector. Derives spectral gap, algebraic connectivity, Cheeger constant lower bound, and Friedman ratio.

• Phase A (N=20): Spectral + Ollivier-Ricci curvature for direct comparison
• Phase B (N=50): Spectral only (cross-reference with experiment 05)

bash experiments/06_spectral_geometry/run.sh
# → results/sounio/spectral_phase_transition.csv

Experiment 07: Scale — N=100 (All Methods) + N=200 (Spectral)

Scales beyond the N=50 barrier. N=500 BFS all-pairs ([i64; 250000]) proved infeasible in the bytecode VM (~6 hours per k-value). Practical design:

• Phase A (N=100): curvature + Q4 embedding + spectral for k ≤ 18, spectral-only for k > 18
  • k_crit = √(2.5 × 100) ≈ 15.8 — curvature spans the full transition
• Phase B (N=200): spectral + BFS metrics only — matches Julia reference

bash experiments/07_scale_n500/run.sh
# → results/sounio/scale_n500.csv

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Key Results

Phase Transition Data

Network	N	⟨k⟩	⟨k⟩²/N	κ_mean	Geometry
Sparse	200	3	0.05	-0.287	Hyperbolic
Medium	200	22	2.42	-0.013	Transition
Dense	200	30	4.50	+0.073	Spherical

Full data: results/experiments/phase_transition_pure_julia.json

Semantic Networks

Language	N	⟨k⟩	⟨k⟩²/N	κ	Prediction
Spanish	9,246	3.0	0.001	-0.155	Hyperbolic ✓
English	10,571	3.1	0.001	-0.258	Hyperbolic ✓
Chinese	8,857	3.2	0.001	-0.214	Hyperbolic ✓
Dutch	2,962	61.6	1.280	+0.125	Spherical ✓

Key Insight: All semantic networks have ⟨k⟩²/N << 1, explaining universal hyperbolicity!

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Documentation

Main Docs

• CHANGELOG.md - Version history
• DEVELOPMENT.md - Development guide

Scientific Reports

• lean/FORMALIZATION_STATUS.md - Lean 4 formalization status
• lean/HyperbolicSemanticNetworks/PREPRINT.md - Formalization preprint
• manuscript/main.md - Main manuscript

Full Index

• experiments/ - Sounio experiment lanes 01-07

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Implementation Status

Complete

• Julia reference implementation (N=200, 11 networks)
• Rust performance implementation (Sinkhorn + null models)
• Sounio graph module (in Sounio repo)
• Phase transition discovery and validation
• Sounio Experiments 01-07 (phase transition, null model, Forman, uncertainty, hypercomplex, spectral, N=100/200 scale)
• Scientific documentation

In Progress

• Cross-language benchmarking (Julia vs Sounio numerical agreement)
• Publication: "Network Geometry in Sounio"

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Citation

This Work

@software{hyperbolic_semantic_networks_julia_rust,
  title = {Hyperbolic Semantic Networks: Julia/Rust/Sounio Implementation},
  author = {Agourakis, Demetrios C.},
  year = {2025},
  doi = {10.5281/zenodo.17655231},
  url = {https://zenodo.org/records/17655231},
  version = {2.0.0}
}

Phase Transition Discovery

@article{agourakis2024phase,
  title = {Universal Phase Transition in Network Geometry},
  author = {Agourakis, Demetrios C.},
  journal = {In preparation},
  year = {2024},
  note = {Transition at $\langle k \rangle^2 / N \approx 2.5$}
}

SWOW Dataset

@article{de2019small,
  title={The Small World of Words English word association norms for over 12,000 cue words},
  author={De Deyne, Simon and Navarro, Danielle J and Perfors, Amy and Brysbaert, Marc and Storms, Gert},
  journal={Behavior Research Methods},
  volume={51},
  pages={987--1006},
  year={2019}
}

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Sounio Integration

The network geometry module has been implemented in the Sounio programming language at stdlib/graph/, showcasing:

• ✅ Effect system: Explicit tracking of Alloc, Random, Confidence
• ✅ Epistemic computing: Automatic uncertainty propagation
• ✅ Units of measure: Dimensional type safety
• 🔜 Refinement types: SMT-verified network properties
• 🔜 GPU acceleration: First-class GPU effects
• 🔜 Parallel computing: Effect-tracked parallelism

See GitHub Issue #13 for implementation details.

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License

• Code: MIT License
• Data: CC BY 4.0
• Manuscript: CC BY 4.0

See LICENSE for details.

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Contact

Demetrios Chiuratto Agourakis Email: demetrios@agourakis.med.br ORCID: 0000-0002-8596-5097 GitHub: @agourakis82

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Acknowledgments

• Small World of Words project team
• Sounio programming language development
• Julia and Rust communities

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Version History

See CHANGELOG.md for detailed version history.

Current Version: v2.0.0 (see CHANGELOG.md)

Previous: v0.2.0 (Phase Transition + Sounio Implementation), v0.1.0 (Initial Julia/Rust implementation)