Cohomological Risk Scoring

Author: Idriss Bado

A sophisticated Python package for financial risk assessment using cohomological methods, persistent homology, and sheaf theory. This framework provides a mathematically rigorous approach to identifying systemic risks and anomalies in financial transaction networks.

Overview

This package implements the Persistence of Cohomological Risk (PCR) scoring methodology, which uses algebraic topology to detect structural inconsistencies in financial networks. Unlike traditional graph-based methods, this approach captures higher-order relationships and cyclic dependencies that often indicate fraudulent behavior or systemic risk.

Key Features

• Sheaf-Theoretic Framework: Model financial data as sheaves over simplicial complexes
• Persistent Cohomology: Compute topological features across multiple scales
• Risk Scoring: Assign cohomological risk scores to entities based on structural inconsistencies
• Risk Class Identification: Automatically identify Cohomological Risk Classes (CRCs)
• Visualization Tools: Generate persistence diagrams and risk score distributions
• Network Analysis: Build and analyze financial transaction networks

Mathematical Foundation

The package is built on three key mathematical concepts:

1. Simplicial Complexes: Financial networks are modeled as simplicial complexes where vertices represent entities, edges represent transactions, and higher-dimensional simplices capture multi-party relationships.

2. Sheaf Theory: Data is organized as a sheaf, with stalks at each simplex and restriction maps encoding consistency conditions.

3. Persistent Cohomology: Cohomological features are tracked across a filtration, revealing persistent structural anomalies that indicate risk.

Installation

From PyPI (when published)

pip install cohomological-risk-scoring

From Source

git clone https://github.com/idrissbado/cohomological-risk-scoring.git
cd cohomological-risk-scoring
pip install -e .

Development Installation

pip install -e ".[dev]"

Quick Start

import networkx as nx
from cohomological_risk_scoring import PCRScorer
from cohomological_risk_scoring.utils import create_example_network

# Create or load your financial network
G, vertex_features, edge_features = create_example_network(n_vertices=30)

# Initialize the scorer
scorer = PCRScorer(max_dim=2, filtration_param='weight')

# Fit the model
scorer.fit(G, vertex_features, edge_features)

# Compute risk scores
scores = scorer.compute_all_scores()

# Identify high-risk entities
high_risk = {v: s for v, s in scores.items() if s > 0.7}
print(f"High-risk entities: {high_risk}")

# Get risk classes
risk_classes = scorer.get_risk_classes(threshold=0.1)

# Visualize results
scorer.visualize_persistence(save_path='persistence_diagram.png')

# Generate detailed report
print(scorer.generate_report())

Core Components

FinancialSheaf

The FinancialSheaf class implements the mathematical foundation:

• Complex Construction: Build simplicial complexes from financial graphs
• Sheaf Data: Define vertex and edge stalks with restriction maps
• Cohomology Computation: Calculate coboundary matrices and cohomology groups
• Persistent Cohomology: Track cohomological features across filtrations

PCRScorer

The PCRScorer class provides the main interface:

• Fit: Build the sheaf model from financial data
• Score Computation: Calculate PCR scores for all entities
• Risk Classification: Identify Cohomological Risk Classes
• Visualization: Generate persistence diagrams and score distributions
• Reporting: Create comprehensive analysis reports

Use Cases

• Anti-Money Laundering (AML): Detect circular transaction patterns
• Fraud Detection: Identify inconsistent financial behavior
• Credit Risk: Assess systemic risk in lending networks
• Market Manipulation: Detect coordinated trading schemes
• Regulatory Compliance: Monitor for suspicious activity patterns

Requirements

• Python >= 3.8
• NumPy >= 1.20.0
• SciPy >= 1.7.0
• NetworkX >= 2.6.0
• GUDHI >= 3.5.0
• Persim >= 0.3.0
• Matplotlib >= 3.4.0

Documentation

Full documentation is available at https://cohomological-risk-scoring.readthedocs.io

Examples

See the examples/ directory for detailed usage examples:

• basic_usage.py: Introduction to the package
• advanced_analysis.py: Custom restriction functions and advanced features
• real_world_data.py: Working with real financial datasets

Testing

Run the test suite:

pytest tests/

With coverage:

pytest --cov=cohomological_risk_scoring tests/

Contributing

Contributions are welcome! Please feel free to submit a Pull Request. For major changes, please open an issue first to discuss what you would like to change.

Citation

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

@software{bado2025cohomological,
  author = {Bado, Idriss},
  title = {Cohomological Risk Scoring: A Topological Approach to Financial Risk Assessment},
  year = {2025},
  url = {https://github.com/idrissbado/cohomological-risk-scoring}
}

License

This project is licensed under the MIT License - see the LICENSE file for details.

Author

Idriss Bado

• GitHub: @idrissbado
• Email: idrissbadoolivier@gmail.com

Acknowledgments

• The GUDHI library team for persistent homology tools
• The NetworkX community for graph analysis capabilities
• Researchers in topological data analysis and financial mathematics

Roadmap

• Support for directed graphs
• GPU acceleration for large-scale networks
• Integration with popular fraud detection frameworks
• Real-time monitoring capabilities
• Interactive web-based visualization
• Extended documentation with tutorials
• Benchmark datasets for evaluation

References

1. Carlsson, G. (2009). "Topology and data." Bulletin of the American Mathematical Society, 46(2), 255-308.
2. Ghrist, R. (2014). "Elementary Applied Topology." Createspace.
3. Hansen, J., & Ghrist, R. (2019). "Toward a spectral theory of cellular sheaves." Journal of Applied and Computational Topology, 3(4), 315-358.