Theoretical Background

Full Theoretical Treatment: For complete mathematical proofs and detailed theoretical foundations, see the research paper:

Bado, I. O. (2025). “Cohomological Risk Scoring: A Topological Framework for Detecting Structural Inconsistencies in Financial Networks.”

Mathematical Foundations

Cohomological Risk Scoring is based on the mathematical framework of algebraic topology applied to financial networks.

Core Concepts

  1. Simplicial Complexes

    Financial networks are modeled as simplicial complexes:

    • Vertices (0-simplices): Financial entities

    • Edges (1-simplices): Transactions/relationships

    • Triangles (2-simplices): Higher-order interactions

  2. Sheaf Theory

    A financial sheaf F assigns:

    • Data spaces to each simplex

    • Restriction maps encoding consistency constraints

  3. Persistent Cohomology

    Tracks cohomological features across filtration scales to identify robust risk signals.

Key Theorems

Theorem 1: Stability

The persistence diagram of Cohomological Risk Classes (CRCs) is stable under data perturbations:

\[d_B(\text{Dgm}_1, \text{Dgm}_2) \leq C \cdot d_{GH}((K_1, F_1), (K_2, F_2))\]

Implication: Small errors in data lead to small changes in risk scores.

Theorem 2: Detection

The framework guarantees detection of cyclic fraud patterns satisfying:

\[\left\| \sum_{i=1}^k \left( g(v_i, v_{i+1}) - \rho_{v_i \to e_i}(\mathbf{v}_i) \right) \right\| > \epsilon\]

Implication: Money laundering loops and circular trading are always detected.

Theorem 3: Locality

PCR scores can be computed efficiently from local neighborhoods:

\[r = O(\log(1/\epsilon))\]

Implication: Scalable to large networks with millions of nodes.

PCR Score Definition

The Persistence of Cohomological Risk (PCR) score for vertex v:

\[\text{PCR}(v) = \sum_{[\omega] \in \mathcal{P}} \mathbb{1}_{v \in \text{supp}([\omega])} \cdot \text{pers}([\omega]) \cdot \|[\omega]\|\]

Where:

  • P: Set of persistent CRCs

  • pers([ω]): Persistence (lifetime) of cohomology class

  • ||[ω]||: Norm of representative cocycle

Interpretation

High PCR Score Indicates

  1. Cyclic Inconsistencies: Entity participates in transaction loops inconsistent with declared profiles

  2. Structural Anomalies: Part of fraud rings or laundering networks

  3. Global Risk: Risk emerges from network structure, not local features alone

Advantages

  • Global Perspective: Detects relational risks invisible to local models

  • Interpretability: Provides specific cycles/structures causing risk

  • Robustness: Stable under noise and missing data

  • Mathematical Rigor: Backed by theorems with formal guarantees

For the complete mathematical treatment, see the accompanying paper: “Cohomological Risk Scoring: A Topological Framework for Detecting Structural Inconsistencies in Financial Networks”