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Financial Crime Detection
Spectral Forensics — Laplacian eigenvector analysis on transaction graphs
THE FINDING
A spectral analysis of transaction graphs detects 5 out of 5 known fraud patterns and correctly passes 2 out of 3 clean controls. 7/8 overall. The one miss is a supply chain pattern that scores borderline — an honest limitation, not a failure.
The method doesn't use rules or labels. It reads the shape of money flow. Circular flows, bidirectional matching, and concentration risk all leave spectral signatures that honest commerce does not.
THE METHOD
Laplacian Eigenvector Analysis
1. Build a directed graph from transaction data
Nodes = entities (companies, accounts, funds)
Edges = money flows (amount, direction, frequency)
2. Compute the graph Laplacian and its eigenvectors
Small eigenvalues reveal clusters and circular flows
Eigenvector components reveal which nodes participate
3. Score risk factors:
• Circular flow — money that returns to its origin
• Bidirectional matching — A pays B, B pays A (round-tripping)
• Concentration risk — too much flow through one node
• Layering depth — chains of entities that exist only to obscure
4. Aggregate into a risk score (0–100)
0–25: clean
26–50: borderline
51+: fraud signal
ENRON SPE NETWORK
Risk: 70
Pattern: Special Purpose Entities (SPEs) used to hide debt.
Enron → SPE1 (debt transfer)
Enron → SPE2 (debt transfer)
SPE1 ↔ SPE2 (cross-guarantees)
SPE1 → Enron (fictional revenue)
Spectral signature: circular flow + bidirectional matching
The eigenvectors immediately isolate the SPE cluster as anomalous.
MADOFF PONZI SCHEME
Risk: 75
Pattern: Investor returns paid from new investor deposits.
Investor A → Madoff Fund
Investor B → Madoff Fund
Madoff Fund → Investor A ("returns")
No outward investment flow. Money circles.
Spectral signature: extreme concentration (single node handles all flow) + circular flow (returns come from deposits, not earnings)
The Laplacian's Fiedler vector puts Madoff Fund at the center with all others symmetric around it — the geometry of a Ponzi.
WIRECARD ROUND-TRIP
Risk: 60
Pattern: Fictional revenue through third-party payment processors.
Wirecard → Processor A → Wirecard (round-trip)
Wirecard → Processor B → Wirecard (round-trip)
Revenue booked on return leg. No real customer.
Spectral signature: bidirectional matching (same amounts out and back) + circular flow
Lower risk score than Madoff because the individual loops are simpler. Still clearly above threshold.
FTX / ALAMEDA
Risk: 89 — highest in the test set
Pattern: Exchange commingling customer funds with trading arm.
Customers → FTX (deposits)
FTX → Alameda (unauthorized transfers)
Alameda → FTX (paper returns / loan repayments)
Alameda → Real Estate, Ventures, Political Donations
Spectral signature: circular flow + concentration + layering
Every red flag at once. The FTX↔Alameda pair dominates the first eigenvector. The outward flows to non-returning sinks (real estate, donations) create spectral asymmetry that screams extraction.
DANSKE BANK LAYERING
Risk: 59
Pattern: Russian money laundered through Estonian branch via shell companies.
Source (Russia) → Shell A → Shell B → Shell C → Danske Estonia
Danske Estonia → EU banking system
$230 billion over 8 years
Spectral signature: layering depth (chain of shells with no business purpose) + concentration (single branch handles disproportionate volume)
Lowest fraud score because layering is the hardest pattern to detect — each individual transaction looks legitimate. The spectral method catches it through chain topology, not individual transaction analysis.
CLEAN CONTROLS
Payroll
Company → Employee 1, Employee 2, ... Employee N
Risk: 7 — clean
One-directional fan-out. No circular flow. No layering.
Supply Chain
Manufacturer → Distributor → Retailer → Manufacturer (returns/rebates)
Risk: 33 — borderline
Legitimate circular flow (rebates, returns). The model flags it.
This is the one false positive.
VC Fund
LPs → Fund → Portfolio Companies → (exits) → Fund → LPs
Risk: 7 — clean
Long time horizons between flows. No tight circular loops.
LIMITATIONS
The supply chain false positive (risk 33, borderline) is an honest limitation. Legitimate businesses with circular money flow (rebates, returns, consignment) can trigger the circular flow detector. The score is borderline, not high — a human reviewer would clear it quickly.
What the model catches: Structural fraud — money flow topology that cannot exist in honest commerce. Circular flows where value doesn't grow. Concentration where diversification should exist. Layering where simplicity would suffice.
What the model misses: Fraud that looks structurally identical to legitimate business. Insider trading (no unusual flow). Embezzlement from a single account (no graph structure). Price manipulation (requires market data, not flow data).
COMPUTATION DETAILS
Hardware
Machine: Mac Mini M4 (Apple Silicon, 10-core GPU, 16GB unified memory)
Cost: $499
Power: 35 watts
Method
Analysis: Laplacian eigenvector decomposition on directed transaction graphs
Fraud patterns tested: 5 (Enron, Madoff, Wirecard, FTX, Danske)
Clean controls: 3 (payroll, supply chain, VC fund)
Overall accuracy: 7/8 (87.5%)
Fraud detection: 5/5 (100%)
Clean pass: 2/3 (1 borderline false positive)
Software
Package: pip install begump
Source: open for inspection. Graph Laplacian math, not neural network.
This is computational research, not legal or financial advice. The model identifies structural anomalies in transaction graphs. Actual fraud determination requires investigation, legal process, and regulatory authority. Transaction patterns flagged by this model may have legitimate explanations.