Identifying Ultimate Beneficial Owners (UBOs) in complex corporate structures is critical for financial transparency and preventing economic crimes. Recursive cycles in ownership networks exacerbate this challenge by increasing computational complexity. This article proposes a model based on weighted directed graphs, where nodes represent individuals or legal entities and edges represent ownership percentages. Integrating graph theory and geometric series eficiently resolves ownership cycles, providing a mathematical framework for calculating efective ownership. Direct ownership is computed as the product of weights along paths, while cycles are addressed using recursive algorithms and convergence factors derived from geometric series. The methodology combines graph modeling, algorithmic design (including a DFS version), and experimental validation. Preliminary results demonstrate that the model significantly reduces computational complexity (from O(n!) to O(n+m)), transforming intricate corporate networks into compact UBO tables with their total ownership. While its efectiveness depends on data quality, this work lays the foundation for scalable corporate transparency systems, with applications in financial regulation and compliance.
transparency and combating economic crime worldwide. In recent years, international organizations such as FATF and the European Union have established strict guidelines for disclosing this information, recognizing its critical role in preventing money laundering, terrorism financing, and other illicit activities.
However, this task faces significant computational challenges due to complex ownership networks
featuring multi-level indirect ownership, circular relationships, and exponential search space growth,
especially in multinational structures [
The key contributions are: a weighted directed graph model for ownership representation, a DFS algorithm with geometric series for cycle resolution, a mathematical framework for ownership calculation, and experimental validation with real-world data, together enabling eficient UBO identification in complex corporate networks while reducing computational complexity from (!) to ( + ).
Ownership is defined as the share of a company held directly (via direct shareholding) or indirectly (through ownership chains). According to GAFI1, UBOs are individuals deriving economic benefits or exercising control, as per FATF guidelines (see Figure 1).
Ownership information for companies and trusts serves as a countermeasure to crime. Clear data
is essential for combating illicit financial flows, and governments increasingly commit to beneficial
ownership transparency as a key law enforcement tool [
GAFI [
Beyond whether UBO registries should be public, a more pressing concern is ensuring their accuracy and integrity. The UK registry, maintained by Companies House, is a prime example of a well-intentioned but unreliable database. UBO registries aim to ensure that the ownership of assets held through legal structures, such as companies or trusts, is known at least to the registry administrator.
Within this framework, a statement may refer to one of three core elements of a beneficial ownership network: • Entities: Corporations, trusts, and various legal arrangements. • Persons: Individuals who own, control, or benefit from entities.
• Relationships: Connections representing interests between an entity and a stakeholder.
The proposed approach transforms corporate networks into directed graphs, resolving cycles using
nested functions and geometric series. This method reduces the complexity of ownership structures
through mathematical modeling, algorithm design, and experimental validation (see Figure 2) [
Based on BODS representation, the model uses graph theory to represent corporate structures:
• Nodes: Individuals or legal entities.
• Edges: Ownership relationships, weighted by ownership percentages, forming a graph =
(, , ), where = { ∪ }, ⊆ × , and : → [
To handle these structures:
– Nested functional equations model recursive ownership by propagating contributions
through the network. This involves a recursive definition where a function depends on
itself, such as () = + (()) [
As the name suggests, explore as deeply as possible in a graph. In our model, DFS traverses weighted
edges to identify ownership paths to UBOs [
When applying a DFS algorithm to a corporate structure graph to determine the real ownership of individuals as shareholders in a reference company, we find that open paths represent ownership links between a shareholder and an entity. Direct ownership is a direct connection, calculated by multiplying ownership percentages. Indirect ownership involves intermediaries and is the total of multiplied percentages along each indirect path (see Figure 3).
Consider two companies (1, 2) and UBOs 1, 2 (see Figure 3). Applying DFS for 1: • UBO 1: – Open path 1 → 1, with direct ownership of 1 – Open path 1 → 2 → 1, with indirect ownership of 4 2 – Open path 2 → 2 → 1, with indirect ownership of 3 2
Closed paths (cycles) occur exclusively among entities, introducing mathematical complexity through recursivity. This manifests in two key efects: (1) individual-to-entity edges maintain fixed ownership percentages, while (2) entity-to-entity edges generate converging sums via geometric series, with values distributed to individuals according to defined edge weights.
Consider two companies (1, 2) with a Cycle 1 → 2 → 1 and UBOs 1, 2 (Figure 4).
– Open path 1 → 1, with direct ownership of 1 – Open path 1 → 2 → 1, with indirect ownership of 4 2
– Open path 2 → 2 → 1, with indirect ownership of 3 2
A closed path 1 → 2 → 1, introduces recursivity, requiring 2 and 5, to be computed as convergent sums via geometric series, forming nested functions: 2 = 2(3 + 4 + 5) 5 = 5(1 + 2)
1 = 1 + 2 1 = 1 + 2(3 + 4) + 25 1 = 1 + 2(3 + 4) + 25( 1 + 2)
Substituting 5 from (2) and rearranging from (2) to (3): Substituting 5: This extends to , where → ∞: Since (1 + 2) repeats in (4), it can be substituted as a nested function: 1 = 1 + 2(3 + 4) + 25( 1 + 2(3 + 4) + 25(1 + 2)) 1 = (1 + 2(3 + 4))(1 + 25 + (25)2) + · · · + ( 25) + (25)+1(1 + 2)
The last term converges to zero because (25)+1 < 1 The remaining terms form a geometric series: (25)+1(1 + 2) = 0
(1 + 25 + (25)2 + · · · + ( 25)) = (1 − 25)−1 1 = (1 + 2(3 + 4))(1 − 25)−1 (1 + 2(3 + 4))−1 = (1 − 25)−1
• UBO 1: (1 + 24)(1 + 2(3 + 4))( − 1) • UBO 2: (23)(1 + 2(3 + 4))( − 1)
The Total Ownership of each UBO is the sum of the product of ownership percentages along each independent path, divided by the sum of all independent paths for all UBOs multiplied by their ownership percentages. This reduces the complexity order to ( + ).
(9) (10) (11)
Algorithm Total Ownership (): Input: Weighted directed graph = (, , ) Output: Table of UBOs and their total ownership 1. Initialize as empty 2. For each node ∈ : 3. Run DFS from to identify paths and cycles 4. For each acyclic path from to : 5. Compute = ∏︀ () for ∈ 6. Compute the cyclic coefficient as the sum of Direct and Indirect
Ownership 7. Compute Total Ownership for each UBO 8. Return T
Consider three companies (1, 2, 3) with a Cycle 1 → 3 → 2 → 1 and UBOs 1, 2, 3 (Figure 5). Applying DFS for 1: • UBO 1
– Open path 1 → 1, with direct ownership of 1. • UBO 2
– Open path 2 → 2 → 1, with direct ownership of 4 2. • UBO 3
– Open path 3 → 3 → 1, with direct ownership of 6 5. • UBO 1: 1(1 + 42 + 65)−1 • UBO 2: 42(1 + 42 + 65)−1 • UBO 3: 65(1 + 42 + 65)−1
Consider three companies (1, 2, 3) with a Cycle 1 → 2 → 1, 1 → 3 → 1 and UBOs 1, 2, 3 (Figure 6). Applying DFS for 1: • UBO 1 • UBO 2 • UBO 3 – Open path 1 → 1, with direct ownership of 1. – Open path 2 → 2 → 1, with direct ownership of 4 2. – Open path 3 → 3 → 2 → 1, with direct ownership of 6 3 2.
– Open path 3 → 3 → 1, with direct ownership of 6 5.
• UBO 1: 1(1 + 42 + 632 + 65)−1 • UBO 2: 42(1 + 42 + 632 + 65)−1 • UBO 3: (632 + 65)(1 + 42 + 632 + 65)−1
The proposed graph-based model tackles the tricky computational challenges of identifying Ultimate
directed graphs and cleverly handling recursive cycles through geometric series, this approach turns what could be an NP-hard problem into one that operates with linear complexity (( + )). Utilizing Depth-First Search (DFS) allows for a thorough exploration of ownership paths, while the underlying mathematical framework ensures accurate calculations of total ownership. Plus, the model’s alignment with the Beneficial Ownership Data Standard (BODS) makes it even more relevant for regulatory and compliance purposes.
Accurately identifying Ultimate Beneficial Owners (UBOs) in complex corporate structures is essential for financial transparency and combating economic crime.
with international standards such as the Beneficial Ownership Data Standard (BODS), making it suitable for regulatory and compliance applications.
When it comes to future research, there are some exciting areas to explore. We could look into how to integrate financial risk detection, tackle the challenges of cross-border data inconsistencies, and ifne-tune algorithms for real-time analysis. It would also be beneficial to validate findings across various economic sectors and examine how legal representatives, proxies, and other types of beneficiaries influence the graph model.