View source for Adjacency Matrix of the Shareholding Graph

=== Introduction ===
The adjacency matrix serves as a fundamental tool in analyzing the shareholding structures of multinational enterprises (MNEs). It provides a detailed representation of ownership percentages, facilitating the understanding of complex corporate networks.

=== Theoretical Background of Adjacency Matrix ===

==== Definition and Structure ====
In the context of MNE shareholding, an adjacency matrix is a square matrix where each element ''a''(''i'',''j'') denotes the percentage of shares entity ''i'' holds in entity ''j'', represented as a decimal between 0 and 1.

==== Properties ====

* Column Sum: The sum of values in each column equals 1, when indicating complete ownership distribution among entities.
* Directionality: Typically, the matrix is asymmetric, reflecting the directional nature of shareholding relationships.

==== Application in MNE Shareholding Structures ====

===== Representing Ownership =====
The adjacency matrix quantifies ownership relationships within an MNE, revealing the distribution of ownership for each entity.

===== Example of an Adjacency Matrix =====
Consider a simplified MNE with entities A, B, C, and D. Their shareholding relationships are:

* A owns 50% of B and 30% of D.
* B owns 20% of D.
* C owns no other entities.
* D owns 30% of A.

The adjacency matrix is:
{| class="wikitable"
!
!A
!B
!C
!D
|-
|A
|0
|0.5
|0
|0.3
|-
|B
|0
|0
|0
|0.2
|-
|C
|0
|0
|0
|0
|-
|D
|0.3
|0
|0
|0.1
|}

=== Interpreting the Adjacency Matrix ===

==== Identifying Parent-Subsidiary Relationships ====

* Single Parent: Entity B has a single parent, A, since only one non-zero entry (0.5) is present in its column.
* Multiple Parents: Entity D has multiple parents, A (0.3), B (0.2) and itself D (0.1), indicating ownership by three entities.

==== Detecting Subsidiaries ====

* Single Subsidiary: Entity B has a single subsidiary, D, as indicated by the single non-zero entry (0.2) in its row.
* Multiple Subsidiaries: Entity A has multiple subsidiaries, B and D, as it has two non-zero entries (0.4 and 0.2) in its row.

==== Finding Self-Ownership links ====

* Self-ownership: Entity D owns shares in itself, this situation is easily spored with values located on the diagonal of the Matrix here (0.1).

==== Finding Isolated Entities ====

* Isolation: Entity C is isolated in terms of ownership, with no entities owning its shares and it owning no shares in others, indicated by zeroes in both its row and column.

==== Identifying Missing Links ====

* Missing Links: The sum of the matrix columns do not reach 1, denoting the fact that the Adjacency Matrix lacks several ownership links that may be owned by entities not listed.

=== Advantages of Adjacency Matrix Representation in Computational Efficiency ===
The use of an adjacency matrix for representing shareholding structures in multinational enterprises provides a gateway to employing advanced mathematical algebra, particularly matrix products and sums, to facilitate and expedite computations. This approach leverages the efficiency and speed of computers in handling matrix operations, a cornerstone in linear algebra.

==== Matrix Products and Sums ====
Matrix multiplication is a fundamental operation that, when applied to adjacency matrices, can reveal complex multi-step relationships within a corporate network. For example, by multiplying the matrix by itself (squaring), one can uncover relationships two steps away, such as indirect ownership or influence. Continuing this process allows for the exploration of progressively distant connections across the network.

Similarly, matrix sums can be used to aggregate different levels of connections, providing a comprehensive view of the network's structure. These operations are particularly efficient on computers, which are optimized for handling large-scale matrix calculations.

==== Geometric Series and Convergence ====
The concept of geometric series in the context of adjacency matrices is especially relevant for analyzing shareholding structures with recursive or looping ownership patterns. By summing the powers of the adjacency matrix (first power, square, cube, etc.), it is possible to approximate the total influence exerted by each entity over time, considering both direct and indirect ownership. This process, akin to constructing a geometric series, converges to a limit that represents the complete picture of influence within the network.

This is the initial step toward the full Baldone ownership computation and its correction term, that we use for the [[Indirect ownership computation using Baldone ownership]].

==== Computational Efficiency ====
Modern computers excel at performing matrix operations, making them ideal for handling large and complex adjacency matrices. Algorithms for matrix multiplication and addition are highly optimized in computational linear algebra, allowing for rapid processing of data. This efficiency is crucial when dealing with the extensive and intertwined shareholding structures typical of multinational corporations, where manual analysis would be prohibitively time-consuming and prone to errors.

By harnessing the power of matrix algebra and the computational efficiency of modern computing, adjacency matrices serve as a potent tool in corporate analysis, revealing intricate patterns and relationships within a corporate structure efficiently and accurately.

This computational efficiency allowed us to optimize the performance of our algorithms.

=== Conclusion ===
The adjacency matrix is an invaluable tool for unraveling the complexities of MNE shareholding structures. It allows for a clear understanding of direct ownership relationships, identification of parent-subsidiary dynamics, detection of isolated entities, and recognition of missing ownership links. This level of analysis is critical in the context of corporate governance, financial analysis, and regulatory compliance.