In corporate economics, the use of company ownership graphs has become instrumental in solving many critical problems for central banks, financial regulators, and national statistics agencies. In particular, National Central Banks (NCBs) treat and, sometimes, own company data for their key institutional goals in a variety of fields, e.g., anti-money laundering, or economic and statistical research. This paper aims at leveraging our experience with Automated Reasoning in Banca d'Italia, focusing on four real use cases typical in the financial domain: (i) Integrated Ownership, (ii) Company Control, (iii) Ultimate Controller, and (iv) Close Links. For each problem, we ofer a formalization, providing practical and real-world examples based on Bank of Italy's company ownership graph. Finally, we express each problem in the form of compact and eficient deductive rules in the Vadalog language, allowing us to obtain a trade-of between computational time and expressive power compared to standard query languages.
Contribution and Overview1. In this paper, we
illustrate four problems, i.e. Integrated Ownership, Company
Company ownership graphs are critical items in corpo- Control, Ultimate Controller and Close Links. These are
rate economics, with central banks, financial regulators, recurrent problems in the financial domain of company
and national statistics agencies relying on them heav- ownership. For each such problem, we report the
comily. The essential notion in these graphs is ownership: monly accepted definition, and we present and describe
edges are ownership links labelled with the proportion a possible formalization, in the form of deductive rules in
of shares a business or person owns of a company , the Vadalog language, that allows both to have a
comwhile nodes are companies and people. Company graphs pact encoding of the problem and to address an eficient
are employed in various contexts, including calculating a solution to real and concrete problems and interests for
company’s total ownership of another, (chains of) control our Institution. The remainder of the paper is organized
relationships, collusion phenomena, collateral eligibility, as follows. In Section 2 we introduce the background
etc. National Central Banks (NCBs) deal with company of company ownership graph representations, as well
data in order to achieve key institutional goals in a variety as the Vadalog approach. In Section 3 we present the
of fields, including banking supervision, credit-worthiness Integrated Ownership concept. In Section 4 we define and
evaluation, anti-money laundering, insurance fraud de- give rules for the Company Control problem. Section 5
tection, economic and statistical research, and more. The describes the formalization for the Ultimate Controller
Bank of Italy, as a supervisory authority, is intensely in- problem, while in Section 6 we investigate the Close Links
terested in studying and extracting valuable insights from use case. Section 7 concludes the paper.
the corporate ownership network. The Italian Central
Bank owns the database of Italian companies, provided
by the Italian Chambers of Commerce. It contains high- 2. Preliminaries
quality, fine-grained data of Italian non-listed companies,
including information such as legal name, legal address, To present the use cases of interest, let us introduce some
incorporation date, shareholders, the composition of the general notions that will be used throughout the paper.
company board, historical data, and many others. Despite Definition 2.1 (Company Ownership Graph). A
Comthe database’s vastness and depth, it has been shown [
node ; • : → R, ∈ (0, 1] is a total weight function
for edges.
1The views and opinions expressed in this paper are those of the authors and do not necessarily reflect the oficial policy or position of Banca d’Italia. (a) A simple indirect ownership.
(b) Indirect ownership with a selfloop.
(c) Indirect ownership with a strongly connected component.
The weight (, ) is the weight of edge (, ); an in the work of [
In this paper, we formalize the Integrated Ownership, to all its shareholders. The actual percentage of the divthe Company Control, Ultimate Controller and Close Links idends received by a shareholder , part of the same problems by encoding them as sets of reasoning rules in ownership structure of , is equal to the accumulated the Vadalog language. Vadalog is based on the Warded ownership from a company to a company . Datalog± family that generalizes Datalog by allowing Figure 1 shows three cases of ownership graphs. Figthe existential quantification in the rule head while guar- ure 1a shows that company receives dividends from anteeing decidability and tractability in the presence of company 1 proportionally to the owned shares. In turn, existential quantification and recursion. A rule is a first- company 1 also receives dividends from the profit of comorder sentence of the form ∀¯∀¯( (¯, ¯) → ∃¯ (¯, ¯)) , pany 2. Then, such last dividends are distributed again where (the body) and (the head) are conjunctions among all the shareholders of company 1, which in the of atoms. For brevity, we omit universal quantifiers and example is only . Therefore, will eventually receive denote conjunction by comma. The semantics of a set of a percentage of company 2’s dividends as well, and it is rules is defined by the well-known chase procedure. indicated as the dashed pink edge from to 2. A more
In the reasoning rules, atoms can be either extensional interesting case is shown 1b. The number of shares that (EDB) when they are immediately available in data stores ifrm 1 holds of itself (i.e., the self-loop in the ownership (e.g. relational databases, graph databases, NoSQL stores, graph) are, de facto, removed from those available on RDF stores, etc.) or intentional (IDB) when they are the market. Therefore, the real percentage of shares of generated when needed as a consequence of a reasoning company 1 held by is greater because the number of process. In the formalizations that will follow in this shares efectively available on the market is less than work, we adopt the convention of colouring extensional 100%. The example in Figure 1c is an even more complex atoms in blue and intensional ones in red. scenario. In fact, the ownership relationships realize a strongly connected component, i.e., a cyclical structure, that behaves like self-loops do: it increases the actual 3. Integrated Ownership amount of shares held by the companies involved in the cycle and, therefore, of all the accumulated ownership relationships (e.g., → 2, → 3, → 4) that flow through the cycle.
In the realm of complex global economic systems, it appears evident that companies could not be considered as stand-alone entities. The concept of Integrated Ownership helps quantify the ownership involvement of companies in complex economic structures such as networks and conglomerates. While the simple notion of ownership identifies the direct connection from a company x to a company y, the Integrated Ownership encompasses the accumulated ownership from a company to a company , considering all the current ownership along with all direct and indirect links. The Integrated Ownership problem has been extensively investigated in the literature, as
Integrated Ownership is at the basis of all the subsequent use cases that we will present in the remainder of the paper, such as company control. A possible approach for the Integrated Ownership computation is the one that aims at formalizing the definition of directed paths in the company ownership graph. This allows defining Baldone ownership, which we will also refer to as Integrated (a) (b) (c) such that (, + 1) ∈ for every = 1, . . . , . For a node ∈ , we call +() the set of edges of incoming into and − () the set of edges of outgoing from . We define the
weight ( ) of a path as ( ) = Π (, )∈ (, ).
paths whose weight is higher than a fixed threshold.
ing directed paths. To this purpose, we introduce the − Baldone path.
Definition 3.2. ( ). and ( ) > , with ∈ R + and 0 < ≤ path [, 1, . . . , , ] such that ̸= for = 1, . . . , more, we denote the weight of an − Baldone path as
An − Baldone path from to is a
Definition 3.3.
The -Baldone ownership of a company on a company in a graph is a function (, ) : (, ) → R defined as ∑︀ ∈ set of all possible -Baldone paths from to . (), where is the
This latter is our Integrated ownership. by letting → 0 in the definition of -Baldone ownership. for all (, ) ∈ , and (ii) if certain topological conditions are present that are peculiar to the company ownership graphs we dealt with.
Definition 3.5.
The Baldone ownership (, ) of a company on a company converges if (, ) ≤ 1. Theorem 3.1. For a given company ownership graph (, , ), the Baldone ownership for all (, ) ∈ if and only if for each strongly connected component of , there exists at least one node (, ) converges ∈ such that ∑︀
()∈ +() (, ) ≤ 1.
The computation of the Baldone ownership of a company over powers of adjacency matrix . It is known that the -power of gives all the path of length of the graph ; for example, in cell , of matrix 2 we have the sum of the weight of path of length 2 and so on. If we sum all the matrices, we have the sum of the accumulated ownership of all the paths in each cell leading from node to node : + 2 + 3 + . . . = ∑︁ − 1 =1
of company over company can be written as: (, ) = (, ) + ∑︁ ˆ(, )(, )
̸= (, ) → R defined as lim →0 (, ).
Definition 3.4.
The Baldone ownership of a company on a company in a graph is a function (, ) : ing form: lowing, Integrated Ownership is essential its computation.
The convergence of Baldone ownership or, in the fol- that can be solved with respect to as:
guaranteed: (i) if it converges for if (, ) converges
= ( − ()) +
= ((( − )− 1)− 1( − )− 1
More details can be found in [
(b) Ultimate Control with two intermediary controlled company.
(c) Close Links scenario.
3.3. The Reasoning Approach Central Banks, are all concerned with the company
control problem. It entails determining who takes decisions
Although the matrix approach provides a compact and in a vast corporate network, i.e., who has the majority of
elegant formulation for calculating the Integrated Own- votes for each individual firm as, it is a generally accepted
ership, it relies on matrix multiplication and inversion assumption [
Definitions 3.1-3.4 can be formalized as reasoning rules A formulation of the company control problem that
in the Vadalog language, as follows: follows is a widely accepted model, and it has been
alOwn(, , ), > , = sum(), ready introduced in the logic and database literature [
= [x,y] → IOwn(, , , ). (1)
IOwn(, , 1, 1), IOwn(, , 2, 2), = 1|2, BPath(, , ), = sum(1 × 2),
pany) controls a company , if: (i) directly owns more than 50% of ; or, (ii) controls a set of companies that → IOwn(, , , ). (2) jointly (i.e., summing their shares), and possibly together with itself, own more than 50% of .
In Rule 1, whenever the amount of shares of company held (through direct ownership) by exceeds the threshold , then path p is a valid -Baldone path and v is the weight of the direct path from to . Instead, in Rule 2, we can compose the integrated ownership from to and the one from to if for the entire path p (the symbol "|" denotes the path concatenation operator), from x to y, the Definition 3.3 holds. The integrated ownership is increased by the product of the two paths weights (i.e. 1 × 2). The extensional atom BPath represents whether Definition 3.2 holds.
aries, regulatory and supervisory authorities, such as
In Figure 2a, a straightforward case of direct control is shown: node directly owns more than the majority of the shares of company 3. In Figure 2b, through the direct possession of 30% of the share of company 3, cannot exert control. However, directly controls company 1, which owns 31% of company 3. Together with the direct share → 3, therefore also controls 31% owned by company 1, totalling 61% of the share of company 3 controlled by . The case shown in Figure 2c is even more complex. controls company 1 by directly owning more than 50% of its total equity. With the contribution of the share that 1 owns of 2, acquires indirect control over company 2. Also, indirectly controls company 3 by contributing shares owned by 2. Finally, controls 4 even though it does not own any direct share. In fact, the sum of the shares of 4 owned b 1,2 and 3 is greater than other company or person does not control it. In Figure 3b, 50%. Since controls the three intermediate companies, realizes control over company 2 through the shares it has the majority of the decision-making power over 4. held by companies 1 and 3 over which exerts direct
Definition 4.1 can be formulated a set of compact Vada- control. is the head of the three simple control chains log reasoning rules. (i.e., →1, →3, →2), so he also assumes the role of ultimate controller. In general, whenever an individual Company() → Control(, ) (1) has control over a firm, it is also its ultimate controller.
In fact, by definition, no natural person can be owned, Control(, ), Own(, , ), in any percentage, by another entity in the graph and = sum(, ⟨⟩), > 0.5 neither controlled.
→ Control(, ) (2) The formalization of the ultimate controller problem
can be given starting from the Control intensional rela
The given formulation is recursive. In the base case, tionships derived with the program shown in Section 4. we assume that every company has control on itself (Rule 1) 2. Then, inductively, we define the control of on Control(, ) → Controlled() (1) by summing the shares of owned by companies , over all companies controlled by (Rule 2). Control(, ), Controlled() → UltC(, ) (2)
The formalization of the company control problem as a
Vadalog reasoning task has been tested for performance
both on real data (i.e. the Italian company graph) and
synthetic graphs [
company in any control relationships (Rule 1). Then, we define the ultimate controller x for the firm y as the one that has the control over y but, in turn, it is not controlled by any other company (Rule 2).
Since control over a firm can also be obtained indirectly, it is not always the case that a firm’s parent is necessarily independent in exerting control over the firm. In the ifnancial world, in fact, there exist situations (e.g., typically for business groups) of chains of control in which a single individual or firm resides on top of it. This subject is defined as the Ultimate Controller for all the companies of the chain. In fact, it is the only one who is able to push his or her own decisions independently across the underlying firms in the chain of control.
In the economic literature [
lem of collateral eligibility takes on particular relevance.
It involves calculating the risk of granting a specific loan
to a firm that is backed by collateral issued by another
company . The Eurosystem provides credit only against
adequate collateral, i.e., only if eligible [
the natural definition but commonly assumed in the literature as it is more compact and formally equivalent.
, an investor (either a company or individual) is said to be the ultimate controller of if: (i) is the head of a chain of companies among which there is ; and, (ii) directly or indirectly controls all the companies in the chain without being controlled by any other investor.
a close link relationship with its guarantor if: (i) the total, either direct or indirect, ownership of held by
Two examples of the ultimate controller relationships is above 20% of the equity of ; or, the vice-versa, (ii) the are shown in Figures 3a and 3b. In both scenarios, in- total accumulated ownership of held by is above 20% dividual has direct or indirect control over all other of the equity of ; or finally, (iii) a common third party ifrms. In Figure 3a, company 1 directly controls company owns, either directly or indirectly, 20% or more of the 2 but is not its ultimate controller. In fact, company 1 equity of both the counter-party and the guarantor . is part of the chain of control (i.e., → 1 → 2) but is not at the top of that chain. Therefore, the ultimate controller in this scenario is the shareholder since any
ship that a company owns both directly and indirectly of another one. That is the definition of integrated ownership that we introduced in Section 3.
A sample ownership graph for illustrating the close links scenario is shown in Figure 3c. We consider the pair of firms 1 and 2. It exists an arc of direct owner- the guarantor, allows the evaluation of collateral eligiship that shows the possession of shares of company 2 bility. In this paper, we first described the background, from part of company 1. Not being there other paths of the main definitions, and examples to provide an adeownership between these two companies, the amount quate overview of each of the above problems. Then, we of share directly owned by company 1 is equivalent to formally characterized each problem, and we explained the total amount of share that it owns of company 2 (i.e. the eficient and compact encoding in the form of deducintegrated ownership). Since the total quota exceeds the tive rules in the Vadalog language. The approach based threshold of 20%, in agreement with the given definition, on reasoning rules showed great potential and ease of we can assert that companies 1 and 2 are in relation of adoption in the financial domain. close links. The same considerations apply for the pairs of companies 1-3 and 3-4. Companies 2 and 3 are also in a close link relationship because of the third point of the References definition. In fact, a common third-party (i.e. company 1) owns, considering all the possible direct or indirect paths, more than 20% of the total shares of both the two companies. The Figure 3c also shows that close links relationships are undirected.
We formalize the close links problem as a set of deductive rules whose input (i.e. IOwn) is directly taken from the reasoning rules in Section 3.
IOwn(, , ), ̸= , ≥ 0.2 → CLinks(, ) (1) IOwn(, , ), ̸= , ≥ 0.2 → CLinks(, ) (2)
IOwn(, , ), IOwn(, , ),
̸= , ≥ 0.2, ≥ 0.2 → CLinks(, ), CLinks(, ) (3)
There is a two-way correspondence between the reasoning rules shown above and the points in the Definition 6.1, hence the interpretation of the rules is quite straightforward. However, with Rule 1 we derive the existence of a close links relationship between and if the total (i.e. integrated) ownership of held by is equal or greater than 20%. The second rule is symmetrical to the ifrst one and generates a close link relationship between and if the accumulated ownership of held by is more than 20%. Finally, Rule 3 considers the last described scenario in which a common third party owns (either directly or indirectly) more than 20% of both and .
Company ownership graphs are helpful in many recurrent problems in the financial domain. One interesting problem is the Integrated Ownership problem, where the goal is to determine the accumulated ownership in complex economic entities. In the Company Control problem, the focus is on finding which entity controls a company of interest. An even more challenging problem is the Ultimate Controller problem, where it is requested to individuate the head of a chain of companies. The Close Links computation between two entities, the counter-party, and