=Paper= {{Paper |id=Vol-3135/EcoFinKG_2022_paper7 |storemode=property |title=Ownership Graphs and Reasoning in Corporate Economics |pdfUrl=https://ceur-ws.org/Vol-3135/EcoFinKG_2022_paper7.pdf |volume=Vol-3135 |authors=Davide Magnanimi,Michela Iezzi |dblpUrl=https://dblp.org/rec/conf/edbt/MagnanimiI22 }} ==Ownership Graphs and Reasoning in Corporate Economics== https://ceur-ws.org/Vol-3135/EcoFinKG_2022_paper7.pdf

Ownership Graphs and Reasoning in Corporate Economics
Davide Magnanimi1,2 , Michela Iezzi1
1
Banca d’Italia
2
Politecnico di Milano, Milan, Italy

Abstract
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 offer 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 efficient deductive rules in
the Vadalog language, allowing us to obtain a trade-off between computational time and expressive power compared to
standard query languages.

1. Introduction Contribution and Overview1 . In this paper, we illus-
trate 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 com-
ily. 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 𝑥 owns of a company 𝑦, the Vadalog language, that allows both to have a com-
while nodes are companies and people. Company graphs pact encoding of the problem and to address an efficient
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 Com-
the database’s vastness and depth, it has been shown [1] pany Ownership Graph 𝐺(𝑁, 𝐸, 𝑤) is a directed weighted
that many of the issues of interest are difficult to tackle graph, such that:
with standard query languages. However, they can be
succinctly expressed as reasoning rules. • 𝑁 = {𝑝0 , . . . , 𝑝𝑛 } is a set of nodes;
• 𝐸 a set of edges of the form (𝑖, 𝑗), from node 𝑖 to
node 𝑗;
Published in the Workshop Proceedings of the EDBT/ICDT 2022 Joint
Conference (March 29-April 1, 2022), Edinburgh, UK • 𝑤 : 𝐸 → R, 𝑤 ∈ (0, 1] is a total weight function
$ davide.magnanimi@polimi.it (D. Magnanimi); for edges.
michela.iezzi@bancaditalia.it (M. Iezzi)
1
0000-0002-6560-8047 (D. Magnanimi) The views and opinions expressed in this paper are those of the
© 2022 Copyright for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0). authors and do not necessarily reflect the official policy or position
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org) of Banca d’Italia.
(a) A simple indirect ownership. (b) Indirect ownership with a self- (c) Indirect ownership with a
loop. strongly connected component.
Figure 1: Cases of integrated ownership. Nodes with letters are people, nodes with numbers are companies, solid edges are
direct ownership relationships while dashed pink edges represent integrated ownership.

The weight 𝑤(𝑝𝑖 , 𝑝𝑗 ) is the weight of edge (𝑖, 𝑗); an in the work of [2], and various approaches can be found.
edge (𝑖, 𝑗) exists if and only if 𝑤(𝑝𝑖 , 𝑝𝑗 ) ̸= 0; further-As mentioned in a recent work [3], one could interpret
more self-loops are allowed, i.e. 𝑖 = 𝑗. In our context, Integrated Ownership as a notion of cumulative flow from
nodes represent companies or people, edges (𝑖, 𝑗) repre- one target company to another. Another way to see this
sent ownership with share 𝑤(𝑝𝑖 , 𝑝𝑗 ). For ease of repre- problem is to think at the cash flow when dividends of
sentation, we denote 𝑤(𝑝𝑖 , 𝑝𝑗 ) = 𝑤(𝑖, 𝑗). a company 𝑦 are distributed backwards and recursively
In this paper, we formalize the Integrated Ownership, to all its shareholders. The actual percentage of the div-
the Company Control, Ultimate Controller and Close Links idends received by a shareholder 𝑥, part of the same
problems by encoding them as sets of reasoning rules in ownership structure of 𝑦, is equal to the accumulated
the Vadalog language. Vadalog is based on the Warded ownership from a company 𝑥 to a company 𝑦.
Datalog± family that generalizes Datalog by allowing Figure 1 shows three cases of ownership graphs. Fig-
the existential quantification in the rule head while guar- ure 1a shows that company 𝐴 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 com-
order sentence of the form ∀𝑥 ¯ ∀𝑦¯(𝜙(𝑥
¯ , 𝑦¯) → ∃𝑧¯ 𝜓(𝑥 pany 2. Then, such last dividends are distributed again
¯ , 𝑧¯)),
where 𝜙 (the body) and 𝜓 (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 𝐴. Therefore, 𝐴 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 𝐴 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 firm 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 𝐴 is greater because the number of
process. In the formalizations that will follow in this shares effectively 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
In the realm of complex global economic systems, it ap-
relationships (e.g., 𝐴 → 2, 𝐴 → 3, 𝐴 → 4) that flow
pears evident that companies could not be considered as
through the cycle.
stand-alone entities. The concept of Integrated Ownership
helps quantify the ownership involvement of companies
in complex economic structures such as networks and 3.1. Definitions of Paths and Convergence
conglomerates. While the simple notion of ownership Integrated Ownership is at the basis of all the subsequent
identifies the direct connection from a company x to a use cases that we will present in the remainder of the
company y, the Integrated Ownership encompasses the paper, such as company control. A possible approach
accumulated ownership from a company 𝑥 to a company for the Integrated Ownership computation is the one that
𝑦, considering all the current ownership along with all aims at formalizing the definition of directed paths in
direct and indirect links. The Integrated Ownership prob- the company ownership graph. This allows defining Bal-
lem has been extensively investigated in the literature, as done ownership, which we will also refer to as Integrated
(a) (b) (c)
Figure 2: Sample ownership graphs where 𝐴, directly and indirectly,controls other nodes. Nodes are entities; solid edges are
direct ownerships; dashed green edges are control relationships.

Ownership. First, let us define the Directed Path. for all (𝑠, 𝑡) ∈ 𝐸, and (ii) if certain topological conditions
are present that are peculiar to the company ownership
Definition 3.1 (Directed Path). A directed path 𝑃 is graphs we dealt with.
a finite or infinite sequence [𝑝1 , . . . , 𝑝𝑘 ] of nodes in 𝑁
such that (𝑖, 𝑖 + 1) ∈ 𝐸 for every 𝑖 = 1, . . . , 𝑛. For Definition 3.5. The Baldone ownership 𝒪𝐺 (𝑠, 𝑡) of a
a node 𝑝𝑖 ∈ 𝑁 , we call 𝛿 + (𝑖) the set of edges of 𝐸 company 𝑠 on a company 𝑡 converges if 𝒪𝐺 (𝑠, 𝑡) ≤ 1.
incoming into 𝑝𝑖 and 𝛿 − (𝑖) the set of edges of 𝐸 outgoing Theorem 3.1. For a given company ownership graph
from 𝑝𝑖 . We define the weight 𝑤(𝑃 ) of a path 𝑃 as 𝐺(𝑁, 𝐸, 𝑤), the Baldone ownership 𝒪𝐺 (𝑖, 𝑗) converges
𝑤(𝑃 ) = Π(𝑝𝑖 ,𝑝𝑗 )∈𝑃 𝑤(𝑝𝑖 , 𝑝𝑗 ). for all (𝑖, 𝑗) ∈ 𝐸 if and only if for each strongly con-
nected component∑︀𝑆 of 𝐺, there exists at least one node
A second step is the definition of the set of directed 𝑝𝑖 ∈ 𝑆 such that (𝑗)∈𝛿+ (𝑝𝑖 ) 𝑤(𝑖, 𝑗) ≤ 1.
paths whose weight is higher than a fixed 𝜖 threshold.
This allows restricting the set of interests of all exist- The proof of the theorem and further insights are be-
ing directed paths. To this purpose, we introduce the yond the scope of this paper.
𝜖−Baldone path.
3.2. The Matrix Approach
Definition 3.2. An 𝜖−Baldone path 𝑃 from 𝑠 to 𝑡 is a
The computation of the Baldone ownership of a company
path [𝑠, 𝑝1 , . . . , 𝑝𝑘 , 𝑡] such that 𝑠 ̸= 𝑝𝑖 for 𝑖 = 1, . . . , 𝑛
and 𝑤(𝑃 ) > 𝜖, with 𝜖 ∈ R+ and 0 < 𝜖 ≤ 1. Further- 𝑠 over 𝑡 can be obtained in closed-form by approximation
more, we denote the weight of an 𝜖−Baldone path as over powers of adjacency matrix 𝑊 . It is known that the
𝑤𝜖 (𝑃 ). 𝑟-power of 𝑊 gives all the path of length 𝑟 of the graph
𝐺; for example, in cell 𝑖, 𝑗 of matrix 𝐴2 we have the sum
We are now ready to define the 𝜖-Baldone ownership, of the weight of path of length 2 and so on. If we sum
i.e., the summation of all the possibly infinite 𝜖-Baldone all the matrices, we have the sum of the accumulated
paths from 𝑠 to 𝑡. ownership of all the paths in each cell leading from node
Definition 3.3. The 𝜖-Baldone ownership of a company 𝑖 to node 𝑗:
𝑠 on a company 𝑡 in a∑︀ graph 𝐺 is a function 𝒪𝜖𝐺 (𝑠, 𝑡) : 𝑛−1
∑︁ 𝑖
(𝑠, 𝑡) → R defined as 𝑃𝑖 ∈𝐵𝜖 𝑤𝜖 (𝑃𝑖 ), where 𝐵𝜖 is the 𝑊 + 𝑊2 + 𝑊3 + ... = 𝑊 (1)
set of all possible 𝜖-Baldone paths from 𝑠 to 𝑡. 𝑖=1
We have to exclude initial cycles; our Baldone ownership
Its generalization, the Baldone ownership is obtained of company 𝑠 over company 𝑡 can be written as:
by letting 𝜖 → 0 in the definition of 𝜖-Baldone ownership. ∑︁
This latter is our Integrated ownership. 𝒪𝐺 (𝑠, 𝑡) = 𝑤(𝑠, 𝑡) + 𝑤
ˆ (𝑠, 𝑘)𝑤(𝑘, 𝑡) (2)
𝑘̸=𝑠
Definition 3.4. The Baldone ownership of a company As in [2], Equation 2 can be manipulated into the follow-
𝑠 on a company 𝑡 in a graph 𝐺 is a function 𝒪𝐺 (𝑠, 𝑡) : ing form:
(𝑠, 𝑡) → R defined as lim𝜖→0 𝒪𝜖 (𝑠, 𝑡).
𝒪𝐺 = (𝐼 − 𝑑𝑖𝑎𝑔(𝒪𝐺 ))𝑊 + 𝒪𝐺 𝑊 (3)
The convergence of Baldone ownership or, in the fol- that can be solved with respect to 𝒪𝐺 as:
lowing, Integrated Ownership is essential its computation.
We give two theorems that assure their convergence is 𝒪𝐺 = (𝑑𝑖𝑎𝑔((𝐼 − 𝑊 )−1 )−1 (𝐼 − 𝑊 )−1 𝑊 (4)
guaranteed: (i) if it converges for 𝐺 if 𝒪𝐺 (𝑠, 𝑡) converges More details can be found in [2].
(a) Ultimate Control in a control (b) Ultimate Control with two (c) Close Links scenario.
chain. intermediary controlled com-
pany.
Figure 3: Sample ownership graphs with Ultimate Control and Close Link scenarios. Nodes are entities; solid edges are direct
ownerships; dashed edges represent different types of relationships: green stands for control; yellow for ultimate control; pink
for integrated ownership and blue for close links relationships.

3.3. The Reasoning Approach Central Banks, are all concerned with the company con-
trol 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 [2] that, there exists a one-to-one correspon-
operations. These operations are known [4, 5] to become
dence between voting rights and company shares.
more and more computationally expensive as the matrix
Control can be direct or indirect. A direct control occurs
size increases. Ownership graphs collect information of
when 𝑥 directly owns the majority of the shares of 𝑦 (i.e.,
many companies, typically at a national and even at an
it is a shareholder of 𝑦). An indirect control occurs when
international level, so the matrix approach may be unsuit-
𝑥 controls, directly or indirectly, a group of companies
able in many cases. For this reason, we provide a more
that collectively own the majority of the shares of y. This
computationally efficient approach based on reasoning
latter is a recursive definition of the company control
rules while keeping the problem formulation compact.
and makes its computation by no means trivial.
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 al-
Own(𝑥, 𝑦, 𝑤), 𝑤 > 𝜖, 𝑣 = sum(𝑤), ready introduced in the logic and database literature [6]
𝑝 = [x,y] → IOwn(𝑥, 𝑦, 𝑣, 𝑝). (1) and also adopted in technical contexts [7].
Definition 4.1 (Company Control). A person (or a com-
IOwn(𝑥, 𝑧, 𝑤1 , 𝑝1 ), IOwn(𝑧, 𝑦, 𝑤2 , 𝑝2 ),
pany) 𝑥 controls a company 𝑦, if: (i) 𝑥 directly owns more
𝑝 = 𝑝1 |𝑝2 , BPath(𝑝, 𝑣, 𝜖), 𝑣 = sum(𝑤1 × 𝑤2 ), than 50% of 𝑥; or, (ii) 𝑥 controls a set of companies that
→ IOwn(𝑥, 𝑦, 𝑣, 𝑝). (2) jointly (i.e., summing their shares), and possibly together
with 𝑥 itself, own more than 50% of 𝑦.
In Rule 1, whenever the amount of shares of company 𝑦
held (through direct ownership) by 𝑥 exceeds the thresh- In Figure 2a, a straightforward case of direct control is
old 𝜖, then path p is a valid 𝜖-Baldone path and v is the shown: node 𝐴 directly owns more than the majority of
weight of the direct path 𝑝 from 𝑥 to 𝑦. Instead, in Rule the shares of company 3. In Figure 2b, through the direct
2, we can compose the integrated ownership from 𝑥 to 𝑧 possession of 30% of the share of company 3, 𝐴 cannot
and the one from 𝑧 to 𝑦 if for the entire path p (the sym-exert control. However, 𝐴 directly controls company
bol "|" denotes the path concatenation operator), from 1, which owns 31% of company 3. Together with the
x to y, the Definition 3.3 holds. The integrated owner- direct share 𝐴 → 3, 𝐴 therefore also controls 31% owned
ship is increased by the product of the two paths weights by company 1, totalling 61% of the share of company 3
(i.e. 𝑤1 × 𝑤2 ). The extensional atom BPath represents controlled by 𝐴. The case shown in Figure 2c is even
whether Definition 3.2 holds. more complex. 𝐴 controls company 1 by directly owning
more than 50% of its total equity. With the contribution
of the share that 1 owns of 2, 𝐴 acquires indirect control
4. Company Control over company 2. Also, 𝐴 indirectly controls company 3
by contributing shares owned by 2. Finally, 𝐴 controls 4
Banks, financial intelligence units, financial intermedi-
even though it does not own any direct share. In fact, the
aries, regulatory and supervisory authorities, such as
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 𝐴 controls the three intermediate companies, 𝐴 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 𝐴 exerts direct
Definition 4.1 can be formulated a set of compact Vada- control. 𝐴 is the head of the three simple control chains
log reasoning rules. (i.e., 𝐴→1, 𝐴→3, 𝐴→2), so he also assumes the role of
ultimate controller. In general, whenever an individual
Company(𝑥) → Control(𝑥, 𝑥) (1) has control over a firm, it is also its ultimate controller.
In fact, by definition, no natural person can be owned,
Control(𝑥, 𝑦), Own(𝑦, 𝑧, 𝑤),
in any percentage, by another entity in the graph and
𝑣 = sum(𝑤, ⟨𝑦⟩), 𝑣 > 0.5 neither controlled.
→ Control(𝑥, 𝑧) (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 𝑥 on 𝑧 Control(𝑥, 𝑦) → Controlled(𝑦) (1)
by summing the shares of 𝑧 owned by companies 𝑦, over
Control(𝑥, 𝑦), 𝑛𝑜𝑡 Controlled(𝑥) → UltC(𝑥, 𝑦) (2)
all companies 𝑦 controlled by 𝑥 (Rule 2).
The formalization of the company control problem as a We collected all companies y that appears as controlled
Vadalog reasoning task has been tested for performance company in any control relationships (Rule 1). Then, we
both on real data (i.e. the Italian company graph) and define the ultimate controller x for the firm y as the one
synthetic graphs [8]. that has the control over y but, in turn, it is not controlled
by any other company (Rule 2).
5. Ultimate Controller
Since control over a firm can also be obtained indirectly,
6. Close Links
it is not always the case that a firm’s parent is necessarily In the context of creditworthiness evaluation, the prob-
independent in exerting control over the firm. In the lem of collateral eligibility takes on particular relevance.
financial world, in fact, there exist situations (e.g., typi- It involves calculating the risk of granting a specific loan
cally for business groups) of chains of control in which a to a firm 𝑥 that is backed by collateral issued by another
single individual or firm resides on top of it. This subject company 𝑦. The Eurosystem provides credit only against
is defined as the Ultimate Controller for all the companies adequate collateral, i.e., only if eligible [10]. European
of the chain. In fact, it is the only one who is able to Central Bank regulations [11] for monetary policy define
push his or her own decisions independently across the a set of criteria that National Central Banks must adopt
underlying firms in the chain of control. to assess the eligibility of specific assets. For instance,
In the economic literature [9], the Ultimate Controller for accessing the credit, National Central Banks of the
problem is formally defined as follows: Eurosystem do not allow a counter-party 𝑥 to submit
Definition 5.1 (Ultimate Controller). Given a company a collateral issued by a guarantor entity to which it is
𝑦, an investor 𝑥 (either a company or individual) is said linked via a close links relationship. A close links situation
to be the ultimate controller of 𝑦 if: (i) 𝑥 is the head of is defined as follows:
a chain of companies among which there is 𝑦; and, (ii) Definition 6.1 (Close Links). A counter-party 𝑥 is in
𝑥 directly or indirectly controls all the companies in the a close link relationship with its guarantor 𝑦 if: (i) the
chain without being controlled by any other investor. total, either direct or indirect, ownership of 𝑦 held by 𝑥
Two examples of the ultimate controller relationships is above 20% of the equity of 𝑥; or, the vice-versa, (ii) the
are shown in Figures 3a and 3b. In both scenarios, in- total accumulated ownership of 𝑥 held by 𝑦 is above 20%
dividual 𝐴 has direct or indirect control over all other of the equity of 𝑦; or finally, (iii) a common third party
firms. In Figure 3a, company 1 directly controls company 𝑧 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 𝑥 and the guarantor 𝑦.
is part of the chain of control (i.e., 𝐴 → 1 → 2) but The definition is based on the concept of total owner-
is not at the top of that chain. Therefore, the ultimate ship that a company 𝑥 owns both directly and indirectly
controller in this scenario is the shareholder 𝐴 since any of another one. That is the definition of integrated own-
2
This formalization of the base case is slightly different from ership that we introduced in Section 3.
the natural definition but commonly assumed in the literature as it A sample ownership graph for illustrating the close
is more compact and formally equivalent. 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 eligi-
ship 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 ade-
ownership 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 efficient and compact encoding in the form of deduc-
integrated 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
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IOwn(𝑥, 𝑦, 𝑞), 𝑥 ̸= 𝑦, 𝑞 ≥ 0.2 → CLinks(𝑥, 𝑦) (1) D. Magnanimi, A. Muci, E. Sallinger, Covid-19
and company knowledge graphs: assessing golden
IOwn(𝑦, 𝑥, 𝑞), 𝑥 ̸= 𝑦, 𝑞 ≥ 0.2 → CLinks(𝑥, 𝑦) (2) powers and economic impact of selective lockdown
via ai reasoning, arXiv preprint arXiv:2004.10119
IOwn(𝑧, 𝑥, 𝑞), IOwn(𝑧, 𝑦, 𝑤), (2020).
𝑥 ̸= 𝑦, 𝑞 ≥ 0.2, 𝑤 ≥ 0.2 [4] A. J. Stothers, On the complexity of matrix multi-
→ CLinks(𝑥, 𝑦), CLinks(𝑦, 𝑥) (3) plication (2010).
[5] A. M. Davie, A. J. Stothers, Improved bound for
There is a two-way correspondence between the rea- complexity of matrix multiplication, Proceedings
soning rules shown above and the points in the Defini- of the Royal Society of Edinburgh Section A: Math-
tion 6.1, hence the interpretation of the rules is quite ematics 143 (2013) 351–369.
straightforward. However, with Rule 1 we derive the ex- [6] S. Ceri, G. Gottlob, L. Tanca, Logic programming
istence of a close links relationship between 𝑥 and 𝑦 if the and databases, Springer, 2012.
total (i.e. integrated) ownership of 𝑦 held by 𝑥 is equal or [7] A. Gulino, S. Ceri, G. Gottlob, E. Sallinger, L. Bel-
greater than 20%. The second rule is symmetrical to the lomarini, Distributed company control in company
first one and generates a close link relationship between shareholding graphs (to appear), in: ICDE, 2021.
𝑥 and 𝑦 if the accumulated ownership of 𝑥 held by 𝑦 [8] L. Bellomarini, D. Magnanimi, M. Nissl, E. Sallinger,
is more than 20%. Finally, Rule 3 considers the last de- Neither in the programs nor in the data: Mining
scribed scenario in which a common third party 𝑧 owns the hidden financial knowledge with knowledge
(either directly or indirectly) more than 20% of both 𝑥 graphs and reasoning, in: Workshop on Mining
and 𝑦. Data for Financial Applications, Springer, 2020, pp.
119–134.
[9] O. Staff, OECD handbook on economic globalisa-
7. Conclusion tion indicators, OECD, 2005.
[10] E. C. Bank, The use of credit claims as col-
Company ownership graphs are helpful in many recur- lateral for eurosystem credit operations, 2013.
rent problems in the financial domain. One interesting URL: https://www.ecb.europa.eu/pub/pdf/scpops/
problem is the Integrated Ownership problem, where the ecbocp148.pdf.
goal is to determine the accumulated ownership in com- [11] E. C. Bank, Guideline (eu) 2015/510 of the european
plex economic entities. In the Company Control problem, central bank of 19 december 2014 on the implemen-
the focus is on finding which entity controls a company tation of the eurosystem monetary policy frame-
of interest. An even more challenging problem is the work (ecb/2014/60), 2014. URL: http://data.europa.
Ultimate Controller problem, where it is requested to indi- eu/eli/guideline/2015/510/oj.
viduate the head of a chain of companies. The Close Links
computation between two entities, the counter-party, and