#  $\alpha$-Indirect Control in Onion-like Networks

Kirill<EMAIL_ADDRESS>Skolkovo Institute of Science and
Technology, 121205 Moscow, Russia Nikita Pospelov  Institute for Advanced
Brain Studies, Lomonosov Moscow State University, 119234 Moscow, Russia
Dmitriy Skougarevskiy  European University at St. Petersburg, 191187 St.
Petersburg, Russia

###### Abstract

Tens of thousands of parent companies control millions of subsidiaries through
long chains of intermediary entities in global corporate networks. Conversely,
tens of millions of entities are directly held by sole owners. We propose an
algorithm for identification of ultimate controlling entities in such networks
that unifies direct and indirect control and allows for continuous
interpolation between the two concepts via a factor damping long paths. By
exploiting onion-like properties of ownership networks the algorithm scales
linearly with the network size and handles circular ownership by design. We
apply it to the universe of 4.2 mln UK companies and 4 mln of their holders to
understand the distribution of control in the country. Furthermore, we provide
the first independent evaluation of the control identification results. We
reveal that the proposed $\alpha$-ICON algorithm identifies more than 96% of
beneficiary entities from the evaluation set and supersedes the existing
approaches reported in the literature. We refer the superiority of
$\alpha$-ICON algorithm to its ability to correctly identify the parents long
away from their subsidiaries in the network.

##  1 Introduction

Economists have long studied ownership patterns [1, 2], tax haven use by
multinational corporations [3, 4, 5], profit-shifting [6, 7], or how ownership
concentration and cross-ownership affects welfare and risk-taking behaviour
[8, 9, 10, 11]. All this requires foreknowledge of the ultimate controlling
entities in the global ownership network. Even though equity ownership is an
observed variable (cash-flow rights over firms are recorded in public
registers), entity control (the ultimate ability to influence the firm’s
decision-making) has to be inferred from the latent power relations in the
corporate ownership network [12].

A striking universal characteristic of the corporate ownership network is a
bow-tie structure or an onion-like topology [13, 14]. It features the dense
core of cross-owned entities surrounded by consecutive layers of nodes with a
lower degree. Such a structure and communities therein reflect the
concentration of ownership and power [15, 16]. For network science it
exemplifies a real-world system with an explicit hierarchical core-shell
organisation.

In this paper we introduce $\alpha$-ICON (Indirect Control in Onion-like
Networks) — a tuneable algorithm for the identification of the direct/indirect
controlling entities that builds on the onion-like (core-shell) properties of
the ownership network [17]. We approach the problem of control in the paradigm
of the network flow, also known as “control by transitivity”: the weights on
incident edges are simply multiplied by the given connectivity of the oriented
ownership graph. Continuous interpolation between direct and indirect
influence is achieved via a damping factor (penalty) $\alpha$ for the length
of the flow-paths, inspired by Katz centrality [18].

We exploit the properties of the onion-like topology to propose a time-
efficient back-propagation algorithm providing dramatic acceleration for the
task of ultimate control identification. To simplify such a topology, we peel
off the hanging vertices in the network recursively until a subgraph of nodes
with at least one in- and one out-edge remains. This procedure is equivalent
to the 2-core decomposition for undirected graphs [19, 20]. This is a robust
approach that demonstrates the existence of the bow-tie motif [14]. The core-
shell decomposition results in a small core in real-world ownership networks,
four orders smaller than the whole network. Small core is the fundamental
feature of the control network due to restrictions on circular ownership in
some jurisdictions.

Our decomposition naturally sorts all the entities into four fundamental
classes, reflecting their topological role in the network: super-holders
(nodes that are not owned by any other nodes), super-targets (nodes that do
not own any other nodes), intermediaries (nodes that own and are owned by
other nodes), and core entities. Specifically, super-holders and super-targets
are removed at the first step of our iterative peeling procedure, the
intermediaries get removed during the following steps, while the core nodes
represent entities that are persistent to the peeling. We note that the
ultimate owners (i.e. beneficial owners, [21]) are topologically equivalent to
super-holders in the control network, thus, the problem of the ultimate
control identification is addressed on the class of super-holders.

We apply $\alpha$-ICON to the universe of 8.1 mln UK entities and their
controlling participants from the People with Significant Control (PSC)
register maintained by the UK Government. This network has only 517 entities
at its core, enabling us to identify the ultimate controlling entities much
faster than the randomised approximations previously suggested in the
literature.

Finally, we evaluate the performance of $\alpha$-ICON against the existing
approaches to identify the controlling entities. Public companies have to
disclose the information on their controlling interest in subsidiary
organisations to the regulatory agencies. We gather such disclosures of the UK
companies with debt or securities traded on the US financial markets to build
an authoritative list of 1007 subsidiaries and their 279 ultimate controlling
entities. This list allows us to compute the recall @ $k$ of any algorithm:
how many subsidiary organisations have the actual controlling entity among the
top-$k$ controlling entities identified by the algorithm. We show that
$\alpha$-ICON is superior to the previous approaches in terms of recall at any
$k$. This advantage is due to efficient detection of super-holder parents
located far away from their super-target subsidiaries in the network. To the
best of our knowledge, this is the first formal evaluation of the controlling
entity identification algorithms. We invite further research by open-sourcing
our data and code.222http://github.com/eusporg/alphaicon

##  2 Methods

###  2.1 Unifying direct and indirect control

Two literatures have emerged to identify the controlling entities in a
directed graph of participant-entity relations where edge weights are the
corresponding equity shares. Statistical physicists consider the flow of power
and control in such graphs by studying the topological importance of nodes
[22]: centrality [23], PageRank [24, 25], coreness [26, 27], integrated
ownership [28], network control [14], control capacity [29], control
contribution [30], Helmholtz-Hodge potential [31]. Another literature coming
from operations research and political science measures effective control in
ownership networks by treating them as coalitional games and computing power
indices of entities [32, 33]. Randomised approximations such as the Network
Power Index (NPI) have recently appeared and are applied to large-scale
ownership networks [34]. The existing approaches can be sorted into one of the
two classes: they either rely on the direct ownership or aim at the
computation of indirect control propagating via all possible routes in the
graph of the direct ownership. Even though the indirect measures are
considered to better capture the real distribution of control in the ownership
networks, it is still unclear how to balance the contribution coming from the
direct and indirect ownership.

####  2.1.1 Direct control

The equity (ownership) network can be represented as an oriented weighted
network. In this model organisations and their participants resemble $N$ nodes
while the weights of $E$ edges between them reflect the corresponding equity
shares. The edges are oriented due to the non-symmetric nature of the
organisation-participant relationship: each edge $\left(i,k\right)$ is
oriented from participant $i$ to organisation $k$ and has the corresponding
weight $0\leq\mathbf{W}_{ik}\leq 1$. Normalisation of the shares requires that
$\sum_{i}\mathbf{W}_{ik}=1$ for any organisation $k$. By construction, we do
not allow direct self-ownership, $\mathbf{W}_{ii}=0$. Matrix $\mathbf{W}$ is a
non-symmetric adjacency matrix, corresponding to the oriented weighted
network.

Control over an organisation can be defined as voting or cash flow rights that
a participant has in an organisation. As the matrix $\mathbf{W}_{ij}$ reflects
the distribution of direct ownership among the players, it is typically used
to approximate the direct control $\mathbf{A}_{ij}$. At the foundation of this
model lies the one-share-one-vote assumption [35]. Even though this assumption
is broadly accepted in the literature, several other approaches based on
transformations on the original matrix $\mathbf{W}$ to the quorum threshold
and the relative control were proposed in [14].

It is worth mentioning a conceptually separate group of models of the direct
control coming from the game theory [33]. These approaches rely on the
assessment of the influence potential among the shareholders and are based on
calculation of the power indices in coalitions, such as Shapley-Shubik or
Banzhaf [32, 36]. These indices were originally proposed to assess the
relative importance of the amount of votes held by each player in a weighted
voting system. However, these methods have been criticised in the corporate
control literature, since they do not account for widespread heterogeneity of
shareholders distribution [1] and due to issues in handling cycles in
shareholdings [12]. Even though no game-theoretic model has reached universal
adoption in the corporate finance literature, there is no denying that such
models provide a self-consistent and straightforward algorithm to take into
account consolidation of voting rights among direct shareholders.

In what follows we juxtapose the two approaches in our quantitative model of
the direct control: (i) the original matrix of normalised equity shares
$\mathbf{A}=\mathbf{W}$ and (ii) the matrix of the Direct Power Indices (DPI)
being the randomised approximation of the Shapley-Shubik weights,
$\mathbf{A}=\mathbf{W}^{DPI}$.

####  2.1.2 Indirect control

It is well-known that direct relationships are not sufficient to reveal hidden
participants with ultimate control over organisations. Thus, one should take
into account the indirect influence a participant $i$ might exert on the
organisation $j$ through all other participants in the network. In the
paradigm of control by transitivity, the indirect control is proportional to
the net flow from $i$ to $j$ given the network connectivity. This can be
formally defined as the sum of the powers of the adjacency matrix $\mathbf{A}$
of the direct control

$\sum_{l=1}^{\infty}\mathbf{A}^{l}=\mathbf{A}+\sum_{k}\mathbf{A}_{ik}*\mathbf{A}_{kj}+\sum_{k,l}\mathbf{A}_{ik}*\mathbf{A}_{kl}*\mathbf{A}_{lj}+\ldots,$
(1)

where the first term is the contribution from the direct control, the second
term is the contribution of paths of length 2, the third term is the
contribution of paths of length 3 etc. The metric in Eq. 1 was proposed in
[14] for calculation of the network control. However, there are three main
issues with this approach.

First, the series in Eq. 1 is divergent for matrices with the spectral radius
$\rho\left(\mathbf{A}\right)\geq 1$. In particular, this is the case for the
stochastic matrices of the direct control, $\sum_{i}\mathbf{A}_{ij}=1$ for
each $j$, which have the spectral radius $\rho\left(\mathbf{A}\right)=1$
according to the Gershgorin circle theorem [37]. Thus, in strongly connected
components of the ownership network one has to heuristically solve the issues
with cycles, a problem which has become classical in the field of the
corporate control [33, 14].

Second, in cases when Eq. 1 is non-singular, the calculation of control
involves inverting the matrix $\mathbf{I}-\mathbf{A}$, which is a
computationally demanding task for the whole network of several millions of
entities. The authors in [14] suggested a complicated auxiliary algorithm to
separate the network into several strongly connected components, which is
found to be not efficient when applied to the bow-tie topologies of real-world
ownership networks.

Third, the metric of indirect control by Eq. 1 intrinsically assumes that all
paths allowed by the network connectivity contribute equally to the final
control, including the paths of length $l=1$ corresponding to the direct
control. Still, there is a debate in the field about a quantitative measure
that would be optimally balanced between direct and indirect contributions of
control [38]. In this regard, Eq. 1 is an absolutely indirect measure being an
opposite to the pure matrix $\mathbf{A}$ of the direct control (the first term
only).

Here we propose a modification of Eq. 1 that aims at resolving the three
aforementioned issues. We introduce a measure that builds on the ideas of the
Katz centrality [18] and is defined as follows

$\mathbf{T}=\sum_{l=1}^{\infty}\alpha^{l-1}\mathbf{A}^{l}=\mathbf{A}\left(\mathbf{I}-\alpha\mathbf{A}\right)^{-1},\quad\alpha<\rho\left(\mathbf{A}\right)^{-1}.$
(2)

The Katz-like network control in Eq. 2 weighs the paths of length $l$ with the
factor $\alpha^{l-1}$, where $0\leq\alpha<1$ is a tuning parameter of the
model, penalising long paths. Thus, this measure continuously interpolates
between direct and indirect control. For $\alpha=0$ the control is purely
direct, $\mathbf{T}=\mathbf{A}$, while in the limit $\alpha\to 1$ the weights
of all possible paths in the network are equal and the purely indirect control
Eq. 1 is recovered. Note that for stochastic matrices $\mathbf{A}$ the series
Eq. 2 is convergent for any $\alpha<1$. Therefore, we (i) circumvent the
problem of cycles by taking into account the infinite number of paths with
exponentially small weight assigned to the long paths and (ii) propose a
metric that continuously interpolates between the direct and indirect control,
allowing to fine-tune the model for the particular system under study.

####  2.1.3 Core-shell decomposition

Matrix inversion in Eq. 2 is not computationally efficient in large-scale
ownership networks with millions of nodes. We suggest a recursive approach
that is based on the core-periphery decomposition of the network.

Note that the control $T_{ij}$ of $i$ over $j$ can be propagated to $k$, a
shareholder of $i$, by increasing the length of each path from $i$ to $j$ by
one. The propagated value would be the portion of control of $k$ over $j$
(through the node $i$). If one rewrites Eq. 2 as

$\mathbf{T}=\sum_{l=1}^{\infty}{\tilde{\mathbf{T}}}^{(l)};\quad{\tilde{\mathbf{T}}}^{(l)}=\alpha{\tilde{\mathbf{T}}}^{(l-1)}\mathbf{A},$
(3)

then Eq. 3 becomes a recursive expression for the control
${\tilde{\mathbf{T}}}^{(l)}$ over paths of certain length $l$. Using Eq. 3,
one can propagate the control upstream from organisations to participants.

Effective propagation of control is conditioned on the absence of cycles in
the graph. Following this idea, we propose a procedure similar to the $2$-core
decomposition for non-oriented graphs [19, 20]. Specifically, we iteratively
remove the nodes and all the incident edges if the node has only in-going or
only out-going edges, i.e. either in-degree $k_{in}^{i}=0$ or out-degree
$k_{out}^{i}=0$. At the very first step we remove the participants that are
not owned by anyone ($k_{in}=0$) and organisations that do not own anyone
($k_{out}=0$). We classify such entities as super-holders (SH) and super-
targets (ST), respectively. Such removal of nodes at the first step results in
formation of new “hanging” vertices with $k_{in}*k_{out}=0$, which have
previously been linked to super-holders and super-targets. After several
iterations we converge to a subgraph $C$ of nodes that have at least one in-
going and at least one out-going edge. By analogy with the $k$-core
definition, the subgraph $C$ can be called $1,1$-core, since this is the
maximal subgraph in the network with at least one in-going and one out-going
edge at every node. We call this subgraph $C$ “the core” which, by definition,
is filled with the core entities. Finally, all other nodes in the graph which
were removed at step 2 onward are called the intermediaries (I).

##### Example

Consider a weakly connected subgraph in Figure S.1 as an example. At the first
iteration all green (super-holders) and blue (super-targets) nodes are peeled
off from the network. After this iteration the two intermediaries, “Document
Outsourcing Group” and “OPUS Property Finance Limited”, become the effective
super-targets and are to be removed at the second step. However, the network
still possesses hanging vertices: the intermediaries “OPUS Trust Marketing
Limited” and “OPUS 107 Limited” are peeled off at the third and at the fourth
iteration steps, respectively. After four iterations the network is left with
four organisations for which $k_{in}*k_{out}\neq 0$ (marked orange). They form
the $1,1$-core of the network.

##### Complexity

The described decomposition is of $\mathcal{O}(N)$ computational complexity
and ensures absence of cycles in the subgraph of non-core entities (the shell:
super-holder, super-targets and intermediaries), allowing for back-propagation
of control along all possible paths from the core to the shell. Notably, the
size of the core is four orders of magnitude smaller than the size of the
whole network for the UK ownership network we consider in this paper. This
allows for dramatic acceleration of the algorithm: while the matrix
$\mathbf{T}$ can be straightforwardly computed for the core (demanding only
$\mathcal{O}(|C|^{3})$ in the worst-case scenario of a dense core), the effect
of the rest of the network is calculated via back-propagation to the shell. In
this procedure we iteratively compute the control of the intermediaries over
the core organisations unfolding them layer by layer in the reverse direction
using transitive relations in Eq. 3. Control over the intermediaries and
super-targets, in turn, is computed simultaneously as they appear in the
matrix $\mathbf{T}$.

The computational complexity of the back-propagation step is of order of the
number of edges in the shell, which is well-approximated by the total number
of edges in the ownership network under study, $\mathcal{O}(E)$. Real-world
ownership networks reside in the sparse regime (there is of order of one edge
attached to each node on average), so the complexity is $\mathcal{O}(N)$. The
total complexity of the control calculation is $\mathcal{O}(N+|C|^{3})$, where
both terms are of the same order for the network considered in the paper.

Figure 1: Principal steps in the $\alpha$-ICON algorithm

####  2.1.4 Identifying ultimate controlling entities

Having estimated $\mathbf{T}$ given $\alpha$ for all organisations in the
network using back-propagation from the core, we can rank every direct and
indirect participant in every organisation by its $\alpha$-ICON share stored
in the matrix $\mathbf{T}$. The ultimate controlling entity, by the nature of
the ultimate control, is the super-holder with the largest $\alpha$-ICON
share. Importantly, these shares are computed as a sum over all possible paths
through intermediaries, thus, super-holders cannot be analysed independently
without considering their control channelling through other entities in the
network.

In order to identify the ultimate controlling entities for each organisation
we sort its super-holders by their $\alpha$-ICON shares. If no super-holder is
present among the direct or indirect participants we consider any other entity
with the largest share. The outcome of this step — a list of length $k$ with
top-$k$ controlling entities — is the final result of our algorithm which we
designate $\alpha$-ICON SH (see Figure 1).

###  2.2 Data

##### PSC

The UK government maintains a public register identifying the people who can
influence the firms’ decision-making. Every entity, legal person, or
individual holding over 25% of shares or voting rights in the company or
having the power to appoint or remove the majority of the board of directors
has to be reported to Companies House, an executive agency of the government
[39]. This register, called People with Significant Control (PSC), was
established by the Small Business, Enterprise, and Employment Act 2015. It is
the companies’ responsibility to report and update the information on its
direct owners: they are required to file a confirmation statement at least
once a year [40].

Companies House makes the PSC register publicly available. We use its August
2, 2021 snapshot and merge it with the basic company data for live companies
(also coming from Companies House) to consider only active (non-dormant)
companies in our analysis. PSC register data has gaps in unique participants
identifiers. When a participant in organisation is a UK entity we restore its
company number by searching for its name in the snapshot of active firms,
accounting for namesakes. After data cleaning and removing 282 instances of
self-ownership (when the company and its participant are the same entity) we
are left with information on ownership of 4 199 626 active companies by 4 042
087 direct participants (including 4 843 147 individuals, 407 224 entities,
and 5 971 legal persons) forming $E=5\,256\,342$ participant-entity links in
the ownership network. Our $\mathbf{W}\in\mathbb{R}^{N\times N}$ with
$N=8\,087\,056$.

An important limitation of the PSC data is that equity shares are reported as
intervals: $\left[25\%\ldots 50\%\right)$, $\left[50\%\ldots 75\%\right)$,
$\left[75\%\ldots 100\%\right]$. We use midpoints of those intervals and
normalise them to $[0...1]$ range for each organisation. Even though it might
be worthwhile to treat this share as an interval censored random variable
instead, we consider the midpoint normalisation due to its simplicity.333Our
normalisation is not innocuous when there are errors in source equity shares
data. When a company has a sole owner with 25% and many minority shareholders
it has one entry in the PSC with $\left[25\%\ldots 50\%\right)$ interval and
its mid point of 37.5%. This is the theoretical minimum, and after
normalisation this owner will have a share of 1. The theoretical maximum total
equity sum per company reaches 150% when it has 4 owners with 25% each, or 4
instances of $\left[25\%\ldots 50\%\right)$ interval reduced to its midpoint
37.5%. However, we have 25 160 companies in PSC data with total equity share
exceeding 150%, with the maximum of 2562.5% (SIMON MIDCO LIMITED). We ignore
those errors in summation and normalise equity shares to one weighing by their
midpoints.

##### SEC filings

Even though several algorithms to identify ultimate controlling entities
exist, no formal evaluation has been performed to date, to the best of our
knowledge. Primary reason are data requirements. Any evaluation implies
knowing the identities of the true ultimate owners of organisations. Gathering
such information is labour-intensive. In this paper we benefit from the fact
that organisations with debt or securities traded on the US financial markets
have to disclose material information to the US Securities and Exchange
Commission. In particular, the companies are required to list all their
subsidiaries, directly or indirectly controlled, in Exhibit 21 of their SEC
10-K filings. We use those lists from most recent filings (for 2020 and 2021)
gathered by CorpWatch [41], to identify 1362 parent–subsidiary relations
between 304 parents located anywhere in the world and their 1328 subsidiaries
— active UK companies present in Companies House data.

In theory, all such parent-subsidiary relations should exist in the PSC
network since all UK companies are required to disclose their ultimate owners.
However, we fail to establish 355 parent-subsidiary paths from the SEC data in
the PSC network, meaning that the parent and the subsidiary are not connected
in the network even though the subsidiary itself exists.444There are two
reasons for such discrepancy. First, we matched 10-K Exhibit 21 filings with
PSC data on entity names (excluding namesakes). This procedure may fail to
identify the ultimate owners, especially outside the UK, when major
differences in the names provided to the SEC and to the PSC arise. Second,
companies may intentionally over- or under-report the information to the SEC
or Companies House. After ignoring the unreachable paths we end up with the
authoritative list of 1007 subsidiaries and their 279 ultimate controlling
entities. Importantly, the distribution of the parent-subsidiary path lengths
$l$ on the list is non-degenerate. Only 391 of 1007 subsidiaries (39%) are
directly held by their parents ($l=1$), and 209 subsidiaries (21%) are more
than 3 intermediaries away from their parent ($l>3$). The longest relation in
our evaluation data is $l=12$, meaning that the subsidiary is 11
intermediaries away from its parent.

We use the constructed list to compute the recall @ $k$ of each algorithm: how
many subsidiary organisations have the true parent among the top-$k$
controlling entities identified by this algorithm. The recall ranges from 0
(meaning that the algorithm did not return the correct parent for any of the
1007 subsidiaries among its top-$k$ holders) to 1 (when all the subsidiaries
have the correct parent among the top-$k$ holders suggested by the algorithm).
The recall can be also decomposed by path length $l$: how many of the
subsidiaries $l-1$ intermediaries away from its parent have its true parent
among the top-$k$ ultimate controlling entities suggested by the algorithm.
This decomposition is useful in comparing the performance of algorithms for
the short (direct, $l=1$) and long (indirect, $l>1$) ownership relations.

##  3 Results

###  3.1 Decomposition of the UK ownership network

We start with the decomposition of the UK’s People with Significant Control
ownership network into the core and the shell. Namely, we aim at
identification of $1,1$-core, which is the maximal subgraph with all vertices
having at least one in-going edge and one out-going edge. Clearly, such a
subgraph contains cycles, since one can walk along the edges of such subgraph
infintely long, while the subgraph is finite. In contrast, the rest of the
network is a subgraph without cycles due to the recursive peeling procedure
(see Methods).

The described iterative procedure turns out to be swiftly convergent and
results in isolation of the core in 12 iterations. At the very first iteration
the largest component with 98% of nodes disassembles: $N_{SH}=3\,887\,370$
nodes with no in-going edges (super-holders) and $N_{ST}=4\,044\,969$ nodes
without out-going edges (super-targets) in the network. In the remaining 11
steps $N_{I}=154\,140$ nodes are removed, despite them having $k_{in}\geq 1$
and $k_{out}\geq 1$ in the original network (intermediaries), see the summary
statistics in Table 1.

It turns out that the 1,1-core is populated with just 517 companies forming a
set of isolated connected components (see Figure S.2). The absolute majority
(84%) of the core entities are dimers, i.e. companies directly shareholding
each other. There are 219 dimers at the core of the British ownership network;
we call such components “trivial” and do not report them in the Figure S.2.
The smallest and most widely represented (9.3%) non-trivial component is a
triad, in which three companies are connected by at least 3 oriented edges. We
also observe 5 tetrads (4.6%), four of which form a simple 4-cycle and the
last one is a hybrid of one triad and one dimer, connected together at one
node (Unipart Group of Companies). The largest components are a pentad (Ice
Futures) and a hexad (Alpha Financial Markets Consulting), accounting for 2.1%
of the core in sum. All these isolated components for the UK network are
strongly connected, i.e. each company controls any other company within the
same component. However, in principle, core components might be be weakly
connected in some other networks. The smallest theoretically possible weakly
connected component of the 1,1-core is a tetrad in which two dimers are
connected by a directed link. However, we do not observe such structures in
the PSC ownership network.

Sectoral distribution of companies turns out to be quite different at the core
compared to the intermediaries or super-targets. 30.6% of companies at the
core are in the Professional, scientific and technical activity sector, while
in the universe of the UK companies only 13.6% are in this sector, a 2-fold
difference. Almost 24% of intermediaries belong to the Financial And Insurance
Activities sector, while only 3.4% of all the UK companies operate in this
sector, see Table 1. Finance/insurance sector together with
Professional/Science activities comprise almost 50% of the core and have a
similar share among the intermediaries. At the same time, among the super-
targets the respective two sectors comprise only 17% of organisations (which
is close to the share among all live companies in the country).

The lion’s share of super-holders are individuals (98.2%). Since it is
impossible for an individual to be a super-target/intermediary or at the core,
this is not surprising. An inherent feature of an ownership network is
presence of many direct super-holder-super-target relations where super holder
is an individual and super-target is an entity. When we compare the mean age
or sectoral distribution of super-targets with that in the universe of UK
companies in Table 1 we see that they are very close, confirming the ubiquity
of a simple super-holder-super-target relation.

Table 1: Summary statistics of the UK’s People with Significant Control
register

| Position in ownership network | All companies | Evaluation set
---|---|---|---
| Core | Intermediary | Super-holder | Super-target
Entity age, years | $14.58$ | $13.32$ | $16.97$ | $7.55$ | $8.41$ | $20.46$
Is public limited company, % | $0.58$ | $0.64$ | $0.03$ | $0.06$ | $0.12$ | $2.15$
Average in-degree | $1.25$ | $1.36$ | $0$ | $1.25$ | $1.25$ | $0.80$
Average out-degree | $1.84$ | $1.99$ | $1.27$ | $0$ | $0.08$ | $0.99$
# Entities/individuals, incl.: | $517$ | $154\,140$ | $3\,887\,370$ | $4\,044\,969$ | $4\,970\,026$ | $1286$
Individual, % | $0$ | $0$ | $98.23$ | $0$ | $0$ | $0$
Profess./Science activ, % | $30.56$ | $19.24$ | $0.07$ | $14.19$ | $13.63$ | $20.14$
Trade, % | $4.06$ | $5.18$ | $0.02$ | $13.11$ | $12.12$ | $4.43$
Construction, % | $4.45$ | $7.47$ | $0.02$ | $10.97$ | $10.23$ | $0.78$
IT, % | $4.45$ | $5.23$ | $0.02$ | $9.11$ | $8.57$ | $8.94$
Admin./Support activ, % | $14.31$ | $8.67$ | $0.04$ | $8.78$ | $8.48$ | $8.63$
Real estate, % | $2.90$ | $9.97$ | $0.03$ | $8.22$ | $7.98$ | $1.40$
Not available, % | $10.44$ | $4.42$ | $1.33$ | $4.08$ | $6.39$ | $25.58$
Other service, % | $3.09$ | $2.22$ | $0.03$ | $4.81$ | $4.75$ | $1.32$
Hospitality/Food, % | $0.77$ | $1.90$ | $0.01$ | $4.86$ | $4.70$ | $0.31$
Manufacturing, % | $5.03$ | $4.43$ | $0.02$ | $4.78$ | $4.58$ | $8.86$
Health/Social work, % | $0.58$ | $2.09$ | $0.02$ | $3.86$ | $3.70$ | $0.31$
Finance/Insurance, % | $17.02$ | $23.95$ | $0.09$ | $2.76$ | $3.37$ | $17.19$
Transportation/Storage, % | $0.39$ | $1.03$ | $0$ | $3.45$ | $3.30$ | $0.86$
Arts/Entertainment, % | $0.39$ | $1.26$ | $0.02$ | $2.28$ | $2.35$ | $0$
Households as employers, % | $0$ | $0.17$ | $0$ | $1.11$ | $2.20$ | $0$
Education, % | $0.77$ | $0.71$ | $0.02$ | $1.72$ | $1.79$ | $0$
Agriculture, % | $0$ | $0.69$ | $0$ | $0.83$ | $0.78$ | $0$
Water/Sewerage/Waste, % | $0$ | $0.33$ | $0$ | $0.34$ | $0.34$ | $0$
Electricity/Gas/Steam, % | $0$ | $0.63$ | $0$ | $0.32$ | $0.31$ | $0.08$
Mining/Quarrying, % | $0.39$ | $0.34$ | $0$ | $0.21$ | $0.21$ | $1.17$
Public admin/Defence, % | $0.39$ | $0.05$ | $0$ | $0.18$ | $0.17$ | $0$
Extraterritorial orgs, % | $0$ | $0.02$ | $0$ | $0.02$ | $0.03$ | $0$

Note: this table reports the means or percentages of entity/individual
characteristics by node type in the People with Significant Control register
as of August 2, 2021. “All companies” are all the live companies in the
Companies House register as of August 1, 2021. “Evaluation set” are the parent
or subsidiary companies forming the set of subsidiaries and their respective
parents in the evaluation data.

###  3.2 Identification of ultimate control

In order to determine the ultimate owners in the PSC network, we make use of
the core-shell decomposition and calculate the $\alpha$-ICON shares of
indirect transitive control with almost no damping of possible paths,
$\alpha=0.999$ (computation of the control for $\alpha=1$ is not possible due
to the divergence of the series Eq. 2). As described in the Methods, first we
explicitly calculate the control matrix of the core (square, $517\times 517$)
by Eq. 2 and then we iteratively add the nodes from consecutive layers of the
shell in the reverse order and compute their control over existing companies
in the control matrix.

Table 2: Top-25 super-holders with significant control in the United Kingdom

Super-holder with significant control | Rank | Sum of controlled shares | Network size by node type
---|---|---|---
|  $\alpha$-ICON | NPI | DPI |  $\alpha$-ICON | NPI | DPI | Total | SH | ST | I | C
SPECSAVERS OPTICAL SUPERSTORES LTD | $1$ | $1$ | $42\,441$ | $1482.2$ | $1453$ | $4$ | $1767$ | $188$ | $1159$ | $420$ | $0$
MR PETER VALAITIS | $2$ | $2$ | $1$ | $1392$ | $1392$ | $1391$ | $1393$ | $1$ | $1390$ | $2$ | $0$
PARTNERS GROUP HOLDING AG | $3$ | $3$ | $2$ | $1166.5$ | $1166.5$ | $1164.5$ | $1170$ | $2$ | $1167$ | $1$ | $0$
MR AVTAR SINGH | $4$ | $4$ | $4$ | $1051$ | $1051$ | $1051$ | $1055$ | $3$ | $1052$ | $0$ | $0$
MR ANDREW DAVIS | $5$ | $6$ | $2788$ | $881.3$ | $869.5$ | $14.5$ | $1020$ | $107$ | $895$ | $18$ | $0$
MR ROBERT JARRETT | $6$ | $5$ | $5$ | $873$ | $873$ | $873$ | $874$ | $1$ | $873$ | $0$ | $0$
LLOYDS BANKING GROUP PLC | $7$ | $8$ | $24\,872$ | $735.6$ | $670.5$ | $5$ | $956$ | $35$ | $596$ | $325$ | $0$
MR CHRIS HADJIOANNOU | $8$ | $7$ | $6$ | $730$ | $730$ | $730$ | $731$ | $1$ | $730$ | $0$ | $0$
BRIDGEPOINT GROUP PLC | $9$ | $10$ | $5149$ | $669.3$ | $591.5$ | $11$ | $808$ | $13$ | $536$ | $259$ | $0$
MR READE GRIFFITH | $10$ | $9$ | $491\,331$ | $657.5$ | $636.5$ | $1.5$ | $868$ | $25$ | $374$ | $469$ | $0$
PERSIMMON PLC | $11$ | $11$ | $2911$ | $586$ | $578.4$ | $14$ | $617$ | $4$ | $561$ | $52$ | $0$
INTERTRUST N.V. | $12$ | $12$ | $887\,094$ | $519.7$ | $511.8$ | $1$ | $530$ | $4$ | $353$ | $173$ | $0$
IWG PLC | $13$ | $13$ | $10$ | $481.8$ | $481.7$ | $422$ | $497$ | $6$ | $473$ | $18$ | $0$
BARRATT DEVELOPMENTS P L C | $14$ | $17$ | $587$ | $481.1$ | $473.6$ | $31$ | $530$ | $9$ | $446$ | $75$ | $0$
FRASERS GROUP PLC | $15$ | $16$ | $2982$ | $480.1$ | $474.8$ | $14$ | $484$ | $2$ | $438$ | $44$ | $0$
MR GRZEGORZ SZEWCZYK | $16$ | $14$ | $8$ | $480$ | $480$ | $480$ | $481$ | $1$ | $480$ | $0$ | $0$
MR DARREN SYMES | $17$ | $15$ | $9$ | $475.5$ | $475.5$ | $474.5$ | $478$ | $2$ | $475$ | $1$ | $0$
PETS AT HOME GROUP PLC | $18$ | $19$ | $1\,226\,949$ | $468$ | $389.1$ | $1$ | $770$ | $216$ | $514$ | $40$ | $0$
BGF GROUP PLC | $19$ | $23$ | $6219$ | $462.1$ | $343$ | $10$ | $674$ | $99$ | $378$ | $197$ | $0$
OCTOPUS CAPITAL LIMITED | $20$ | $18$ | $1624$ | $414.5$ | $411.7$ | $19$ | $431$ | $4$ | $310$ | $117$ | $0$
M&G PLC | $21$ | $20$ | $17\,408$ | $404$ | $386.6$ | $6$ | $469$ | $14$ | $298$ | $157$ | $0$
BP P.L.C. | $22$ | $26$ | $1757$ | $377.8$ | $336.4$ | $18$ | $493$ | $5$ | $390$ | $96$ | $2$
IVC ACQUISITION PIKCO LTD | $23$ | $21$ | $1\,535\,931$ | $370$ | $366.3$ | $1$ | $371$ | $1$ | $341$ | $29$ | $0$
SUN LIFE FINANCIAL INC. | $24$ | $22$ | $2\,086\,664$ | $364.7$ | $358.2$ | $1$ | $469$ | $10$ | $242$ | $217$ | $0$
PLACES FOR PEOPLE GROUP LIMITED | $25$ | $25$ | $458$ | $345.8$ | $338.7$ | $35.5$ | $387$ | $11$ | $324$ | $52$ | $0$

Note: this table reports the 25 super-holders with the largest total
$\alpha$-ICON transitive share in all entities when $\alpha=0.999$. Also
reported are the sums and ranks of their respective NPI and DPI indices. We
also provide the information on the topological types of nodes sharing the
same sub-network with this super-holder. Ownership data comes from the People
with Significant Control register as of August 2, 2021.

We also compute the NPI and DPI indices [34] for the entire ownership network
using our own implementations of those algorithms.555Quota = 0.5, 10 000
iterations. For NPI we discarded the first 1000 iterations and set original
label restoration probability $\epsilon=0.01$. As the baseline approach we
identify the controlling entities using the direct equity shares, i.e. the
original matrix of shares $\mathbf{W}$ before normalisation. We compare the
three approaches to the results of $\alpha$-ICON algorithm by examining the
top-25 super-holders in Table 2.

Surprisingly, the two sets of top-25 holders in the ownership network
determined by their $\sum\alpha$-ICON and $\sum$NPI — another algorithm of
indirect control — closely match each other. Furthermore, the first four
super-holders are ranked the same by the two methods. Interestingly, the first
four super-holders (but the first one) have equal total sums of shares by
$\alpha$-ICON and NPI. The leading super-holder, Specsavers Optical
Superstores Ltd, is predicted to control roughly 30 companies more by
$\sum\alpha$-ICON than by $\sum$NPI. Even though NPI does identify fewer
subsidiary companies held by the Specsavers, this does not change the ranking
order. Note, however, that further down the list lower $\sum$NPI in relation
to $\sum\alpha$-ICON does change the rank of the subsequent holders.

Naturally, the $\sum$DPI ranks outlined in Table 2 are much less compatible
with $\sum\alpha$-ICON (when $\alpha=0.999$) or $\sum$NPI, since DPI provides
a measure of direct control. Still it is instructive to cross-compare the
results of the two approaches. The second, the third and the fourth super-
holders by both $\sum\alpha$-ICON and $\sum$NPI have $\sum$DPI shares with
almost identical absolute values of control. This means that these super-
holders exert control over their subsidiaries directly, without
intermediaries. In contrast, the first super-holder has the $\sum$DPI rank 42
441, meaning that it has a distributed network of subsidiaries not accounted
for by DPI. We depict a similar difference between direct and indirect control
in Figure S.1(b) where the super-holder The Berkeley Group PLC directly
controls 39 entities and 177 entities in total, if we include indirect
control.

Another case in point is Mr Reade Griffith, ranked 10th by $\sum\alpha$-ICON,
9th by $\sum$NPI, and 491 331st by $\sum$DPI. He exerts his control
predominately indirectly compared to Mr Peter Valaitis, ranked 2nd by
$\sum\alpha$-ICON and $\sum$NPI, and 1st by DPI, who is the only super-holder
in his network of control. This is consistent with the classification of
entities within the connected components controlled by top-ranked super-
holders in the Table 2. Mr Peter Valaitis represents the central node in his
star-like connected component with 1392 leaves, which correspond to the values
of $\sum\alpha$-ICON and $\sum$NPI. However, as we see, $\sum$DPI identifies
only 1391 subsidiaries, because one of them is at distance 2 from Mr Peter
Valaitis and, thus, is hidden from the algorithms of direct control. On the
other hand, in the component of the first holder, Specsavers Optical
Superstores Ltd, there are more than twice as many intermediaries as super-
holders. Despite it being the primary beneficiary holder in the UK, only 4 out
of 1400+ of its subsidiaries are identified by DPI. This is an example of a
strongly distributed ownership network in which determination of the leading
beneficiary holder is non-trivial. In Figure S.1(a) we plot this network where
the super-holder Specsavers Optical Superstores Ltd (in green) owns a large
stake in intermediary Specsavers UK Holdings Limited (depicted in grey in the
centre) that, in turn, co-owns many super-targets (in blue) alongside many
other super-holders (also in green).

Interestingly, the networks of all top-25 super-holders, but one (BP P.L.C.),
do not contain cycles. This is reflected in the absence of the core nodes
therein. As a rule, the control networks of the main UK super-holders
represent trees. From the practical standpoint, this drastically simplifies
determination of the ultimate control in these networks.

###  3.3 Evaluation

Figure 2: Recall of controlling entity identification algorithms

(a) Recall @ $k$ across algorithms

(b) Recall @ $k$ for $\alpha$-ICON SH,
$\alpha\in\left(0.1,\ldots,0.999\right)$

(c) Recall @ 1 across algorithms by path length

(d) Recall @ 5 across algorithms by path length

Note: detailed evaluation results from all models are reported in Table S.1.

We engage in an evaluation study to understand the validity of various control
identification algorithms using the ground-truth set of parent-subsidiary
relations from SEC 10-K Exhibit 21 filings of companies with debt/equity
traded on the US financial markets (see Data section). Note that this list is
not topologically representative of the whole ownership network or the UK
companies (compare column “Evaluation set” and “All companies” in Table 1).
Notwithstanding, it allows us to compare the control identification
algorithms.

We make use of the core-shell decomposition and calculate the $\alpha$-ICON
shares of indirect transitive control for a series of
$\alpha=0,0.1,0.2,\ldots,0.8,0.9,0.999$. For the quantitative comparison of
the algorithms we estimate how many subsidiary companies have the correct
parent among its top-$k$ super-holders identified by an algorithm (recall @
$k$). Results of this computation for different $k$, $\alpha$ and various
algorithms of control are reported in Figure 2(a)–(b) and Table S.1(a).

We immediately see from Figure 2(a) that DPI (light blue line) with recall of
38.5% @ 1 performs the worst in the evaluation, highlighting the need to
account for indirect ownership. In contrast, $\alpha$-ICON SH (for
$\alpha=0.999$, dark green line) is superior to all conventional algorithms
and yields 96.1% of correctly determined ultimate holders at recall @ 1 and
99.8% at recall @ 5. In particular, the $\alpha$-ICON SH is superior to NPI
(dark blue line): at recall @ 1 the fraction of correctly determined ultimate
holders is 6.7 p.p. lower for NPI.

The performance gain of $\alpha$-ICON is due to the final step of our
algorithm when we search for entities with the largest share only among the
super-holders. Absent this filter, $\alpha$-ICON (light green line) yields
worse results. This is expected since the entities with the largest transitive
share are the core entities and the intermediaries. As a result, there are
owners that exert control over them, despite having lower transitive share in
the network. The problem of the ultimate control is different from the
identification of the ultimate owner in the way that the former should be
analysed in the class of super-holders. When we consider the
$\mathbf{W}^{DPI}$ weights in lieu of the normalised $\mathbf{W}$ equity
shares $\alpha$-ICON SH demonstrates somewhat worse results, Table S.1(a)
suggests. However, it still registers better performance than NPI. We conclude
that the transitive control computed by $\alpha$-ICON SH produces more
coherent results.

$\alpha$-ICON SH has one tuning parameter. In Figure 2(b) we compute its
recall at different $\alpha$s. One could immediately notice non-monotonic
relationship between algorithm quality and $\alpha$ at different $k$. At $k=1$
the highest recall is achieved by $\alpha\in\left[0.9,\ldots,0.999\right]$,
with recall for $\alpha=0.8$ being the close second. Lower values of $\alpha$
produce lower recall @ 1. The situation changes starting from recall @ 3.
$\alpha<0.8$ produces better results in such settings. This might hint at the
existence of optimal damping parameter $\alpha^{*}$ that maximises the recall.

To further understand the source of discrepancy between the algorithms, it is
instructive to look at the evaluation performance at different subsidiary-
parent path lengths $l$, see Figure 2(c)–(d) and Table S.1(b). $\alpha$-ICON
SH algorithm (dark green line) displays monotonically increasing recall @ 1
starting from $l=2$, while the performance of NPI (dark blue line) is clearly
not stable with the path length. This analysis suggests that NPI fails to
correctly identify ultimate super-holders located at large distance $l$ from
their subsidiaries in the ownership graph. With increase of $k$ this effect
essentially vanishes, Figure 2(d) suggests.

##  4 Discussion

Our algorithm to determine the ultimate controlling entities in ownership
networks scales linearly with network size and supersedes previous approaches
in terms of its recall. We observe that $\alpha$-ICON with $\alpha$ very close
to unity takes into account all possible paths of control in the ownership
network, overestimating the shares of distal super-holders along the ownership
graph. At the same time, NPI overestimates the weight of nearby holders by
allowing them to form coalitions with other participants. This observation is
crucial for the discussion to follow.

##### Cycles

Previous algorithms had to develop ad hoc procedures to overcome cycles
induced by circular ownership. Our algorithm, in contrast, is built around
cycles by isolating them at the network’s $1,1$ core. This approach allows for
a dramatic speed-up because cycles are rare in real-world ownership networks.
In fact, we are surprised to find any cycles in the UK PSC network given that
subsidiaries cannot be members of its holding company.666Companies Act 2006,
Section 136(1)(a). However, previous studies of the PSC data reached similar
conclusions regarding the circular ownership [42]. With explicit prohibition
of circular ownership in place it is reasonable to expect the UK ownership
network to feature a small core, which is indeed the case. The smaller the
core, the faster the $\alpha$-ICON algorithm.

##### Quota

Heuristic approaches and algorithms rooted in game theory require setting
multiple parameters, including the quota, or the equity share threshold to
influence the firm’s decision-making. While most authors opt for the simple
majority quota of 50%, this assumption may dramatically alter the control
identification. Consider the example of an ownership network of Boparan
Holdings Ltd in Figure S.1(c). This is an involved ownership network super-
held by two individuals: Ranjit Boparan (with his 50–75% stake) and Bajlinder
Boparan (with her 25–50% stake). Since the stake of the latter super-holder is
less than the 50% quota she does not control any firm in the network per DPI
or NPI. Conversely, under the transitive ownership of $\alpha$-ICON when
$\alpha=0.999$ she has the total control of 24 in comparison to Ranjit
Boparan’s 73. By lowering $\alpha$ we lower her total control up until it
reaches $\nicefrac{{37.5}}{{37.5+62.5}}=0.375$ when $\alpha=0$ (direct control
only after midpoint normalisation). In principle, one can easily integrate the
quota in $\alpha$-ICON by equating edges with weight less than the specified
threshold to zero in adjacency matrix $\mathbf{W}$.

##### Optimal $\alpha^{*}$

The higher the value of $\alpha$ the more weight is given to distant ownership
relations. This might be undesirable in networks where there is a legal
restriction on the maximal ownership path length. If the maximal path allowed
is $l_{max}$, the optimal $\alpha^{*}$ can be assessed from
$l_{max}=-\nicefrac{{1}}{{\ln\left(\alpha^{*}+1\right)}}$. An alternative
strategy is to gather a validation data set with the true subsidiary-parent
relationships that represents the network in terms of its topological and non-
topological characteristics. Then $\alpha^{*}$ is the value at which the
recall reaches maximum in this validation data set. While this is an
attractive way forward, it is not deprived of well-known challenges. Gathering
a representative validation set is a daunting task when the majority of
relationships in real-world ownership networks are simple super-holder-super-
target links. In a network populated only with such links we do not need
anything more than $\alpha^{*}=0$ (i.e. direct control only) to capture the
control in its entirety. In a network where the majority of links are the
simple SH-ST but there exists a handful of extremely complicated sub-networks
of indirect ownership setting $\alpha^{*}=0$ would result in a severe under-
estimation of the influence of indirect players. Therefore, it might be
worthwhile to build a validation dataset that accurately represents both
direct and indirect ownership in the network. Using this benchmark one could
pick $\alpha^{*}$ that results in the highest cumulative average of recalls @
$k$ for every path length $l$. Under this approach one can achieve the balance
between penalising long paths and fostering the indirect control.

##### Limitations

For a dense core network, the computational complexity of $\alpha$-ICON is
cubic in the size of the core. In networks with large and dense strongly
connected components amounting to tens of thousands of nodes in the core
computation of the transitivity might become unfeasible. However, an
approximation by several first components of the transitivity matrix (in the
spirit of the eigenvector centrality) can significantly reduce the
computational complexity. We leave this to future research. Another limitation
relates to the interpretation of our findings on control shares. The top-25
super-holders in Table 2 include the individual holders of thousands of firms,
retail chains with complicated structures, and multinational corporations. It
is hard to believe that the supermajor oil company BP PLC (market
capitalisation of USD 84.1 bln) ranked 22th per $\sum\alpha$-ICON exerts less
influence than Pets at Home PLC (market capitalisation of USD 3.5 bln) ranked
18th. Our transitive ownership shares are all but the starting point in
determination of the total control. They, for instance, can be used as weights
in summation of net assets or market capitalisation of companies held by the
super-holders.

##  5 Conclusion

In this paper we have proposed $\alpha$-ICON — a robust numerical algorithm
for identification of ultimate control in onion-like ownership networks that
may include consisting of millions of nodes.

Our algorithm borrows from the ideas of Katz centrality to calculate the
network flow with the additional damping factor, exponentially weighting the
probability of long paths of transitive control. In contrast to previous
methods, our approach allows for the continuous interpolation between fully
direct control (when only direct shareholders can exert their influence) for
$\alpha=0$ and purely transitive network control in the limit $\alpha\to 1$
(when control paths of all lengths in the network are equally likely to
exist). In order to satisfy the domain constraint ruling out intermediaries
from the ultimate holders, our algorithm determines top-$k$ beneficiaries for
each company by ranking its super-holders by the computed $\alpha$-ICON
shares.

We are first to offer formal evaluation of controlling entity identification
algorithms. We gather such disclosures of the UK companies with debt or
securities traded on the US financial markets to build an authoritative list
of subsidiaries and their ultimate controlling entities. We show that
$\alpha$-ICON is superior to the previous approaches. In particular, we find
that the recently proposed indirect control algorithm, NPI, though being close
in performance to $\alpha$-ICON, fails to correctly identify _distal_ parents
of subsidiaries. This is because NPI is based on particular game-theoretic
model of control by consolidation of votes, which is quite a debatable
approach in corporate control literature due to heterogeneity in distribution
of shareholders [1] and circular shareholding [12].

We reveal the universal core-shell topological organisation of the ownership
network with a small core (4 orders of magnitude smaller than the whole
network), which allows for dramatic acceleration of calculation of the
transitive control. We also note that the widely known bow-tie structure of
real-world ownership networks is a reflection of their fundamental degeneracy
and sparsity. This inherent property corresponds to the $1,1$-core
decomposition ($k_{in}=1$ and $k_{out}=1$) of the oriented graphs. We find
that only less than 0.01% of the whole ownership graph belongs to (in general,
weakly, but strongly in practice) connected components with cycles (the core),
while the remaining 99.99% of entities grow on trees (the shell). Our
algorithm exploits this feature to supply dramatic optimisation via recursive
back-propagation of control from the core to the shell along the tree-like
paths.

Importantly, the restriction of long transitive paths at $\alpha<1$ suggests a
universal approach to the problem of votes consolidation in determination of
the ultimate control. While conventional methods taking into account
consolidation rely on a particular game-theoretic model (such as Shapley-
Shubik), resulting in the model-specific typical path length between
organisation and its beneficiary in the ownership graph, our approach is
advantageous as it is capable to explore the beneficiaries at all path lengths
(resolutions) simultaneously. We note that the possibility of coalition
formation is the particular reason for the transitive network control (at
$\alpha=1$) to be not optimal.

In general, the optimal value of $\alpha^{*}$, reflecting some sort of balance
between direct and indirect control, should be chosen guided by the data. In
this paper, based on a novel evaluation procedure, we demonstrate that the
values $\alpha>0.8$ overestimate the role of long paths in the control
ranking, though still showing better performance than NPI and DPI. To make any
credible conclusions on the optimal $\alpha^{*}$ inherent to the whole
network, the evaluation dataset should be topologically and economically
representative to the network. In practice, the latter condition is not always
possible to satisfy, which, thus, poses an interesting problem for future
research.

##  6 Acknowledgements

Authors are grateful to Interdisciplinary Scientific Center Poncelet, European
University at St. Petersburg, and Skoltech for organisational support at the
initial stages of this work. KP and NP acknowledge partial support of RSF
grant 21-11-00215.

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## Supplementary materials

Table S.1: Results of evaluation on SEC 10-K Exhibit 21 subsidiaries data

|  Recall @ $k$
---|---
| 1 | 3 | 5 | 10
Baseline | $0.384$ | $0.388$ | $0.388$ | $0.388$
DPI | $0.385$ | $0.388$ | $0.388$ | $0.388$
NPI | $0.894$ | $0.971$ | $0.971$ | $0.971$
NPI, SH | $0.894$ | $0.971$ | $0.971$ | $0.971$
$\alpha$-ICON, $\mathbf{W}$, $\alpha=0$ | $0.384$ | $0.388$ | $0.388$ | $0.388$
$\alpha$-ICON, $\mathbf{W}$, $\alpha=0.999$ | $0.492$ | $0.868$ | $0.993$ | 1.000
$\alpha$-ICON, $\mathbf{W}$, $\alpha=0.999$, SH | 0.961 | 0.986 | 0.998 | 1.000
$\alpha$-ICON, $\mathbf{W}^{DPI}$, $\alpha=0.999$, SH | $0.911$ | $0.985$ | $0.997$ | $0.999$

(a) Recall @ top-$k$ ultimate controlling entities

|  Recall @ 1 for subsidiary-parent path length $l$
---|---
| 1 | 2 | 3 | 4 | 5 | 10 | 12
Baseline | $0.990$ | $0.000$ | $0.000$ | $0.000$ | $0.000$ | $0.000$ | $0.000$
DPI | 0.992 | $0.000$ | $0.000$ | $0.000$ | $0.000$ | $0.000$ | $0.000$
NPI | $0.977$ | $0.916$ | $0.894$ | $0.874$ | $0.944$ | $0.062$ | $0.000$
NPI, SH | $0.977$ | $0.916$ | $0.894$ | $0.874$ | $0.944$ | $0.062$ | $0.000$
$\alpha$-ICON, $\mathbf{W}$, $\alpha=0$ | $0.990$ | $0.000$ | $0.000$ | $0.000$ | $0.000$ | $0.000$ | $0.000$
$\alpha$-ICON, $\mathbf{W}$, $\alpha=0.999$ | 0.992 | $0.097$ | $0.300$ | $0.310$ | $0.083$ | $0.000$ | $0.000$
$\alpha$-ICON, $\mathbf{W}$, $\alpha=0.999$, SH | 0.992 | $0.921$ | 0.922 | 0.954 | 1.000 | 1.000 | 1.000
$\alpha$-ICON, $\mathbf{W}^{DPI}$, $\alpha=0.999$, SH | 0.992 | 0.930 | $0.917$ | $0.920$ | $0.944$ | $0.062$ | $0.000$
# subsidiaries at path length | 391 | 227 | 180 | 87 | 36 | 16 | 1

(b) Recall @ 1 for subsidiaries $l$ intermediaries away from their parents

Figure S.1: Core-periphery decomposition of an ownership network

Note: colour codes the inferred node types. Edge texts are equity shares.
Numbers in parentheses are iterations when the respective nodes are peeled
off.

Figure S.2: All non-trivial components (excl. dimers) at the core of the UK’s
People with Significant Control register (node size is proportional to total
holdings)

Figure S.3: Selected ownership networks from the UK’s People with Significant
Control register (node size is proportional to log of direct holdings)

(a) A network with the largest $\alpha$-ICON SH and the largest difference
between direct and indirect control by SH: Specsavers Optical Superstores LTD
($\sum$DPI=4, $\sum$NPI=1453, $\sum\alpha$-ICON=1482)

(b) A network where $\alpha$-ICON and DPI differ for SH:
The Berkeley Group PLC ($\sum$DPI=39, $\sum\alpha$-ICON=177)

(c) A network where $\alpha$-ICON and NPI differ for SH:
Bajlinder Boparan ($\sum$NPI=0, $\sum\alpha$-ICON=24)