Backbone of complex networks of corporations: The flow of control

Abstract

We present a methodology to extract the backbone of complex networks based on the weight and direction of links, as well as on nontopological properties of nodes. We show how the methodology can be applied in general to networks in which mass or energy is flowing along the links. In particular, the procedure enables us to address important questions in economics, namely, how control and wealth are structured and concentrated across national markets. We report on the first cross-country investigation of ownership networks, focusing on the stock markets of 48 countries around the world. On the one hand, our analysis confirms results expected on the basis of the literature on corporate control, namely, that in Anglo-Saxon countries control tends to be dispersed among numerous shareholders. On the other hand, it also reveals that in the same countries, control is found to be highly concentrated at the global level, namely, lying in the hands of very few important shareholders. Interestingly, the exact opposite is observed for European countries. These results have previously not been reported as they are not observable without the kind of network analysis developed here.

Full text

arXiv:0902.0878v2 [q-fin.GN] 21 Aug 2009
J.B. Glattfelder and S. Battiston:
Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
Backbone of complex networks of corporations:
The flow of control
J.B. Glattfelder and S. Battiston
Chair of Systems Design, ETH Zurich, Kreuzplatz 5, 8032 Zurich, Switzerland
Abstract
We present a methodology to extract the backbone of complex networks based on the
weight and direction of links, as well as on nontopological properties of nodes. We show how
the methodology can be applied in general to networks in which mass or energy is flowing
along the links. In particular, the procedure enables us to address important questions in
economics, namely, how control and wealth are structured and concentrated across national
markets. We report on the first cross-country investigation of ownership networks, focusing
on the stock markets of 48 countries around the world. On the one hand, our analysis confirms
results expected on the basis of the literature on corporate control, namely, that in Anglo-
Saxon countries control tends to be dispersed among numerous shareholders. On the other
hand, it also reveals that in the same countries, control is found to be highly concentrated at
the global level, namely, lying in the hands of very few important shareholders. Interestingly,
the exact opposite is observed for European countries. These results have previously not been
reported as they are not observable without the kind of network analysis developed here.
PACS numbers: 89.65.Gh, 64.60.aq, 02.50.–r
I Introduction
The empirical analysis of real-world complex networks has revealed unsuspected regularities
such as scaling laws which are robust across many domains, ranging from biology or computer
systems to society and economics [1, 2, 3, 4]. This has suggested that universal or at least
generic mechanisms are at work in the formation of many such networks. Tools and concepts
from statistical physics have been crucial for the achievement of these findings [5, 6].
In the last years, in order to offer useful insights into more detailed research questions, several
studies have started taking into account the specific meaning of the nodes and links in the
various domains the real-world networks pertain to [7, 8]. Three levels of analysis are possible.
The lowest level corresponds to a purely topological approach where the network is described
by a binary adjacency matrix. By taking weights [7], or weights and direction [9], of the links
into account, the second level is defined. Only recent studies have started focusing on the third
level of detail, in which the nodes themselves are assigned a degree of freedom, sometimes also
called fitness. This is a nontopological state variable which shapes the topology of the network
[8, 10, 11].
The physics literature on complex economic networks has previously focused on boards of di-
rectors [12, 13], market investments [10, 14], stock price correlations [15, 16], and international
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J.B. Glattfelder and S. Battiston:
Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
trade [17, 18, 19]. Here we instead present a comprehensive cross-country analysis of 48 stock
markets world wide. Our first contribution is an algorithm able to identify and extract the back-
bone in the networks of ownership relations among firms: the core subnetwork where most of the
value and control of the market resides. Notably, we also provide a generalization of the method
applicable to networks in which weighted directed links and nontopological properties of nodes
play a role. In particular, the method is relevant for networks in which there is a flow of mass
(or energy) along the links and one is interested in identifying the subset of nodes where a given
fraction of the mass of the system is flowing. The growing interest in methods for extracting the
backbone of complex networks is witnessed by recent work in similar direction [20].
Furthermore, given the economic context of the analyzed networks, we contribute a model to
estimate corporate control based on the knowledge of the ownership ties. In order to identify the
key players according to their degree of control, we take the value of the market capitalization
of the listed companies (a good proxy for their size) to be the nontopological state variable of
nodes in the network. Our main empirical results are in contrast with previously held views
in the economics literature [21], where a local distribution of control was not suspected to
systematically result in global concentration of control and vice versa.
II Dataset
We are able to employ a unique data set consisting of financial information of listed companies
in national stock markets. The ownership network is given by the web of shareholding relations
from and to such companies. The analysis is constrained to 48 countries given in Appendix A.
The data are compiled from Bureau van Dijk’s ORBIS database1. In total, we analyze 24 877
corporations (or stocks) and 106 141 shareholding entities who cannot be owned themselves (in-
dividuals, families, cooperative societies, registered associations, foundations, public authorities,
etc.). Note that because the corporations can also appear as shareholders, the network does not
display a bipartite structure. The stocks are connected through 545 896 ownership ties to their
shareholders. The database represents a snapshot of the ownership relations at the beginning of
2007. The values for the market capitalization, which is defined as the number of outstanding
shares times the firm’s market price, are also from early 2007. These values will serve as the
nontopological state variables assigned to the nodes.
We ensure that every node in the network is a distinct entity. In addition, as theoretically the
sum of the shareholdings of a company should be 100%, we normalize the ownership percentages
if the sum is smaller due to unreported shareholdings. Such missing ownership data is nearly
always due to their percentage values being very small and hence negligible.
1http://www.bvdep.com/orbis.html.
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J.B. Glattfelder and S. Battiston:
Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
Figure 1: Schematic illustration of a bow-tie topology: the central area is the strongly connected
component (SCC), where there is a path from each node to every other node, and the left (IN)
and right (OUT) sections contain the incoming and outgoing nodes, respectively.
III Three-Level Network Analysis
Not all networks can be associated with a notion of flow. For instance, in the international trade
network the fact that country A exports to B and B exports to C does not imply that goods
are flowing from A to C. In contrast, in ownership networks the distance between two nodes
(along a directed path) corresponds to a precise economic meaning which can be captured in a
measure of control that considers all directed paths of all lengths (see Sec. III.4). In addition,
the weight of an ownership link has a meaning relative to the weight of the other links attached
to the same node. Finally, the value of the nodes themselves is very important. Therefore, in the
following, we focus on network measures which take these aspects into account, and we do not
report on standard measures such as degree distribution, assortativity, clustering coefficients,
average path lengths, connected components, etc.
III.1 Level 1: Topological analysis
We start from the analysis of strongly connected components. These subgraphs correspond to
sets of corporations where every firm is connected to every other firm via a path of indirect
ownership. Furthermore, strongly connected components may form bow-tie structures, akin to
the topology of the world wide web [22]. Fig. 1 illustrates an idealized bow-tie topology. This
structure reflects the flow of control, as every shareholder in the IN section exerts control and
all corporations in the OUT section are controlled.
We find that roughly two thirds of the countries’ ownership networks contain bow-tie structures
(see also [23]). As an example, the countries with the highest occurrence of (small) bow-tie
structures are KR and TW, and to a lesser degree JP. A possible determinant is the well-known
existence of so-called business groups in these countries (e.g., the keiretsu in JP, and the chaebol
in KR) forming a tightly knit web of cross shareholdings [24, 25]. For AU, CA, GB, and US
we observe very few but large bow-tie structures of which the biggest ones contain hundreds to
thousands of corporations. This raises the question relevant to economics: if the emergence of
these mega-structures in the Anglo-Saxon countries is due to their unique “type” of capitalism
3/24

J.B. Glattfelder and S. Battiston:
Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
i1i2i3
j
sj
Wi1j
Wi2j
Wi3j
Figure 2: Definition of the concentration index sj, measuring the number of prominent incoming
edges, respectively, the effective number of shareholders of the stock j. When all the weights are
equal, then sj=kin
j, where kin
jis the in degree of vertex j. When one weight is overwhelmingly
larger than the others, the concentration index approaches the value one, meaning that there
exists a single dominant shareholder of j.
(the so-called Atlantic or stock market capitalism [26]), and whether this finding contradicts the
assumption that these markets are characterized by the absence of business groups [24].
III.2 Level 2: Extending the notions of degree
In graph theory, the number kiof edges per vertex iis called the degree. If the edges are oriented,
one has to distinguish between the in degree and out degree, kin and kout , respectively. When
the edges ij are weighted with the number wij, the corresponding quantity is called strength [7]:
kw
i:= X
j
Wij.(1)
When there are no weights associated with the edges, we expect all edges to count the same.
If weights have a large variance, some edges will be more important than others. A way of
measuring the number of prominent incoming edges is to define the concentration index [27] as
follows:
sj:= Pkin
j
i=1 Wij 2
Pkin
j
i=1 W2
ij
.(2)
Note that this quantity is akin to the inverse of the Herfindahl index extensively used in eco-
nomics as a standard indicator of market concentration [28]. Notably, a similar measure has
also been used in statistical physics as an order parameter [29]. A recent study [20] employs a
Herfindahl index in their backbone extraction method for weighted directed networks (where,
however, the nodes hold no nontopological information). In the context of ownership networks,
sjis interpreted as the effective number of shareholders of the stock j, as explained in Fig. 2.
Thus it can be understood as a measure of control from the point of view of a stock.
The second quantity to be introduced measures the number of important outgoing edges of the
vertices. For a given vertex i, with a destination vertex j, we first define a measure which reflects
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J.B. Glattfelder and S. Battiston:
Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
i1
j1
j2
j3
i2
i3
Hi1j1
Hi1j2
Hi1j3
Hi2j3
Hi3j3
hi1
Figure 3: The definition of the control index hi, measuring the number of prominent outgoing
edges. In the context of ownership networks this value represents the effective number of stocks
that are controlled by shareholder i. Note that to obtain such a measure, we have to consider
the fraction of control Hij, which is a model of how ownership can be mapped to control (see
the discussion in Appendix B).
the importance of iwith respect to all vertices connecting to j:
Hij := W2
ij
Pkin
j
l=1 W2
lj
.(3)
This quantity has values in the interval (0,1]. For instance, if Hij ≈1 then iis by far the most
important source vertex for the vertex j. For our ownership network, Hij represents the fraction
of control [27] shareholder ihas on the company j. As shown in Fig. 3, this quantity is a way
of measuring how important the outgoing edges of a node iare with respect to its neighbors’
neighbors. For an interpretation of Hij from an economics point of view, consult Appendix B.
From that, we then define the control index,
hi:=
kout
i
X
j=1
Hij.(4)
Within the ownership network setting, hiis interpreted as the effective number of stocks con-
trolled by shareholder i.
III.3 Distributions of sand h
Fig. 4 shows the probability density function (PDF) of sjfor a selection of nine countries (for
the full sample consult [30]). There is a diversity in the shapes and ranges of the distributions
to be seen. For instance, the distribution of GB reveals that many companies have more than
20 leading shareholders, whereas in IT few companies are held by more than five significant
shareholders. Such country-specific signatures were expected to appear due to the differences in
legal and institutional settings (e.g., law enforcement and protection of minority shareholders
[31]).
On the other hand, looking at the cumulative distribution function (CDF) of kout
i(shown for
three selected countries in the top panel of Fig. 5; the full sample is available at [30]) a more
uniform shape is revealed. The distributions range across two to three orders of magnitude.
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J.B. Glattfelder and S. Battiston:
Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
1 5 10 20
10−3
10−2
10−1
100JP
1 5 10 20
10−2
10−1
100DE
1 5 10 20
10−2
10−1
100FR
1 5 10 20 40
10−3
10−2
10−1
100GB
1 5 10 20
10−2
10−1
100SG
1 5 10 20 40
10−2
10−1
100IT
1 5 10 20
10−3
10−2
10−1
100TW
1 5 10 20 4070
10−3
10−2
10−1
100US
1 5 10 20
10−3
10−2
10−1
100IN
ln sln sln s
PDF PDF PDF
Figure 4: Probability distributions of sjfor selected countries; PDF in log-log scale.
Hence some shareholders can hold up to a couple of thousand stocks, whereas the majority
have ownership in less than 10. Considering the CDF of hi, seen in the middle panel of Fig. 5,
one can observe that the curves of hidisplay two regimes. This is true for nearly all analyzed
countries, with a slight country-dependent variability. Notable exceptions are FI, IS, LU, PT,
TN, TW, and VG. In order to understand this behavior it is useful to look at the PDF of hi,
shown in the bottom panel of Fig. 5. This uncovers a systematic feature: the peak at the value
of hi= 1 indicates that there are many shareholders in the markets whose only intention is to
control one single stock. This observation, however, could also be due to a database artifact as
incompleteness of the data may result in many stocks having only one reported shareholder. In
100101102103
10−3
10−2
10−1
100JP
100101102
10−3
10−2
10−1
100DE
100101102103
10−4
10−3
10−2
10−1
100US
10−1 100101102
10−3
10−2
10−1
100
10−1 100
10−3
10−2
10−1
100
10−1 100101102
10−4
10−3
10−2
10−1
100
10−1 100101102
10−3
10−2
10−1
100
10−1 100
10−3
10−2
10−1
100
10−1 100101102
10−4
10−3
10−2
10−1
100
ln hlnhln h
ln hlnhln h
ln kout
ln kout
ln kout
PDF CDF CDF
Figure 5: Various probability distributions for selected countries: (top panel) CDF plot of kout
i;
(middle panel) CDF plot of hi; (bottom panel) PDF plot of hi; all plots are in log-log scale.
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J.B. Glattfelder and S. Battiston:
Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
order to check that this result is indeed a feature of the markets, we constrain these ownership
relations to the ones being bigger than 50%, reflecting incontestable control. In a subsequent
analysis we still observe this pattern in many countries (BM, CA, CH, DE, FR, GB, ID, IN,
KY, MY, TH, US, and ZA; ES being the most pronounced). In addition, we find many such
shareholders to be non-firms, i.e., people, families, or legal entities, hardening the evidence for
this type of exclusive control. This result emphasizes the utility of the newly defined measures
to uncover relevant structures in the real-world ownership networks.
III.4 Level 3: Adding nontopological values
The quantities defined in Eqs. (2) and (4) rely on the direction and weight of the links. However,
they do not consider nontopological state variables assigned to the nodes themselves. In our
case of ownership networks, a natural choice is to use the market capitalization value of firms in
thousand US dollars (USD), vj, as a proxy for their sizes. Hence vjwill be utilized as the state
variable in the subsequent analysis. In a first step, we address the question of how much wealth
the shareholders own, i.e., the value in their portfolios.
As the percentage of ownership given by Wij is a measure of the fraction of outstanding shares
iholds in j, and the market capitalization of jis defined by the number of outstanding shares
times the market price, the following quantity reflects i’s portfolio value:
pi:=
kout
i
X
j=1
Wij vj.(5)
Extending this measure to incorporate the notions of control, we replace Wij in the previous
equation with the fraction of control Hij , defined in Eq. (3), yielding the control value:
ci:=
kout
i
X
j=1
Hijvj.(6)
A high civalue is indicative of the possibility to control a portfolio with a big market capital-
ization value. Recall that the economic meaning of Hij is discussed in Appendix B.
It should be noted that Eq. (6) only considers direct neighbors. To address the question of how
control propagates via all possible direct and indirect ownership paths, the so-called integrated
model has been proposed [32], which we briefly sketch. Consider a sample of nfirms connected
by cross-shareholding relations. Let Aij, with i, j = 1,2, ..., n, be the ownership (Wij) or control
(Hij) that company ihas directly on company j, and A= [Aij ] is the matrix of all the links
between every one of the nfirms. By definition, it holds that
n
X
i=1
Aij ≤1; j= 1, ..., n. (7)
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J.B. Glattfelder and S. Battiston:
Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
When some shareholders of company iare not identified or are outside the sample n, the inequal-
ity becomes strict. The integrated model accounts for direct and indirect ownership through a
recursive computation. The general form of the equation reads
˜
Aij := Aij +X
n
Ain ˜
Anj,(8)
where the tilde denotes integrated ownership or control. This expression can be written in matrix
form as
˜
A=A+A˜
A, (9)
the solution of which is given by
˜
A= (I−A)−1A. (10)
For the matrix (I−A) to be non-negative and non-singular, a sufficient condition is that the
Frobenius root is smaller than one, λ(A)<1. This is ensured by the following requirement: in
each strongly connected component Sthere exists at least one node jsuch that Pi∈S Aij <1.
In an economic setting, this means that there exists no subset of kfirms (k= 1,...,n) that
are entirely owned by the kfirms themselves. A condition which is always fulfilled in ownership
networks [32].
In order to define the integrated control value ˜ciin the same spirit as Eq. (6), we first solve
Eq. (10) for the fraction of control Hij , which yields the integrated fraction of control ˜
Hij. ˜ci
represents the value of control a shareholder gains from companies reached by all direct and
indirect paths of ownership:
˜ci:=
kout
i
X
j=1
˜
Hijvj.(11)
This quantity is used in the next section to identify and rank the shareholders by importance.
IV Identifying the Backbone of Corporate Control
IV.1 Computing cumulative control
The first step of our methodology requires the construction of a Lorenz-like curve in order to
uncover the distribution of the control in a market. In economics, the Lorenz curve gives a
graphical representation of the cumulative distribution function of a probability distribution.
It is often used to represent income distributions, where the xaxis ranks the poorest x% of
households and relates them to a percentage value of income on the yaxis.
Here, on the xaxis we rank the shareholders according to their importance — as measured by
their integrated control value ˜ci, cf., Eqs. (3), (10), and (11) — and report the fraction they
represent with respect to the whole set of shareholder. The yaxis shows the corresponding
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J.B. Glattfelder and S. Battiston:
Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
Figure 6: First steps in computing cumulative control: (top panel) selecting the most important
shareholder (light shading) ranked according to the ˜civalues and the portfolio of stocks owned
at more than 50% (dark shading); in the second step (bottom panel), the next most important
shareholder is added; although there are now no new stocks which are owned directly at more
than 50%, cumulatively the two shareholder own an additional stock at 55%.
percentage of controlled market value, defined as the fraction of the total market value they
collectively or cumulatively control.
In order to motivate the notion of cumulative control, some preliminary remarks are required.
Using the integrated control value to rank the shareholders means that we implicitly assume
control based on the integrated fraction of control ˜
Hij. This however is a potential value reflecting
possible control. In order to identify the backbone, we take a very conservative approach to the
question of what the actual control of a shareholder is. To this aim, we introduce a stringent
threshold of 50%. Any shareholder with an ownership percentage Wij >0.5 controls by default.
This strict notion of control for a single shareholder is then generalized to apply to the cumulative
control a group of shareholders can exert. Namely, by requiring the sum of ownership percentages
multiple shareholders have in a common stock to exceed the threshold of cumulative control. Its
value is equivalently chosen to be 50%.
We start the computation of cumulative control by identifying the shareholder having the highest
˜civalue. From the portfolio of this holder, we extract the stocks that are owned at more than
the said 50%. In the next step, the shareholder with the second highest ˜civalue is selected.
Next to the stocks individually held at more than 50% by this shareholder, additional stocks are
considered, which are cumulatively owned by the top two shareholders at more than the said
threshold value. See Fig. 6 for an illustrated example.
Uin(n) is defined to be the set of indices of the stocks that are individually held above the
threshold value by the nselected top shareholders. Equivalently, Ucu(n) represents the set of
indices of the cumulatively controlled companies. It holds that Uin (n)∩Ucu(n) = ∅. At each step
n, the total value of this newly constructed portfolio, Uin(n)∪Ucu (n), is computed:
vcu(n) := X
j∈Uin(n)
vj+X
j∈Ucu(n)
vj.(12)
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J.B. Glattfelder and S. Battiston:
Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
10−4 10−3 10−2 10−1 100
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
US
GB
JP
DE
IT
CN
KR
IN
ϑ(Market Value, %)
η(Shareholder Rank, %)
Cumulative Control
Figure 7: (Color online) Fraction of shareholders η, sorted by descending (integrated) control
value ˜ci, cumulatively controlling ϑpercent of the total market value; the horizontal line denotes
a market value of 80%; the diagram is in semilogarithmic scale.
Eq. (12) is in contrast to Eq. (5), where the total value of the stocks jis multiplied by the
ownership percentage Wij.
Let ntot be the total number of shareholders in a market and vtot the total market value. We
normalize with these values, defining:
η(n) := n
ntot
, ϑ(n) := vcu(n)
vtot
,(13)
where η, ϑ ∈(0,1].
In Fig. (7) these values are plotted against each other for a selection of countries (the full sample
is in [30]), yielding the cumulative control diagram, akin to a Lorenz curve (with reversed x
axis). As an example, a coordinate pair with value (10−3,0.2) reveals that the top 0.1% of
shareholders cumulatively control 20% of the total market value. The top right corner of the
diagram represents 100% of the shareholders controlling 100% of the market value, and the first
data point in the lower left-hand corner denotes the most important shareholder of each country.
Different countries show a varying degree of concentration of control.
It should be emphasized that our analysis unveils the importance of shareholders: the ranking of
every shareholder is based on all direct and indirect paths of control of any length. In contrast,
most other empirical studies start their analysis from a set of important stocks (e.g., ranked
by market capitalization). The methods of accounting for indirect control (see Sec. III.4) are,
if at all, only employed to detect the so-called ultimate owners of the stocks. For instance, [33]
studies the ten largest corporations in 49 countries, [31] looks at the 20 largest public companies
in 27 countries, [34] analyzes 2980 companies in nine East Asian countries, and [35] utilizes a
set of 800 Belgian firms.
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J.B. Glattfelder and S. Battiston:
Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
Algorithm 1 BB(˜c1,...,˜cn,δ,ˆ
ϑ)
1: ˜c←sort descending(˜c1,...,˜cn)
2: repeat
3: c←get largest(˜c)
4: I←I∪index(c)
5: P F ←stocks controlled by(I) (individually and cumulatively at more than δ)
6: P F V ←value of portf olio(P F )
7: ˜c←˜c\ {c}
8: until P F V ≥ˆ
ϑ·total market value
9: prune network(I , P F )
Finally, note that although the identity of the individual controlling shareholders is lost due
to the introduction of cumulative control, the emphasis lies on the fact that the controlling
shareholders are present in the set of the first nholders.
IV.2 Extracting the backbone
Once the curve of the cumulative control is known for a market, one can set a threshold for
the percentage of jointly controlled market value, ˆ
ϑ. This results in the identification of the
percentage ˆηof shareholders that theoretically hold the power to control this value, if they were
to coordinate their activities in corresponding voting blocks. The subnetwork of these power
holders and their portfolios is called the backbone. Here we choose the value ˆ
ϑ= 0.8, revealing
the power holders able to control 80% of the total market value.
Algorithm (1) gives the complete recipe for computing the backbone. As inputs, the algorithm
requires all the ˜civalues, the threshold defining the level of (cumulative) control δand the
threshold for the considered market value ˆ
ϑ. Steps 1 – 7 are required for the cumulative control
computation and δis set to 0.5. Step 8 specifies the interruption requirement given by the
controlled portfolio value being bigger than ˆ
ϑtimes the total market value.
Finally, in step 9, the subnetwork of power holders and their new portfolios is pruned to eliminate
weak links and further enhance the important structures. For each stock jin the union of these
portfolios, only as many shareholders are kept as the rounded value of sjindicates, i.e., the
(approximate) effective number of shareholders. Although a power holder can be in the portfolio
of other power holders, the pruning only considers the incoming links. That is, if jhas five holders
but sjis roughly three, only the three largest shareholders are considered for the backbone. The
portfolio of jis left untouched. In effect, the weakest links and any resulting isolated nodes are
removed.
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J.B. Glattfelder and S. Battiston:
Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
IV.3 Generalizing the method of backbone extraction
Notice that our method can be generalized to any directed and weighted network in which (1)
a nontopological real value vj≥0 can be assigned to the nodes (with the condition that vj>0
for at least all the leaf nodes in the network) and (2) an edge from node ito jwith weight
Wij implies that some of the value of jis transferred to i. Assume that the nodes which are
associated with a value vjproduce vjunits of mass at time t= 1. Then the flow φientering the
node ifrom each node jat time tis the fraction Wij of the mass produced directly by jplus
the same fraction of the inflow of j:
φi(t+ 1) = X
j
Wijvj+X
j
Wij φi(t),(14)
where PiWij = 1 for the nodes jthat have predecessors and PiWij = 0 for the root nodes
(sinks). In matrix notation, at the steady state, this yields
φ=W(v+φ).(15)
The solution
φ= (1 −W)−1W v , (16)
exists and is unique if λ(W)<1. This condition is easily fulfilled in real networks as it re-
quires that in each strongly connected component Sthere exists at least one node jsuch that
Pi∈S Wij <1. Or, equivalently, the mass circulating in Sis also flowing to some node outside
of S. To summarize, some of the nodes only produce mass (all the leaf nodes but possibly also
other nodes) at time t= 1 and are thus sources, while the root nodes accumulate the mass.
Notice that the mass is conserved at all nodes except at the sinks.
The convention used in this paper implies that mass flows against the direction of the edges. This
makes sense in the case of ownership because although the cash allowing an equity stake in a
firm to be held flows in the direction of the edges, control is transferred in the opposite direction,
from the corporation to its shareholders. This is also true for the paid dividends. Observe that
the integrated control value defined in Eq. (11) can be written in matrix notation as
˜c=˜
Hv = (1 −H)−1H v, (17)
which is in fact equivalent to Eq. (16). This implies that for any node ithe integrated control
value ˜ci=Pj˜
Hijvjcorresponds to the inflow φiof mass in the steady state.
Returning to the generic setting, let U0and E0be, respectively, the set of vertices and edges
yielding the network. We define a subset U⊆U0of vertices on which we want to focus on (in
the analysis presented earlier U=U0). Let E⊆E0then be the set of edges among the vertices
in Uand introduce ˆ
ϑ, a threshold for the fraction of aggregate flow through the nodes of the
network. If the relative importance of neighboring nodes is crucial, Hij is computed from Wij
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Physical Review E 80 (2009)
by the virtue of Eq. (3). Note that Hij can be replaced by any function of the weights Wij that
is suitable in the context of the network under examination. We now solve Eq. (10) to obtain
the integrated value ˜
Hij. This yields the quantitative relation of the indirect connections among
the nodes. To be precise, it should be noted that in some networks the weight of an indirect
connection is not correctly captured by the product of the weights along the path between the
two nodes. In such cases one has to modify Eq. (8) accordingly.
The next step in the backbone extraction procedure is to identify the fraction of flow that is
transferred by a subset of nodes. A systematic way of doing this was presented in Sec. IV.1
where we constructed the curve, (η, ϑ). A general recipe for such a construction is the following.
On the xaxis all the nodes are ranked by their φivalue in descending order and the fraction
they represent with respect to size of Uis captured. The yaxis then shows the corresponding
percentage of flow the nodes transfer. As an example, the first k(ranked) nodes represent the
fraction η(k) = k/|U|of all nodes that cumulatively transfer the amount ϑ(k) = (Pk
i=1 φi)/φtot
of the total flow. Furthermore, ˆηcorresponds to the percentage of top ranked nodes that pipe
the predefined fraction ˆ
ϑof all the mass flowing in the whole network. Note that the procedure
described in Sec. IV.1 is somewhat different. There we considered the fraction of the total value
given by the direct successors of the nodes with largest ˜ci. This makes sense due to the special
nature of the ownership networks under investigation, where every non-firm shareholder (root
node) is directly linked to at least one corporation (leaf node), and the corporations are connected
among themselves.
Consider the union of the nodes identified by ˆηand their direct and indirect successors, together
with the links among them. This is a subnetwork B= (UB, EB), with UB⊂Uand EB⊂E
that comprises, by construction, the fraction ˆ
ϑof the total flow. This is a first possible definition
of the backbone of (U, E). A discussion of the potential application of this procedure to other
domains, and a more detailed description of the generalized methodology (along with specific
refinements pertaining to the context given by the networks) is left for future work. Viable
candidates are the world trade web [8, 17, 36, 37], food webs [4], transportation networks [38],
and credit networks [39].
IV.4 Defining classification measures
According to economists, markets differ from one country to another in a variety of respects
[31, 33]. They may not look too different if one restricts the analysis to the distribution of local
quantities, and in particular to the degree, as shown in Sec. III.3. In contrast, at the level of
the backbones, i.e., the structures where most of the value resides, they can look strikingly
dissimilar. As seen for instance in the case of CN and JP, shown in Fig. 8. In the following, we
provide a quantitative classification of these diverse structures based on the indicators used to
construct the backbones.
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Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
Figure 8: (Top) the backbone of JP; (bottom) the backbone of CN (for the complete set of
backbone layouts consult [30]); the graph layouts are based on [40].
Let nst and nsh denote the number of stocks and shareholders in a backbone, respectively. As
sjmeasures the effective number of shareholders of a company, the average value,
s=Pnst
j=1 sj
nst
,(18)
is a good proxy characterizing the local patterns of ownership: the higher s, the more dispersed
the ownership is in the backbone or the more common is the appearance of widely held firms.
Furthermore, due to the construction of sj, the metric sequivalently measures the local concen-
tration of control.
In a similar vein, the average value
h=Pnsh
i=1 hi
nsh
=nst
nsh
,(19)
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Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
Figure 9: The map of control: illustration of idealized network topologies in terms of local
dispersion of control (s) vs global concentration of control (h); shareholders and stocks are shown
as empty and filled bullets, respectively; arrows represent ownership; region (E) is excluded due
to consistency constraints; (A) does not necessarily need to be a single connected structure; see
Fig. 11 for the empirical results.
reflects the global distribution of control. A high value of hmeans that the considered backbone
has very few shareholders compared to stocks, exposing a high degree of global concentration of
control. Recall that nst and nsh refer to the backbone and not to the original network. Fig. 9
shows the possible generic backbone configurations resulting from local and global distributions
of control.
Remember also that in order to construct the backbones we had to specify a threshold for
the controlled market value: ˆ
ϑ= 0.8. In the cumulative control diagram seen in Fig. (7), this
allows the identification of the number of shareholders being able to control this value. The
value ˆηreflects the percentage of power holders corresponding to ˆ
ϑ. To adjust for the variability
introduced by the different numbers of shareholders present in the various national stock markets,
we chose to normalize ˆη. Let n100 denote the smallest number of shareholders controlling 100%
of the total market value vtot, then
η′:= ˆη
n100
.(20)
A small value for η′means that there will be very few shareholders in the backbone compared
to the number of shareholders present in the whole market, reflecting that the market value
is extremely concentrated in the hands of a few shareholders. In essence, the metric η′is an
emergent property of the backbone extraction algorithm and mirrors the global distribution of
the value.
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Physical Review E 80 (2009)
Figure 10: (Color online) The backbone of CH is a subnetwork of the original ownership network
which was comprised of 972 shareholders, 266 stocks, and 4671 ownership relations; firms are
denoted by red nodes and sized by market capitalization, shareholders are blue, whereas firms
owning stocks themselves are represented by red nodes with thick blue bounding circles, arrows
are weighted by the percentage of ownership value; the graph layouts are based on [40].
V Analyzing the Backbones
How relevant are the backbones and how many properties of the real-world ownership networks
are captured by the classification measure? As a qualitative example, Fig. (10) shows the layout
for the CH backbone network. Looking at the few stocks left in the backbone, it is indeed the case
that the important corporations reappear (recall that the algorithm selected the shareholders).
We find a cluster of shareholdings linking, for instance, Nestl´e, Novartis, Roche Holding, UBS,
Credit Suisse Group, ABB, Swiss Re, and Swatch. JPMorgan Chase & Co. features as the most
important controlling shareholder. The descendants of the founding families of Roche (Hoffmann
and Oeri) are the highest ranked Swiss shareholders at position four. UBS follows as dominant
Swiss shareholder at rank seven.
We can also recover some previous empirical results. The “widely held” index [31] assigns to a
country a value of one if there are no controlling shareholders, and zero if all firms in the sample
are controlled above a given threshold. The study is done with a 10% and 20% cut-off value
for the threshold. We find a 76.6% correlation (and a pvalue for testing the hypothesis of no
correlation of 3.2×10−6) between sin the backbones and the 10% cut-off “widely held” index
for the 27 countries it is reported for. The correlation of sin the countries’ whole ownership
networks is 60.0% (9.3×10−4). For the 20% cutoff, the correlation values are smaller. These
relations should however be handled with care, as the study [31] is restricted to the 20 largest
firms (in terms of market capitalization) in the analyzed countries and there is a 12 year lag
between the data sets in the two studies.
The backbone extraction algorithm is also a good test for the robustness of market patterns. The
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Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
0 0.5 1 1.5 2 2.5
−2
−1
0
1
2
3
4
AE
AR AT
AU
BE BM
CA
CH
CL
CN DE
DK
ES
FI
FR
GB
GR
HK
ID
IE
IL
IN
IS
IT
JO
JP
KR
KW
KY
LU
MX
MY
NLNO
NZ
OM
PH
PT
SA
SE
SG
TH
TN
TR
TW
US
VG
ZA
ln(s)
ln(h)
Figure 11: Map of control: local dispersion of control, s, is plotted against global concentration
of control, h, for 48 countries.
bow-tie structures (discussed in Sec. III.1) in JP, KR, and TW vanish or are negligibly small in
their backbones, whereas in the backbones of the Anglo-Saxon countries (and as an outlier SE)
one sizable bow-tie structure survives. This emphasizes the strength and hence the importance
of these patterns in the markets of AU, CA, GB, and US.
V.1 Global concentration of control
We utilize the measures defined in Sec. IV.4 to classify the 48 backbones. In Fig. 11 the loga-
rithmic values of sand hare plotted against each other. sis a local measure for the dispersion
of control (at first-neighbor level, insensitive to value). A large value indicates a high presence of
widely held firms. his an indicator of the global concentration of control [an integrated measure,
i.e., derived by virtue of Eq. (10), at second-neighbor level, insensitive to value]. Large values
are indicative that the control of many stocks resides in the hands of very few shareholders. The
scoordinates of the countries are as expected [31]: to the right we see countries known to have
widely held firms (AU, GB, and US). Instead, FR, IT, and JP are located to the left, reflecting
more concentrated local control. However, there is a counterintuitive trend in the data: the more
local control is dispersed, the higher the global concentration of control becomes. What looks
like a democratic distribution of control from close up, actually turns out to warp into highly
concentrated control in the hands of very few shareholders. On the other hand, the local concen-
tration of control is in fact widely distributed among many controlling shareholder. Comparing
with Fig. 9, where idealized network configurations are illustrated, we conclude that the empir-
ical patterns of local and global control correspond to network topologies ranging from types
(B) to type (D), with JP combining local and global concentration of control. Interestingly, type
(A) and (C) constellations are not observed in the data.
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Physical Review E 80 (2009)
0 0.5 1 1.5 2 2.5
−1
0
1
2
3
4
5
AE
AR
AT
AU
BE BM
CA
CH
CL
CN
DE
DK ES
FI
FR
GB
GR
HK
ID
IE
IL IN
IS
IT
JO
JP
KR
KW
KY
LU
MX
MY
NL
NO
NZ
OM
PH
PT
SA
SE
SG
TH
TN
TR
TW
US
VG
ZA
ln(s)
ln(η′)
Figure 12: Map of market value: local dispersion of control, s, is plotted against global concen-
tration of market value, η′, for 48 countries.
In Fig. 12 the logarithmic values of sand η′are depicted. η′is a global variable related to
the (normalized) percentage of shareholders in the backbone (an emergent quantity). It hence
measures the concentration of value in a market, as a low number means that very few share-
holders are able to control 80% of the market value. What we concluded in the last paragraph
for control is also true for the market value: the more the control is locally dispersed, the higher
the concentration of value that lies in the hands of very few controlling shareholders and vice
versa.
We realize that the two figures discussed in this section open many questions. Why are there
outliers such as JP in Fig. 11 and VG in Fig. 12? What does it mean to group countries according
to their s,h, and η′coordinates and what does proximity imply? What are the implications for
the individual countries? We hope to address such and similar questions in future work.
V.2 Seat of power
Having identified important shareholders in the global markets, it is now also possible to address
the following questions. Who holds the power in an increasingly globalized world? How important
are individual people compared to the sphere of influence of multinational corporations? How
eminent is the influence of the financial sector? By looking in detail at the identity of the power
holders featured in the backbones, we address these issues next.
If one focuses on how often the same power holders appear in the backbones of the 48 countries
analyzed, it is possible to identify the global power holders. Following is a top-ten list, comprised
of the company’s name, activity, country the headquarter is based in, and ranked according to
the number of times it is present in different countries’ backbones: the Capital Group Compa-
nies (investment management, US, 36), Fidelity Management & Research (investment products
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Physical Review E 80 (2009)
and services, US, 32), Barclays PLC (financial services provider, GB, 26), Franklin Resources
(investment management, US, 25), AXA (insurance company, FR, 22), JPMorgan Chase &
Co. (financial services provider, US, 19), Dimensional Fund Advisors (investment management,
US, 15), Merrill Lynch & Co. (investment management, US, 14), Wellington Management Co.
(investment management, US, 14), and UBS (financial services provider, CH, 12).
Next to the dominance of US American companies we find: Barclays PLC (GB), AXA (FR) and
UBS (CH), Deutsche Bank (DE), Brandes Investment Partners (CA), Soci´et´e G´en´erale (FR),
Credit Suisse Group (CH), Schroders PLC (GB), and Allianz (DE) in the top 21 positions.
The government of Singapore is at rank 25. HSBC Holdings PLC (HK/GB), the world’s largest
banking group, only appears at position 26. In addition, large multinational corporations outside
of the finance and insurance industry do not act as prominent shareholders and only appear in
their own national countries’ backbones as controlled stocks. For instance, Exxon Mobil, Daimler
Chrysler, Ford Motor Co., Siemens, and Unilever.
Individual people do not appear as multinational power holders very often. In the US backbone,
we find one person ranked at ninth position: Warren E. Buffet. William Henry Gates III is next,
at rank 26. In DE the family Porsche/Piech and in FR the family Bettencourt are power-holders
in the top ten. For the tax-haven KY one finds Kao H. Min (who is placed at number 140 in
the Forbes 400 list) in the top ranks.
The prevalence of multinational financial corporations in the list above is perhaps not very
surprising. For instance, Capital Group Companies is one of the world’s largest investment
management organizations with assets under management in excess of one trillion USD. However,
it is an interesting and novel observation that all the above-mentioned corporations appear as
prominent controlling shareholders simultaneously in many countries. We are aware that financial
institutions such as mutual funds may not always seek to exert overt control. This is argued, for
instance, for some of the largest US mutual funds when operating in the US [21], on the basis of
their propensity to vote against the management (although, the same mutual funds are described
as exerting their power when operating in Europe). However, to our knowledge, there are no
systematic studies about the control of financial institutions over their owned companies world
wide. To conclude, one can interpret our quantitative measure of control as potential power
(namely, the probability of achieving one’s own interest against the opposition of other actors
[41]). Given these premises, we cannot exclude that the top shareholders having vast potential
power do not globally exert it in some way.
VI Summary and Conclusion
We have developed a methodology to identify and extract the backbone of complex networks that
are comprised of weighted and directed links and nodes to which a scalar quantity is associated.
We interpret such networks as systems in which mass is created at some nodes and transferred to
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Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
the nodes upstream. The amount of mass flowing along a link from node ito node jis given by
the scalar quantity associated with the node jtimes the weight of the link, Wij vj. The backbone
corresponds to the subnetwork in which a preassigned fraction of the total flow of the system is
transferred.
Applied to ownership networks, the procedure identifies the backbone as the subnetwork where
most of the control and the economic value resides. In the analysis the nodes are associated
with nontopological state variables given by the market capitalization value of the firms, and
the indirect control along all ownership pathways is fully accounted for. We ranked the share-
holders according to the value they can control, and we constructed the subset of shareholders
which collectively control a given fraction of the economic value in the market. In essence, our
algorithm for extracting the backbone amplifies subtle effects and unveils key structures. We
further introduced some measures aimed at classifying the backbone of the different markets in
terms of local and global concentration of control and value. We find that each level of detail in
the analysis uncovers features in the ownership networks. Incorporating the direction of links in
the study reveals bow-tie structures in the network. Including value allows us to identify who is
holding the power in the global stock markets.
With respect to other studies in the economics literature, next to proposing a model for es-
timating control from ownership, we are able to recover previously observed patterns in the
data, namely, the frequency of widely held firms in the various countries studied. Indeed, it
has been known for over 75 years that the Anglo-Saxon countries have the highest occurrence
of widely held firms [42]. The statement that the control of corporations is dispersed among
many shareholders invokes the intuition that there exists a multitude of owners that only hold
a small amount of shares in a few companies. However, in contrast to such intuition, our main
finding is that a local dispersion of control is associated with a global concentration of control
and value. This means that only a small elite of shareholders controls a large fraction of the
stock market, without ever having been previously systematically reported on. Some authors
have suggested such a result by observing that a few big US mutual funds managing personal
pension plans have become the biggest owners of corporate America since the 1990s [21]. On
the other hand, in countries with local concentration of control (mostly observed in European
states), the shareholders tend to only hold control over a single corporation, resulting in the
dispersion of global control and value. Finally, we also observe that the US financial sector holds
the seat of power at an international level. It will remain to be seen, if the continued unfolding
of the current financial crisis will tip this balance of power as the US financial landscape faces
a fundamental transformation in its wake.
Acknowledgements
We would like to express our special gratitude to G. Caldarelli and D. Garlaschelli who provided
invaluable advice to this research especially in its early stages. We would also like to thank F.
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Backbone of complex networks of corporations: The flow of control
Physical Review E 80 (2009)
Schweitzer and M. Napoletano for fruitful discussions. Finally, we are very grateful for the advice
of G. Davis regarding the relevance of our work with respect to issues in corporate governance.
Appendix A Analyzed Countries
Data from the following countries was used: United Arab Emirates (AE), Argentina (AR),
Austria (AT), Australia (AU), Belgium (BE), Bermuda (BM), Canada (CA), Switzerland (CH),
Chile (CL), China (CN), Germany (DE), Denmark (DK), Spain (ES), Finland (FI), France (FR),
United Kingdom (GB), Greece (GR), Hong Kong (HK), Indonesia (ID), Ireland (IE), Israel (IL),
India (IN), Iceland (IS), Italy (IT), Jordan (JO), Japan (JP), South Korea (KR), Kuwait (KW),
Cayman Islands (KY), Luxembourg (LU), Mexico (MX), Malaysia (MY), Netherlands (NL),
Norway (NO), New Zealand (NZ), Oman (OM), Philippines (PH), Portugal (PT), Saudi Arabia
(SA), Sweden (SE), Singapore (SG), Thailand (TH), Tunisia (TN), Turkey (TR), Taiwan (TW),
USA (US), Virgin Islands (VG), and South Africa (ZA).
Countries are identified by their two letter ISO 3166–1 alpha-2 codes (given in the parenthesis
above).
Appendix B Ownership vs. Control or the Interpretation of Hij
While ownership is an objective quantity (the percentage of shares owned), control (reflected in
voting rights) can only be estimated. In this appendix we provide a motivation for our proposed
model of control Hij (defined in Sec. III.2) from an economics point of view and discuss how
our measure overcomes some of the limitations of previous models.
There is a great freedom in how corporations are allowed to map percentages of ownership
in their equity capital (also referred to as cash-flow rights) into voting rights assigned to the
holders at shareholders meetings. However, empirical studies indicate that in many countries the
corporations tend not to exploit all the opportunities allowed by national laws to skew voting
rights. Instead, they adopt the so-called one-share-one-vote principle which states that ownership
percentages yield identical percentages of voting rights [31, 43].
It is however still not obvious how to compute control from the knowledge of the voting rights.
As an example, some simple models introducing a fixed threshold for control have been proposed
(with threshold values of 10% and 20% [31] next to a more conservative value of 50% [44]). These
models can easily be extended to incorporate indirect paths of control vie the integrated model
of Sec. III.4.
Given any model for control, there is always a drawback in estimating real-world control or power:
shareholders do not only act as individuals but can collaborate in shareholding coalitions and give
rise to so-called voting blocks. The theory of political voting games in cooperative game theory
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has been applied to the problem of shareholder voting in the form of so-called power indices [45].
However, the employment of power indices for measuring shareholder voting behavior has failed
to find widespread acceptance due to computational, inconsistency and conceptual issues [45].
The so-called degree of control αwas introduced in [46] as a probabilistic voting model mea-
suring the degree of control of a block of large shareholdings as the probability of it attracting
majority support in a voting game. Without going into details, the idea is as follows. Consider a
shareholder iwith ownership Wij in the stock j. Then the control of idepends not only on the
value in absolute terms of Wij, but also on how dispersed the remaining shares are (measured
by the Herfindahl index). The more they tend to be dispersed, the higher the value of α. So even
a shareholder with a small Wij can obtain a high degree of control. The assumptions underlying
this probabilistic voting model correspond to those behind the power indices. However, αsuffers
from drawbacks. It gives a minimum cut-off value of 0.5 (even for arbitrarily small sharehold-
ings) and hence Eq. (7) is violated, meaning that it cannot be utilized in an integrated model.
The computation of αcan become intractable in situations with many shareholders.
To summarize, our measure of control extends existing integrated models using fixed thresholds
by incorporating insights from probabilistic voting models (the analytical expressions of Hij
and αshare very similar behavior), and, furthermore, ˜
Hij can be computed efficiently for large
networks.
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