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PACS 87.23.Ge; 89.65 -s, 89.65.Gh
Statistical properties of Corporate Board and Director Networks
Stefano Battiston1, Michele Catanzaro 2
1Laboratoire de Physique Statistique ENS, 24 rue Lhomond,
75005 Paris FRANCE 2INFM UdR ROMA1 Dipartimento di Fisica,
Universit`a di Roma “La Sapienza” Piazzale A. Moro 2 00185 Roma, Italy
(Dated: November 20 2003)
The boards of directors of the largest corporations of a country together with the directors form a
dense bipartite network. The board network consist of boards connected through common directors.
The director network is obtained taking the directors as nodes, and a membership in the same
board as a link. These networks are involved in the decision making processes relevant for the
macro-economy of a country. We present an extensive and comparative analysis of the statistical
properties of the board network and the director network for the US Fortune 1000 corporations and
the Italian Stock Market corporations. Some statistical properties are found to be specific to the
director networks and the same in all the different cases of study. Some other statistical properties
are instead found to be specific to the board networks but again the same in all the different cases
of study. In particular the connectivity degree distribution of the director network has always a
power law tail with similar exponent. On the contrary, the connectivity degree distribution of the
board network is always rapidly decreasing. All the considered networks are Small World networks,
assortative, highly clustered and dominated by a giant component. The presence of lobbies in boards
turns out to be a macroscopic phenomenon in all cases of study. These results suggest a common
underlying mechanism shaping the corporate control network over time and over different countries
and should be taken into account in models of macroeconomic dynamics.
INTRODUCTION
The boards of directors of the corporations of a country form together with the directors a bi-
partite network. The board network consist of boards connected through common directors. The
director network is the network obtained taking the directors as nodes, and a membership in the
same board as a link. It is well known that the director network of the largest companies in the US
and in other countries has a high degree of interlock, meaning the fact that some directors serve on
several boards at the same time so that many boards are connected by shared directors. Interlock
convey information and power (i.e. banks lending money to a firm can use interlocked directors in
firms of the same industrial sector to get information about the real risk of the loan).
It has been argued that in a capitalistic economy, as a consequence of economic power concentration,
”a special social type emerges spontaneously, a cohesive group of multiple directors tied together
by shared background, friendship networks, and economic interest, who sit on bank boards as rep-
resentative of capital in general” [1]. Now, while part of the public opinion has been since long
ago concerned about the fact that the corporate elite would represent a sort of ”financial oligarchy
controlling the business of the country” [2], stockholders are more concerned about the effectiveness
of boards in overseeing management.
Board’s directors should in fact monitor managers’s strategies and decisions to the interest of
stockholders. Recently, after several cases of bankruptcy in the western countries, the role of boards
in the decision making process is under examination and more sophisticated forms of corporate
control are often advocated in the public opinion.
Two issues raise very naturally about directors interlock networks: the first is the characterization of
the topological properties of the board network and the director network. Because large corporations’
boards are organized in a network leading the economy of a country, the second issue is of course
how the structure of these networks influences the decision making process in which directors are
involved.
Davis and collaborators have shown [3] that the director network and the board network of the
Fortune 1000 corporations has Small World properties in the sense of Watts and Strogatz [4].
Newman et al. [5] have applied on the same data set a random graph model showing that using
the generating function method, it is possible to reproduce very accurately the degree distribution
of the director network. On the contrary, their model fails in predicting the degree distribution of
the board network. In fact the director network turns out to be assortative as observed commonly
in social networks, meaning that directors with high (low) degree tend to be connected to directors
with high (low) degree. As a consequence even if the random graph model predicts the right degree
distribution for the director network it underestimate the number of boards with high number of
2
interlocks and with small number of interlocks.
As a general empirical finding, social networks are characterized by assortativity and high cluster-
ing coefficient cc ( the latter measuring the average fraction of connection between the first neighbors
of a node out of all the possible connections among them).
Newman et al. [7] have recently argued that the presence of groups or communities in a social
network is able to produce alone both assortativity and clustering. They develop a model in which
nodes belong to one or more groups and have probability p to be connected to another node of the
same group. Instead they are never connected to nodes of groups they do not belong to. If groups
have heterogenous size, than nodes who belong to a small group tend to have low degree and are
connected to others in the same group, who also have low degree. This model explains about 40
p.c. of the observed assortativity in the Fortune 1000 network. This means that some additional
sociological mechanism is at work, probably the fact that new board members are more likely to be
recruited among those who are already connected to some of the current board member.
Some recent works have focussed on the influence of the structure of the interlock network on the
decisions made by boards. There are essentially two kinds of decisions a board is faced to. Local
decisions regard topics specific to the board, such as the appointment of a vice president, for which
boards can be assumed not to influence each other. Battiston et al. [9] investigate by means of a
decision making process model how a minority of well connected directors can influence significantly
the decision of the majority.
By contrast, global decisions concern topics of general interest to the economy such as whether
to increase or decrease investments in development or in advertisement, which depend on the belief
in economical growth or recession. In these cases, decisions previously made in some boards might
influence other boards, through the presence of shared directors.
In a recent model, Battiston et al. [10] investigate the conditions under which a large majority
of boards making a same decision can emerge in the network. In their model board directors are
engaged in a decision making dynamics based on ”herd behavior” and boards influence each other
through shared directors. They find that imitation of colleagues and opinion bias due to the interlock
do not trigger an avalanche of identical decisions over the board network, whereas the information
about interlocked boards’ decisions does. There is no need to invoke global public information, nor
external driving forces. This model provides a simple endogenous mechanism to explain the fact
that boards of the largest corporations of a country can, in the span of a few months, take the same
decisions about general topics, despite the a priori uncertainty of the economic trend.
The results of this model find some support in previous studies carried on in sociology. Haunschild
[11] demonstrates the role of inter-organizational imitation of managers in US corporate acquisition
activity in the 80’. Davis and Greve [12] have studied the diffusion of governance practices such as
the so called ’poison pill’ and ’golden parachute’, throughout the board network of the US largest
corporations in the 80’ (The poison pill is a counter measure against hostile takeover allowing “target
shareholders to acquire shares at a 50 % discount if an hostile shareholder passes a certain ownership
threshold”).
Similar issues concern of course not only the boards of large corporations, but many governance
structure in social institutions. Boards and directors networks have the advantage to be a relatively
well defined framework for which data are publicly available.
In this paper we adopt a complex networks approach, thus going far beyond the analysis of the
average quantities involved in the definition of a Small World (average clustering coefficient and
average shortest path).
We report an extensive and comparative analysis of the topological properties of the board network
and the director network of two cases of study: the corporations of Fortune 1000 for the year 1999 and
the companies quoted in the Milan Stock Exchange Market for the years 1986 and 2002. We show
that several statistical properties are common to the different data sets despite the fact that they
refer to different years and countries. These facts suggest that some universal formation mechanism
is at work for this kind of networks, which is not captured in a satisfactory way by the existent
models of network formation.
Our analysis indicates that: all the considered networks are Small World networks assortative
highly clustered. They are all above the percolation threshold in the sense that they have a strongly
connected giant component. The connectivity degree distribution of the director network has always
a power law tail with similar exponent. On the contrary, the connectivity degree distribution of the
board network is always rapidly decreasing. The director networks have common properties in the
different cases of study. The board networks have different properties with respect to the director
networks, but these properties are essentially the same in the different cases of study. The presence
of a lobby [9] in a board, turns out to be a macroscopic phenomenon in all cases of study.
3
DATA ANALYSIS
The data sets we consider span over different countries and over time. We analysed the composition
of the boards of the Fortune 1000 corporations in 1999 (1000 companies) and the boards of the quoted
companies in the Milan Stock Market in two temporal snapshots: 1986 (220 companies) and 2002
(240 companies). Data are taken from technical publications used by stock market operators [13–15].
In this paper we map the set of corporate boards into a bipartite graph and we perform a thorough
statistical analysis: we analyse the average quantities, the statistical distributions and the degree-
degree correlations. Finally, we focus on the econophysical significance of such an analysis, exploring
the picture of the market emerging from it.
The bipartite graph structure
A bipartite graph is composed by nodes belonging to two separate classes, and one edge connects
always a node of one class to one of the other. An example is reported in figure 1. This is a suitable
structure to represent our systems, since each node represents either a director or a company, and
we put one link between them, if the first sits in the board of the second. We call an interlock a
link between a director of one company to another company. In other words, we have an interlock
when the same director sits in the boards of two companies. If two directors of a given board, serve
together as well in another board, we then have a multiple interlock. we call lobby the subset of
directors of a boards who serve on an outside board together with a director of the present board
(after [9]). In fact, the members of such a sub-group will be more strongly connected to each other
than with the other members of the board, and they will have common interests outside the company
under consideration.
As it is well known, two one-mode networks can be extracted from a bipartite graph. Each
of these is composed by all the nodes belonging to one class only, conveniently connected. In
particular, we will join two companies (directors) if they have at least one director (company)
in common. It is worth highlighting that, the existence of interlocks in the bipartite graphs is
a necessary condition for the connectedness of the one-mode projections. For example, if there
weren’t directors administrating more than one company, the projections would be split up into
small clusters completely connected inside themselves, but without any connections between each
other, each corresponding to a company. It is worth noticing, as well, that a concept of weight
naturally arises from the procedure of projection. Indeed, we have very different situations if -for
instance- two companies are connected by only one director, or if they have two or more common
directors. Such a difference can be included in a weight attached to each edge, whose value reflects
the strength of the interaction between the two nodes it connects.
A bipartite graph can be represented in a compact way by means of its adjacency matrix:
Cαi =½1 if αsits in board i
0 otherwise (1)
This is an M×Nmatrix, Mbeing the number of directors, and Nbeing the number of companies.
This is a binary matrix, and in general it is neither square, nor symmetric. For the one-mode
projection relative to the companies, we should take into account that the number of directors
sitting in each of the boards of companies iand j, is equivalent to the number of paths of length
2 connecting iand jin the bipartite graph. Therefore, this number, that gives the weight of the
connection between iand j, can be expressed in terms of the adjacency matrix. In the end, if we
define the adjacency matrix of the projection as
Bij =½wij if iand jare connected with weight wij
0 if iand jare not connected (2)
then, we obtain that its entries are:
Bij =X
α
CαiCαj .(3)
It is easy to derive a general relationship between the adjacency matrix of the bipartite graph and
that of the one-mode projection.
B=CTC. (4)
4
With the same reasoning we derive the corresponding result for the directors’ projection.
Dαβ =X
i
CαiCβ i.(5)
And,
D=CTC. (6)
This representation stores most of the information regarding the networks into a compact matrix. It
will be used in our further studies on bipartite graphs. For the moment, we notice that the matrix
elements have a straightforward interpretation. While the off-diagonal entries are essentially the
weights of the edges, the diagonal entries, are, respectively, the size Bii of the board of the company
i, and the number Dαα of boards director αserves on.
Average quantities
In table I some of the global and average values describing the two projections are reported.
For sake of comparison, we reported in the same table and in the next, the values corresponding to
two well studied networks: cond-mat [17] and A.S.Internet 1999 [16]. The first one is the network
of authors of the papers of condensed matter physics archived at Los Alamos Laboratories. The
second, is the Internet map, as it appeared in 1999, considered at the autonomous systems level.
While the first network is a social one, and we expect to observe some similarity with the ones of
the directors we are studying, the second one is a technological network.
Observing the projections from a global point of view, the first thing to notice is the presence of a
maximal connected component. The fraction Nc/N of nodes belonging to it is over the value of 0.8
for all the projections. Since the connectedness derives from the interlock, we can conclude that the
phenomenon of the interlock is strong enough to bring the projected networks above the percolation
threshold.
Another interesting aspect to be noticed, is that both the projections are much less sparse than
the comparison networks. A measure of the sparsity of a network is the value of its average degree
kcompared with the one it would have if it were completely connected kc(equal to N−1, where
Nis the number of nodes). The value of k/kcfor the boards projection is bigger than that for
the directors projection, which is anyway one order of magnitude higher than that of cond-mat and
Internet.
Finally, the network displays small-world property. The average distance between two nodes of
the maximal connected component is always of the order of few units, thus of the order of log(N), N
being the total number of nodes. Moreover, the clustering coefficient is exceptionally high, reaching
values around 90% in the directors network and around 35% in the boards one. This indicates a
remarkable tendency towards clicquishness in both of them.
Fluctuations in the average quantities can be observed both in space and time. Such a behaviour
can be essentially reduced to the fluctuations in the dimensions and number of links of the networks.
As far as the Italian market is concerned, it should be taken into account that an element of noise
was introduced when collecting data. Indeed, the 1986 data include the so called restricted market,
i.e. that composed by companies listed only in certain cities (for example, only Milan or Rome,
and not in the whole national market). This slice of the market is absent in 2002 data, and this
accounts for most of the fluctuations in the global quantities describing the two Italian networks.
Nevertheless fluctuations do not affect the statistical properties.
Distributions
Let us now move from the average quantities describing the networks toward their statistical
distributions.
In figures 2, 3, the weight distributions are displayed. Weights correspond to the off-diagonal
elements of the adjacency matrixes. A broad distribution can be observed in both the projections,
but, significantly enough, the Italian market show a lower slope, as well as a higher maximum weight
than those displayed by the American one (two boards with six common directors and two directors
with eight common companies). This suggests a stronger interlock and lobbying phenomena in the
first one.
5
The number of companies administrated by each director and the distribution of the size of
the boards (the corresponding quantity in the other projection) show very different distributions
(figure 4,5). These are exactly the distributions of the diagonal matrix elements. The first one is
broad, suggesting a heterogeneous hierarchy in the power of directors (if we take as a measure of
power the number of boards a director sits in). Peaks of influence are reached by directors that
sit simultaneously in a number of boards of the order of 10. The board size distribution, on the
contrary, displays a characteristic scale around the value of 10, and a maximum size around the
value of 30.
Again we find radically different behaviours of the degree distribution for the two projections
(figures 6, 7). While the boards show a rapidly decaying trend, the directors display a power law
tail with slope 2.5. It is necessary to observe that, on the contrary of what happens for cond-mat,
such behaviour appear only above a threshold of around 10 directors: it is very unlikely to find a
director connected with less than 10 colleagues, essentially because this is the characteristic size of
a board. Interestingly enough, the power law tail stretches much further than the maximal size of
the boards. Whenever a director has more than 10 links in general this is the result of the interlock
between different boards.
A similar behaviour between the two projections and between them and cond-mat can be observed
when the site-betweenness distribution is plotted (figure 9 and 8), since a rapidly decaying trend is
always displayed. A positive correlation between site betweenness and degree is displayed in figure
11.10, where the trends are increasing power-laws with slopes 2.2 for the directors and 1.5 for the
boards.
Degree degree correlation
As observed in many recent studies[6], a feature that seems to be characteristic of social network
is the assortativity. In other words, the presence of positive degree correlations. In particular, social
networks tend to be assortative, in the sense that nodes with a certain degree tend to connect with
nodes of similar degree. This tendency can be measured by means of the distribution Knn(k) (the
average nearest neighbour degree of a node of degree k), and by the assortativity coefficient. The
first distribution is given by
Knn(k) = X
k0
k0P(k0|k) (7)
where P(k0|k) gives the probability that a nearest neighbour of a node of degree khas degree k0. This
distribution is increasing, flat or decreasing if, respectively, the network is assortative un-assortative
or disassortative (terms corresponding to positive, null or negative degree correlation). A complete
definition of the assortativty coefficient can be found in ref. [6]. For the purpose of this work, by
the way, it is enough to say that it is proportional to the correlation coefficient of the degrees iand
jof the nodes at the end of an edge. The plots of Knn(k) for the networks under study display a
slight increase, but the values of the assortativity coefficient, are definetely positive (see table II).
CONCLUSIONS: THE EMERGING PICTURE OF THE CORPORATE CONTROL
NETWORK
Before summarizing our main results, one further quantity needs to be added to give the magnitude
of the multiple interlock. We have computed the percentage of companies whose board contains a
lobby of size at least two (we call lobby the subset of directors of a boards who serve on an outside
board together with a director of the present board. Thus a lobby of size 2 consists of two directors
co-administrating another company). It turns out that 35% of US companies and 44% (1986) and
63% (2002) of Italian companies have a lobby of size at least 2. This is a macroscopical phenomenon,
and it has been shown [9] that it can affect the decision making process of boards, making possible
that minorities manage to drive the board decision against the interest of the majority.
From the analysis we have performed, a number of global features seems to characterize the
network of the entities who control the major corporation of a country: all the considered networks
are Small World networks, assortative and highly clustered. They are all above the percolation
threshold in the sense that they have a giant component. The connectivity degree distribution
of the director network has always a power law tail with similar exponent. On the contrary, the
connectivity degree distribution of the board network is always rapidly decreasing. The director
6
networks have the same common properties in the different cases of study. The board networks
have different properties with respect to the director networks, but these properties are essentially
the same in the different cases of study. The presence of a lobby in a board, turns out to be a
macroscopic phenomenon in all cases of study. These results seem stable over different countries
(Italy and US) and over time (Italy 1986 and 2002).
Because economies depend so much on the choice of the corporate elite, it is clear that the observed
features should be taken into account in modelling macro-economical dynamics.
ACKNOWLEDGEMENTS
Data about Fortune 1000 boards were kindly provided by Gerald Davis [3]. Data of boards
of companies quoted in the MIB were collected by Stefano Battiston and Diego Garlaschelli. A
special thank to Guido Caldarelli for advice and coordination. We thank Guido Caldarelli, Diego
Garlaschelli, Vito Servedio e Simone Triglia for precious discussions. This work is supported by
FET-IST department of the European Community, Grant IST-2001-33555 COSIN.
[1] B.Mintz, and M.Schwartz, The Power Structure of American Business,(1985), University of Chicago
Press.
[2] L.D.Brandeis, Other People’s Money: And How the Bankers Use It. New York (1914), Frederick A.
Stokes.
[3] Davis, G.F., Yoo, M., Baker, W.E., The small world of the American corporate elite, 1982-2001, Strategic
Organization 1: 301-326 (2003).
[4] J.D.Watts and S.Strogatz, Collective Dynamics of ”Small World” Networks, Nature 393 (1998) 440-442.
[5] M. E. J. Newman, S. H. Strogatz, and D. J. Watts, Random graphs with arbitrary degree distributions
and their applications, Phys. Rev. E 64, 026118 (2001).
[6] M. E. J. Newman, Assortative mixing in networks, Phys. Rev. Lett. 89, 208701 (2002).
[7] M. E. J. Newman and Juyong Park, Why social networks are different from other types of networks,
Phys. Rev. E, in press.
[8] M. Catanzaro, G. Caldarelli, L. Pietronero, Assortative model for social networks, cond-mat 0308073
v1
[9] Battiston, S., Bonabeau, E., Weisbuch G., Decision making dynamics in corporate boards, Physica A,
322, 567 (2003).
[10] Battiston, S., Weisbuch G., Bonabeau, E., Decision spread in the corporate board network, submitted.
[11] Haunschild, P.R., Interorganizational imitation: the impact of interlocks on corporate acquisition ac-
tivity, Administrative Science Quarterly 38, 564-592 (1993).
[12] Davis, G.F. and Greve, H.R., Corporate elite networks and governance changes in the 1980s, Am. J. of
Sociology, 103, 1-37 (1996).
[13] Mediobanca, Calepino dell’Azionista (Milano) (1986).
[14] Banca Nazionale del Lavoro, La meridiana dell’investitore 2002 (Class Editori, Milano) (2002).
[15] Fortune 1000 (Thanks to Gerald Davis, Michigan University).
[16] A. Vazquez, R. Pastor-Satorras, A. Vespignani, Physical Review E 65, 066130 (2002).
[17] M. E.J. Newman, Physical Review E 64, 016132 (2001).
7
TABLE I: Average and global quantities for The boars’ projections, marked with B, the directors’ projec-
tions, marked with D, cond-mat and Internet. N=number of nodes, E= number of edges, Nc/N=fraction
of nodes belonging to the maximal connected component, k/kc=average degree over N−1, b=average site
betweenness, Cc=average clustering coefficient, d=average distance.
B, 86 B , 02 B, US D, 86 D, 02 D, U S C −M AS, 99
N221 240 916 2378 1906 7680 16725 5287
E1295 636 3321 23603 12815 55437 47594 10100
Nc/N 0.97 0.82 0.87 0.92 0.84 0.89 0.83 −
k/kc(%) 5.29 2.22 1.57 0.84 0.71 0.79 0.03 0.07
b/N 0.736 0.875 1.080 1.116 1.206 1.384 1.932 2.21
Cc 0.356 0.318 0.376 0.899 0.915 0.884 0.327 0.241
d3.6 4.4 4.6 2.7 3.6 3.7 6.4 3.7
TABLE II: Assortativity coefficients
B, 86 B , 02 B, US D, 86 D, 02 D , US C −M AS, 99
r0.12 0.32 0.27 0.13 0.25 0.27 − −
8
B
D
A
C
E
F
G
H
I
J
K
A B C D E F G H I J K
1 2 3 4
FIG. 1: A bipartite graph and its one-mode projection. After Newman et al. 2001
1w
1
10
100
1000
10000
1e+05
#(w)
1
1
10
100
1000
10000
1e+05
IT,86
IT,02
US,99
FIG. 2: Weight distribution of the directors’ projection: broad distribution, with differences visible between
the American and the Italian market.
1w
1
10
100
1000
#(w)
1
1
10
100
1000 IT,86
IT,02
US,99
FIG. 3: Weight distribution of the companies’ projection.
9
1 10
n
1
10
100
1000
#(n)
1 10
1
10
100
1000
IT,86
IT,02
US,99
FIG. 4: Distribution of the number of boards per director: broad distribution.
10
n
1
10
100
#(n)
10
1
10
100 IT,86
IT,02
US,99
FIG. 5: Distribution of the size of the boards: characteristic scale around 10.
10 100
k
0.001
0.01
0.1
P(k)
10 100
0.001
0.01
0.1 IT,86
IT,02
US,99
CM
FIG. 6: Degree distribution of the directors’ projection: power law tail.
10 20 30 40
k
0.01
0.1
1
P’(k)
10 20 30 40
0.01
0.1
1
IT,86
IT,02
US,99
FIG. 7: Rapidly decreasing degree distribution of the companies’ projection.
10
0510 15 20 25
b/N
0.01
0.1
1
P’(b)
0510 15 20 25
0.01
0.1
1
IT,86
IT,02
US,99
C-M
FIG. 8: Rapidly decreasing Site betweenness distribution of the directors’ projection.
0 1 2 3 4 5
b/N
0.1
1
P’(b)
0 1 2 3 4 5
0.1
1
IT,86
IT,02
US,02
FIG. 9: Rapidly decreasing Site betweenness distribution of the companies’ projection.
10 100
k
0.01
1
100
b/N
10 100
0.01
1
100
IT,86
IT,02
US,99
C-M
FIG. 10: Site betweenness-degree correlation in the directors’ projection: increasing power law trend.
10
k
1
b/N
10
1
IT,86
IT,02
US,99
FIG. 11: Site betweenness-degree correlation in the companies’ projection: increasing power law trend.
11
10 k
1
10
100
Knn(k)
10
1
10
100
IT,86
IT,02
US,99
FIG. 12: Increasing average nearest neighbour degree of the nodes of degree kin the directors’ pro jection.
1 10
k
10
Knn(k)
1 10
10
IT,86
IT,02
US,99
FIG. 13: Increasing average nearest neighbour degree of the nodes of degree kin the directors’ pro jection.
