Skip to main content
Log in

Persistence in corporate networks

  • Regular Article
  • Published:
Journal of Economic Interaction and Coordination Aims and scope Submit manuscript

Abstract

We examine the bipartite graphs of German corporate boards in 1993, 1999 and 2005, focusing on their projections onto directors (the “personal” network) and onto companies (the “institutional” network). The novel feature here is our focus on the temporal evolution of the two projections. The personal networks exhibit cores of highly central directors who are densely connected among themselves, while the institutional networks show a persistent core of large corporations whose identity remains mostly the same. This results in the persistent presence of a core network of very large corporations, despite substantial turnover in the identity of directors and significant changes in Germany’s corporate governance during the investigated period. Our findings strongly suggest that core persistence originates from the board appointment decisions of the very largest corporations and is largely independent of personal destinies.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. See Uzzi et al. (2007) for a review of small world networks in the social sciences.

  2. It would appear that the work by Kogut and Walker (2001) and the much earlier study by Mariolis and Jones (1982) account for the entire international literature on the dynamics of the corporate network to date.

  3. See http://www.boerse-frankfurt.de/en/basics+overview/indices/selection+indices for an explanation of the rules on index composition and re-adjustments. Information on the historical index composition is provided under http://dax-indices.com/DE/MediaLibrary/Document/Historical_Index_Compositions.

  4. Some companies in our sample have a board of directors that consists of internal and external directors, as in many Anglo-Saxon countries. Their roles are similar but not identical to Vorstand and Aufsichtsrat in the traditional German system.

  5. Dresdner Bank, for instance, was acquired by Allianz in the financial sector, while VEBA and VIAG merged to EON in the utilities industry.

  6. Notice that our b-cores differ from so-called k-cores, which are constructed using a node’s minimum degree (see Seidman 1983). See also Bellenzier and Grassi (2014), who use a very similar methodology in an analysis of the Italian board network.

  7. See http://www.corporate-governance-code.de/index-e.html.

  8. Current members of the management board must not simultaneously serve on the company’s supervisory board (§  105 AktG), but have routinely been allowed to serve as supervisory board members at other companies.

  9. For a discussion of the finding that power and centrality are not equivalent see Bonacich (1987).

  10. The percentage of reconstructed ties among German companies is four to five times higher than previously observed in the US by Stearns and Mizruchi (1986). From a network core perspective, it would be interesting to clarify whether the percentage of reconstituted ties among the largest (core) corporations in the US is substantially higher than in the original Stearns and Mizruchi sample.

  11. Imagine a director with three mandates and suppose that she is on the board of company A, which manages to place her on the board of company B for strategic reasons, e.g., to oversee A’s interests. If she also serves on the board of a third company C, we consider the link between A and B intentional, while the links AC and BC are unintentional byproducts.

  12. These and other figures are easily calculated from the transition matrices in the “Appendix”.

  13. All three measures of centrality exhibit a slight decrease in average centrality between 1993 and 1999, and an increase between 1999 and 2005, as reported in Table 3. This is in line with the visual inspection of the network structure in Fig. 1, which shows an increasing number of peripheral nodes in 1999, and denser cores in 2005.

  14. The Laplacian is a special form of the adjacency matrix of the network, where the trace of the Laplacian corresponds to the number of links between nodes.

  15. The benefit of this method is that PCA is more parsimonious and transparent than community detection algorithms, and perhaps also better known among social scientists. The foundations of our subsequent analysis can be found in Reichardt (2009).

  16. If the best score of a firm is not greater than the mean plus half a standard deviation of scores within a group, we do not match the firm to the group. The exact tuning of this threshold is of course arbitrary: choosing a much larger threshold level leads to smaller but more homogeneous groups (some firms might not be matched at all because of a single differing link), while a much lower threshold will inflate groups by matching peripheral firms that only show marginal similarity in the link pattern. Since neither group turns out to be large compared to the entire set of firms, our quantile-approach is quite robust, and the exact value of the threshold is not crucial for the overall results. In larger datasets, this parameter could certainly be endogenized.

References

  • Alfarano S, Milaković M (2009) Network structure and \(N\)-dependence in agent-based herding models. J Econ Dyn Control 33:78–92

    Article  Google Scholar 

  • Alfarano S, Milaković M, Raddant M (2013) A note on institutional hierarchy and volatility in financial markets. Eur J Finance 19(6):449–465

    Article  Google Scholar 

  • Barabási A-L, Albert R (1999) Emergence of scaling in random networks. Science 286:509–512

    Article  Google Scholar 

  • Battiston S, Bonabeau E, Weisbuch G (2003) Decision making dynamics in corporate boards. Phys A 322:567–582

    Article  Google Scholar 

  • Battiston S, Catanzaro M (2004) Statistical properties of corporate board and director networks. Eur Phys J B 38:345–352

    Article  Google Scholar 

  • Bellenzier L, Grassi R (2014) Interlocking directorates in Italy: persistent links in network dynamics. J Econ Interact Coord 9:183–202

    Article  Google Scholar 

  • Beyer J, Höpner M (2003) The disintegration of organized capitalism: German corporate governance in the 90s. West Eur Polit 26:179–198

    Article  Google Scholar 

  • Bonacich P (1972) Factoring and weighting approaches to status scores and clique identification. J Math Sociol 2:113–120

    Article  Google Scholar 

  • Bonacich P (1987) Power and centrality: a family of measures. Am J Sociol 92(5):1170–1182

    Article  Google Scholar 

  • Borgatti SP (2005) Centrality and network flow. Soc Netw 27:55–71

    Article  Google Scholar 

  • Borgatti SP (2006) Identifying sets of key players in a social network. Comput Math Organ Theory 12:21–34

    Article  Google Scholar 

  • Borgatti SP, Everett MG (1999) Models of core/periphery structures. Soc Netw 21:375–395

    Article  Google Scholar 

  • Borgatti SP, Foster PC (2003) The network paradigm in organizational research: a review and typology. J Manag 29(6):991–1013

    Google Scholar 

  • Brass DJ, Galaskiewicz J, Greve HR, Tsai W (2004) Taking stock of networks and organizations: a multilevel perspective. Acad Manag J 47(6):795–817

    Article  Google Scholar 

  • Caldarelli G, Catanzaro M (2004) The corporate boards networks. Phys A 338:98–106

    Article  Google Scholar 

  • Conyon MJ, Muldoon MR (2006) The small world of corporate boards. J Bus Finance Account 33(9):1321–1343

    Article  Google Scholar 

  • Davis GF, Greve HR (1999) The money center cannot hold: commercial banks in the U.S. system of corporate governance. Adm Sci Q 44:215–239

    Article  Google Scholar 

  • Davis GF, Yoo M, Baker WE (2003) The small world of the American corporate elite, 1982–2001. Strateg Organ 1(3):301–326

    Article  Google Scholar 

  • Galaskiewicz J (2007) Has a network theory of organizational behaviour lived up to its promises? Manag Organ Rev 3(1):1–18

    Article  Google Scholar 

  • Haunschild P, Beckmann C (1998) When do interlocks matter? Alternate sources of information and interlock influence. Adm Sci Q 43(4):815–844

    Article  Google Scholar 

  • Heemskerk E (2007) Decline of the corporate community. Amsterdam University Press, Amsterdam

    Google Scholar 

  • Holme P (2005) Core-periphery organization of complex networks. Phys Rev E 72:046111

    Article  Google Scholar 

  • Jobson DJ (1991) Applied multivariate data analysis. Springer, New York

    Book  Google Scholar 

  • Kogut B, Walker G (2001) The small world of Germany and the durability of national networks. Am Sociol Rev 66(3):317–335

    Article  Google Scholar 

  • Mariolis P, Jones M (1982) Centrality in corporate interlock networks: reliability and stability. Adm Sci Q 27(4):571–585

    Article  Google Scholar 

  • Milaković M, Alfarano S, Lux T (2010) The small core of the German corporate board network. Comput Math Organ Theory 16(2):201–215

    Article  Google Scholar 

  • Mintz B, Schwartz M (1981) Interlocking directorates and interest group formation. Am Sociol Rev 46(6):851–869

    Article  Google Scholar 

  • Mizruchi S (1996) What do interlocks do? An analysis, critique, and assessment of research on interlocking directorates. Annu Rev Sociol 22:271–298

    Article  Google Scholar 

  • Newman MEJ (2003) The structure and function of complex networks. SIAM Rev 45:167–256

    Article  Google Scholar 

  • Newman MEJ (2010) Networks, an introduction. Oxford U.P, New York

    Book  Google Scholar 

  • Newman MEJ, Girvan M (2004) Finding and evaluating community structure in networks. Phys Rev E 4:026113

    Article  Google Scholar 

  • Newman MEJ, Strogatz SH, Watts DJ (2001) Random graphs with arbitrary degree distributions and their applications. Phys Rev E 64:026118

    Article  Google Scholar 

  • Palmer D (1983) Broken ties: interlocking directorates and intercorporate coordination. Admin Sci Q 28:40–55

    Article  Google Scholar 

  • Provan KG, Fish A, Sydow J (2007) Interorganizational networks at the network level: a review of the empirical literature on whole networks. J Manag 33(3):479–516

    Google Scholar 

  • Reichardt J (2009) Structure in complex networks, lecture notes in physics. 766. Springer, Berlin

    Google Scholar 

  • Robins G, Alexander M (2004) Small worlds among interlocking directors: network structure and distance in bipartite graphs. Comput Math Organ Theory 10:69–94

    Article  Google Scholar 

  • Schnettler S (2009) A structured overview of 50 years of small-world research. Soc Netw 31:165–178

    Article  Google Scholar 

  • Schreyögg G, Papenheim-Tockhorn H (1995) Dient der Aufsichtsrat dem Aufbau zwischenbetrieblicher Koopertationsbeziehungen? Zeitschrift für Betriebswirtschaft 65:205–230

    Google Scholar 

  • Seidman SB (1983) Network structure and minimum degree. Soc Netw 5:269–287

    Article  Google Scholar 

  • Soramäki K, Bech M, Arnold J, Glass R, Beyeler W (2007) The topology of interbank payment flows. Phys A 379:317–333

    Article  Google Scholar 

  • Stearns LB, Mizruchi MS (1986) Broken-tie reconstitution and the functions of interorganizational interlocks: a reexamination. Admin Sci Q 31:522–538

    Article  Google Scholar 

  • Uzzi B, Amaral LAN, Reed-Tsochas F (2007) Small-world networks and management science research: a review. Eur Manag Rev 4:77–91

    Article  Google Scholar 

  • von Luxburg U (2007) A tutorial on spectral clustering. Stat Comput 17:395–416

    Article  Google Scholar 

  • Watts DJ, Strogatz SH (1998) Collective dynamics of ‘small-world’ networks. Nature 393:440–442

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Matthias Raddant.

Additional information

This research was supported by the Volkswagen Foundation through their grant on “Complex Networks As Interdisciplinary Phenomena.” We are indebted to Simone Alfarano for useful discussions and comments.

Appendix

Appendix

1.1 Clustering procedure

There are essentially two streams of literature that deal with the detection of groups in networks (see, e.g., Newman 2010). The traditional approach is called graph partitioning and splits the network into a fixed number of subgraphs, e.g., by spectral decomposition of the so-called Graph Laplacian (see, e.g., von Luxburg 2007).Footnote 14 Graph partitioning algorithms can become very time consuming, but the more serious concern is conceptual because the number of useful partitions is generally not known. Community detection therefore considers algorithmic procedures that endogenize the number of subgraphs in the partitioning process, for instance by iterative edge removal based on the calculation of betweenness scores, as in Newman and Girvan (2004). We employ a mixture of these two approaches here, starting from a principal component analysis (PCA) and combining it with a scoring algorithm that creates groups based on the largest components without predefining the number of subgraphs. A combination of PCA and some matching algorithm represents a reasonable alternative to community detection algorithms in the social sciences, because the size of datasets is generally smaller than in the natural sciences.Footnote 15

First we take the adjacency matrix of the firm network and standardize its columns by subtracting the column-wide mean and dividing by the standard deviation of column entries. This new adjacency matrix is denoted \(\hat{\mathbf {B}}\). The column entries can now be interpreted as the relative weights of the companies’ links, while the row entries resemble the relative attention a firm receives through links from other firms. We measure the correlations of link patterns by calculating the empirical correlation matrix \(\mathbf {R}=c^{-1}\hat{\mathbf {B}}^T \hat{\mathbf {B}}\), allowing us to infer which firms have similar relative weights in their link patterns. Based on this correlation matrix we compute our new variables, the principal components \(\mathbf {F}\), which are linear combinations of the original variables such that \(\mathbf {F}=\mathbf {Y} \hat{\mathbf {B}}\). The column vectors in \(\mathbf {Y}\) carry the weights for each new variable, and it can be shown that the solution to this problem amounts to solving for \(\mathbf {F}=\mathbf {V} \hat{\mathbf {B}}\), where \(\mathbf {Y}=\mathbf {V}\) contains the eigenvectors of the correlation matrix \(\mathbf {R}\) ordered by descending eigenvalues, see also Jobson (1991). The eigenvectors with the highest eigenvalues account for a large amount of the variance in the data, while low eigenvalues stand for eigenvectors and components that contribute very little to the variance and are consequently neglected.

To illustrate the principles that we use to form groups, assume that based on some decomposition we have approximated the adjacency matrix \(\mathbf {B}\) by a dimensionally reduced matrix \(\hat{\mathbf {E}}\), where both matrices would contain ones for links and zeros otherwise, and \(\hat{\mathbf {E}}\) being of lower rank than \(\mathbf {B}\) due to the dimensional reduction. To decide whether \(\hat{\mathbf {E}}\) is a good representation of the original network, we essentially have to compare and score by rewarding the correct matching of links (and non-links) in \(\mathbf {B}\) and \(\hat{\mathbf {E}}\), and symmetrically by punishing the matching of links to non-links in either direction. A weighting scheme will be helpful for the comparison because the adjacency matrices of our corporate networks are sparse, therefore actual links are more informative than non-links. These considerations result in a scoring (or error) function of the form

$$\begin{aligned} s\left( \hat{\mathbf {E}}|\mathbf {B},\varOmega _{1 \ldots 4}\right)= & {} \underbrace{ \sum _{i=1}^c \sum _{j=1}^c \varOmega _{1,ij} \hat{\mathbf {E}}_{ij} \mathbf {B}_{ij}}_{\text {links to links}} - \underbrace{ \sum _{i=1}^c \sum _{j=1}^c \varOmega _{2,ij} \left( 1- \hat{\mathbf {E}}_{ij}\right) \mathbf {B}_{ij}}_{\text {non-links to links}} \nonumber \\&-\underbrace{ \sum _{i=1}^c \sum _{j=1}^c \varOmega _{3,ij} \hat{\mathbf {E}}_{ij} (1 - \mathbf {B}_{ij})}_{\text {links to non-links}} + \underbrace{ \sum _{i=1}^c \sum _{j=1}^c \varOmega _{4,ij} \left( 1- \hat{\mathbf {E}}_{ij}\right) (1 - \mathbf {B}_{ij})}_{\text {non-links to non-links}}\nonumber \\ \end{aligned}$$
(4)

where \(\varOmega _{1 \ldots 4}\) represent weighting matrices. It is common to focus on the matching of links in the reference matrix \(\mathbf {B}\) by equally weighting the (mis)matching of original links, \(\varOmega _{1,ij}=\varOmega _{2,ij} > \varOmega _{3,ij}=\varOmega _{4,ij}\) (see Reichardt 2009).

The algorithmic procedure that we employ here to visualize the groups in the network follows exactly the principles illustrated above. The only difference is that instead of comparing two (binary) adjacency matrices, we compare the adjacency matrix with the principal components (which are not binary) in column-wise fashion. The groups to which we want to match firms are given by the largest components resulting from the PCA. Since \(\hat{\mathbf {B}}\) describes links, the components in \(\mathbf {F}\) describe those links in which companies differ. If we normalize the column vectors in \(\mathbf {F}\) and allow for a sign change in every column, we get a new matrix of link profiles \(\hat{\mathbf {F}}\) with dimensions \(c \times 2c\), where each even column contains the entries of the previous (i.e. original, now odd) column with reversed signs. This sign change is necessary since the principal components describe only a new axis within the variable space, but do not inform us of the direction. As detailed below, we will only use the first few columns of \(\hat{\mathbf {F}}\) and calculate a similarity score for each group (represented by a column in \(\hat{\mathbf {F}}\)) and each firm (represented by a column in \(\mathbf {B}\)). This results in a matrix \(\mathbf {S}\) of scores for all firms and groups with dimension \(c \times 2k\), where k is the number of included principal components (more on k below). Given the sign change, the number of groups will be \(G \le 2k\). The weights of links and non-links can be approximated by the number of links versus non-links in the original network. Since the graph density is only about 0.03, we set \(\varOmega _{3}=\varOmega _{4}=0\). Furthermore, we do not need to differentiate the weight of each single link as it is already contained in the respective values of the link profile \(\hat{\mathbf {F}}\), hence \(\varOmega _{1}=\varOmega _{2}=1\). Notice, however, that in contrast to \(\hat{\mathbf {E}}\) the link profile \(\hat{\mathbf {F}}\) is not a binary matrix (it contains the relative weights of links), therefore we lastly introduce a matrix \(\varvec{\Theta }\) that translates \(\hat{\mathbf {F}}\) into a binary form such that \(\varTheta _{jg} = 1\) if \(\hat{\mathbf {F}}_{jg} > 0\) and zero otherwise:

$$\begin{aligned} \mathbf {S}_{ig}= \sum _{j=1}^c \varvec{\Theta }_{jg}\left( \hat{\mathbf {F}}_{jg} \mathbf {B}_{ji}\right) ^2 - \sum _{j=1}^c (\varvec{\Theta }_{jg} -1)\left( \hat{\mathbf {F}}_{jg} \mathbf {B}_{ji}\right) ^2 \end{aligned}$$
(5)

for all \(i=1,\ldots ,c\) and \(g=1,\ldots ,G\). Thus \(\varvec{\Theta }\) ensures that we sum over all positive entries in \(\hat{\mathbf {F}}\) in the first sum, and over all negative entries in the second sum. For every correct match, that means if a firm i has a link to company j where the link profile would suggest one, we increase the score that firm i obtains for group g by the squared entry in the link profile. Symmetrically, if the firm has a link where we do not expect one, we deduct this score. Each firm is now assigned to the group for which it has the highest score (by identifying the maxima in each row of \(\mathbf {S}\)). There are, of course, quite a few firms that score rather poorly in all of the groups, simply because they do not belong to any. These firms either have very few links to begin with, or have a unique link pattern. To filter out such firms, we set a threshold in our scoring procedure, yet it turns out that the grouping is quite robust with respect to the exact value of this threshold.Footnote 16

A critical point in any PCA analysis concerns the question how many components k to include in the first place. In our context, we find it instructive to check how many components will create groups containing at least three firms, representing a rather conservative criterion for defining a group. Our algorithm iteratively increases the number of principal components and stops when the last included component no longer produces an additional group. It turns out that only a fraction of the firms can be mapped to groups, with the greatest eigenvalues accounting for roughly 10 % of total variance, and the smallest relevant eigenvalues accounting for roughly 3 % of total variance. We never include more than the five largest eigenvalues to create significant groups, that is to say that the inclusion of more than the largest five eigenvalues leads to groups of size smaller than three.

1.2 Adjacency matrices sorted by cliques

See Figs. 9, 10, 11.

Fig. 9
figure 9

Resorted adjacency matrix 1993

Fig. 10
figure 10

Resorted adjacency matrix 1999

Fig. 11
figure 11

Resorted adjacency matrix 2005

1.3 Mandate frequencies and transition matrices

See Tables 4, 5, 6, 7, 8.

Table 4 Overall frequency distribution of mandates
Table 5 Frequency of executives’ supervisory board memberships (as far as these roles were clearly reported in the annual reports)
Table 6 Transition matrix for board membership during 1993–1999
Table 7 Transition matrix for board membership during 1999–2005
Table 8 Transition matrix for board membership during 1993–2005

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Raddant, M., Milaković, M. & Birg, L. Persistence in corporate networks. J Econ Interact Coord 12, 249–276 (2017). https://doi.org/10.1007/s11403-015-0165-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11403-015-0165-5

Keywords

Navigation