## 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.

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## Notes

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

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.

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

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

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.

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.

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

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.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.

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).

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.

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## 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

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:

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

### 1.3 Mandate frequencies and transition matrices

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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

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