Fig 1.
Shareholder-company network example.
An example of our shareholder-company network, which illustrates the relationships between shareholders, the upper blue circles numbered 1 to 12 and companies, the lower red squares labelled A to G. An edge indicates that there is an investment from the shareholder represented by the source node in the company represented by the target node. For example, shareholders 3 and 5 have both invested in company E, and an arrow represents this relation. In addition, shareholder 3 has invested in company A, but no other shareholder.
Fig 2.
The “shareholder network” for the data displayed in Fig 1. This is the projection of the shareholder-company network onto just the shareholder nodes. The nodes here are just the shareholder nodes 1 to 12. An edge indicates that two shareholders have assets in common. For example, shareholders 3 and 5 have both invested in E, therefore, an edge between nodes 3 and 5 exists in this projected graph. Note that a simple network is used, edges have no directions and no weights, and there are no self-edges.
Table 1.
Summary statistics for different types of shareholder.
The different types of shareholder recorded in our data as retrieved from the BvD database. The numbers found in our different data sets are in the righthand columns.
Table 2.
Basic information on the two data sets.
Each data set looks at the companies registered in one country and their shareholders from any country for the year 2014. There is no information on many of the Companies as the numbers above indicate. The number of edges in our shareholder network is based on the information available on the shareholding information. The slope γ of a power-law degree distribution of similar slope, P(k) ∼ k−γ, is a rough characterisation to illustrate the broad distributions. ‘LCC’ is the largest connected component.
Fig 3.
Plots of degree distributions.
The degree distributions P(k) (the frequency of nodes with degree k) against degree k edges on a log-log scale for shareholder networks where the holdings are in (a) Turkish companies, (b) Dutch companies. The red dots are the raw data, the green crosses represent the same data in logarithmic bins, and the blue lines are the best linear fits (P(k) ∼ k−γ) to ranges of k values where we see approximately linear behaviour. The slope of the blue lines, −γ, is 2.6 and 2.7 for Turkey and the Netherlands respectively. A summary of the general statistics of these shareholder networks can be found in Table 2.
Fig 4.
Violin plots of the degree of the most common types of shareholders for the largest connected component of shareholder network of (a) Turkish and (b) Dutch companies. There are too few shareholders for other types of investor. This figure breaks down degree distributions into different types of shareholders. We note that in Turkey, the large degrees are contributed by the banks and insurance while in Netherlands, banks’ average degree is higher than the other types of shareholders. It means Netherlands’ banks co-invested a lot with other shareholders.
Fig 5.
The number of components increases as the number of nodes removed.
(a) Turkish and (b) Dutch companies. Nodes of one shareholder type are chosen at random and removed one by one from the largest connected component of the shareholder network. Results shown here are averaged over 100 realisations. Blue represents Bank shareholders being removed, green represents Corporate and red represents Families. The larger scale plots display the regions in the dashed boxes of the smaller scale plots to more clearly reveal the behaviour for small numbers of node removals. Note in particular the different role of banks (blue) and corporates (green) in Turkey and Netherlands. The small and big plots share the same axis labels.
Fig 6.
Heatmaps of degree assortativity.
The degree assortativity in the LCC for different investor types. It has been normalised for the counts of pair of each edge in LCC. The assortativity coefficient r is 0.17 for Turkey and 0.0081 for Netherlands. The colour bar is set at the same scale.
Fig 7.
Violin plot of diversity index of selected types of shareholders.
(a) Turkey and (b) Netherlands. The information of other type is not listed here because due to the limited information available and the limited amount of the data. The blue space is the diversity index density estimation and compared with a null index (indicated by a green line) which is define as dnull in Eq (4).
Fig 8.
The mean betweenness values for different types of shareholders in the largest connected component of shareholder networks, (a) for Turkey and (b) for the Netherlands. The red dots are the real data and the box plots for the results obtained from 100 degree preserving null models. We note that most betweenness values for Turkey and Netherlands are significantly different from the randomised networks, some types are lower and some types are higher. That means that there is significant network structure on larger scales and the properties are not just controlled by the degree.
Fig 9.
The average closeness indices for different types of shareholders in the LCC of Turkish shareholder network.
In Fig (a) we show the results for each shareholder with the red dots for the original data while the box plots are for the randomised data. In (b) each point shows the average ‘Farness’ (the inverse of closeness) of one shareholder type against log(N/k), where k is the average degree of nodes of that type. The higher blue points are for the original data, the lower orange points are for the randomised network. The lines in (b) are for a linear fit to the points. The slope of this fit to the original data is 0.71, 0.26 for the randomised network and the theoretical value in a random branching model is 0.24.
Fig 10.
The average closeness indices for different types of shareholders in the LCC of Dutch shareholder network.
In Fig (a) we show the results for each shareholder with the red dots for the original data while the box plots are for the randomised data. In (b) each point shows the average ‘Farness’ (the inverse of closeness) of one shareholder type against log(N/k), where k is the average degree of nodes of that type. The higher blue points are for the original data, the lower orange points are for the randomised network. The lines in (b) are for a linear fit to the points. The slope of this fit to the original data is 0.34, 0.16 for the randomised network and the theoretical value in a random branching model is 0.17.
Fig 11.
Illustration of communities in a network.
The same projected graph as in Fig 2 from the network graph shown in Fig 1. The different colours label them as different communities which are the structural characteristics in the network science context. As in the graph, 1-12 shareholders are categorised into 4 communities, 12 belongs to one community, 2,3,4,5,6 belong to another community, 8,9,10,11 are in the third community and 1,7 are in the fourth community.
Table 3.
Statistics of the communities.
Communities are found in the shareholder networks derived from the two data sets using the two different methodologies, Louvain (L) and Infomap (I). ‘Avg. community size (CS)’ is the average number of shareholders in one community. The average community size is defined by as the number of shareholders divided by the number of communities, where the number of communities include the communities whose community size is 1. The average community sizes excluding single nodes are: 3.03 and 3.01 for Turkey and 3.03 and 2.98 for Netherlands.
Fig 12.
Community size frequency distribution.
Community size frequency distribution for (from top to bottom): (a) Turkey and (b) the Netherlands. The figures are plotted on log-log scale. For each country we show the results from two methods; Louvain on the left and Infomap on the right. The blue cross represents the data, the green dot represents the data binned using a logarithmic binning, and the black line is a linear fit to the binned data.
Fig 13.
The number of communities found with a given fraction of one type of shareholder.
The communities are found with Infomap in the shareholder network for Turkey and Netherlands. On the left we have the fraction of Family shareholders in different communities while on the right we have the fraction of Corporate shareholders in each community. The figures in first row includes community of all sizes. The fat-tailed distribution means this is dominated by the large number of small communities, and these are almost always of a single type of shareholder, hence the peak at 1.0. The second row shows the same analysis done when we exclude small communities which have three or less nodes (CS = community size). Similar analysis for the Louvain community detection method is given in the S1 Appendix.