Backbone of complex networks of corporations: The flow of control

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.

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.

Figure 2

Definition of the concentration index $s_j$, measuring the number of prominent incoming edges, respectively, the effective number of shareholders of the stock $j$. When all the weights are equal, then $s_j=k_j^{in}$, where $k_j^{in}$ is 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 $j$.

Figure 3

The definition of the control index $h_i$, 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 $H_{ij}$, which is a model of how ownership can be mapped to control (see the discussion in Appendix ).

Figure 4

Probability distributions of $s_j$ for selected countries; PDF in log-log scale.

Figure 5

(Color online) Various probability distributions for selected countries: (top panel) CDF plot of $k_i^{out}$; (middle panel) CDF plot of $h_i$; (bottom panel) PDF plot of $h_i$; all plots are in log-log scale.

Figure 6

First steps in computing cumulative control: (a) selecting the most important shareholder (light shading) ranked according to the ${\tilde{c}}_i$ values and the portfolio of stocks owned at more than 50% (dark shading); in the second step (b), 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%.

Figure 7

(Color online) Fraction of shareholders $\eta$, sorted by descending (integrated) control value ${\tilde{c}}_i$, cumulatively controlling $\vartheta$ percent of the total market value; the horizontal line denotes a market value of 80%; the diagram is in semilogarithmic scale.

Figure 8

(Color online) (a) the backbone of JP; (b) the backbone of CN (for the complete set of backbone layouts consult [31]); the graph layouts are based on [41].

Figure 9

The map of control: illustration of idealized network topologies in terms of local dispersion of control $\left(\overline{s}\right)$ vs global concentration of control $\left(\overline{h}\right)$; 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.

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 (light grey) nodes and sized by market capitalization, shareholders are blue (dark grey), 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 [41].

Figure 11

(Color online) Map of control: local dispersion of control, $\overline{s}$, is plotted against global concentration of control, $\overline{h}$, for 48 countries.

Figure 12

(Color online) Map of market value: local dispersion of control, $\overline{s}$, is plotted against global concentration of market value, ${\eta }^{\prime }$, for 48 countries.