The Main Methodology: Computing Control in Ownership Networks
In this chapter the mathematical bulk of the thesis is presented. The aim of having an exhaustive account of the methods results in the extensive scope of the chapter. As networks find their mathematical embodiment in adjacency matrices (see Appendix B),
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The Main Methodology: Computing Control in Ownership Networks
The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Fundamentally, we do not know why our theories work so well. (E. Wigner in Wigner 1960)
In this chapter the mathematical bulk of the thesis is presented. The aim of having an exhaustive account of the methods results in the extensive scope of the chapter. As networks find their mathematical embodiment in adjacency matrices (see Appendix B), most of the formalism is comprised of linear-algebraic manipulations. The reader who is mostly interested in the application of the methods, i.e., the empirical network analysis, can directly go to Chaps. 3 and 4. In Chap. 5 a network-evolution model is presented, shedding new light on the micro rules underlying the empirical properties. All these chapters are written to be self-supporting and provide a minimal introduction to the details of the methodology given in the following. Alternatively, a brief summary is found in Sects. 2.9 and 2.11. Chapter 6 also summarizes the results before discussing the relevance and implications of our findings.
2.1 Introduction Given an adjacency matrix of an ownership network, what can be said about the distribution of control? The long answer to this question encompasses the following aspects: 1. the introduction of existing measures; 2. their extension and correction; 3. a reinterpretation and unification using network-theoretic notions.
J. B. Glattfelder, Decoding Complexity, Springer Theses, DOI: 10.1007/978-3-642-33424-5_2, © Springer-Verlag Berlin Heidelberg 2013
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2 The Main Methodology: Computing Control in Ownership Networks
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Fig. 2.1 a Company i has Wi j percent of direct ownership in company j and indirect ownership in companies k and l, through j; the computation of indirect ownership is straightforward as long ik = Wi j W jk ; b in the presence of cycles, it is as the network has no cycles; as an example, W necessary to account for the arising recursive paths of indirect ownership; the model of integrated ownership does this, see Sect. 2.2.1
In a nutshell, the answer is as follows: from the knowledge of the ownership relations the control associated with a shareholder can be estimated. This quantity then needs to be adapted to account for all the indirect paths in the network. Recall from Sect. 1.1.3 that the percentage of ownership firm i has in company j is given by the entry Wi j in the adjacency matrix. The underlying value of the firms are denoted by v j . In Chaps. 3 and 4, v j is taken to be the market capitalization and the operating revenue, respectively. In the next section, a first try at assessing the impact of a network structure on ownership is presented.
2.2 Direct and Indirect Ownership Figure 2.1 illustrates how chains of direct ownership lead to indirect paths of ownership. In the case of tree-like topologies, e.g., Fig. 2.1a, the indirect paths are mul
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