Network Analysis for Economics and Finance: An Application to Firm Ownership

Historically, networks have been studied extensively in graph theory, an area of mathematics. After many applications to a number of different subjects including statistical physics, health science, and sociology, over the last two decades, an extensive body of theoretical and empirical literature was developed also in economics and finance. Broadly speaking, a network is a system with nodes connected by linkages. A node can be, e.g., an individual, a firm, an industry, or even a geographical area. Correspondingly, different types of relationships have been represented as linkages. Indeed, network has become such a prominent cross-disciplinary topic [10] because it is extremely helpful to model a variety of data, even when they are big data [67]. At the same time, network analysis provides the capacity to estimate effectively the main patterns of several complex systems [66]. It is a prominent tool to better understand today’s interlinked world, including economic and financial phenomena. To mention just a few applications, networks have been used to explain the trade of goods and services [39], financial flows across countries [64], innovation diffusion among firms, or the adoption of new products [3]. Another flourishing area of research related to network is the one of social connections, which with the new forms of interaction like online communities (e.g., Facebook or LinkedIn) will be even more relevant in the future [63]. Indeed network analysis is a useful tool to understand strategic interactions and externalities [47]. Another strand of literature, following the 2007–2008 financial crisis, has shown how introducing a network approach in financial models can feature the interconnected nature of financial systems and key aspects of risk measurement and management, such as credit risk [27], counterparty risk [65], and systemic risk [13]. This network is also central to understanding the bulk of relationships that involve firms [12, 74, 70]. In this chapter, we present an application explaining step by step how to construct the network and perform some analysis, in which links are based on ownership information. Firms’ ownership structure is an appropriate tool to identify the concentration of power [7], and a network perspective is particularly powerful to uncover intricate relationships involving indirect ownership. In this context, the connectivity of a firm depends on the entities with direct shares, if these entities are themselves controlled by other shareholders, and whether they also have shares in other firms. Hence, some firms are embedded in tightly connected groups of firms and shareholders, others are relatively disconnected. The overall structure of relationships will tell whether a firm is central in the whole web of ownership system, which may have implications, for example, for foreign direct investment (FDI).

Besides this specific application, the relevance of the network view has been particularly successful in economics and finance thanks to the unique insights that this approach can provide. A variety of network measures, at a global scale, allow to investigate in depth the structure, even of networks including a large number of nodes and/or links, explaining what patterns of linkages facilitate the transmission of valuable information. Node centrality measures may well complement information provided by other node attributes or characteristics. They may enrich other settings, such as standard micro-econometric models, or they may explain why an idiosyncratic shock has different spillover effects on the overall system depending on the node that is hit. Moreover, the identification of key nodes (i.e., nodes that can reach many other nodes) can be important for designing effective policy intervention. In a highly interconnected world, for example, network analysis can be useful to map the investment behavior of multinational enterprises and analyze the power concentration of nodes in strategic sectors. It can also be deployed to describe the extension and geographical location of value chains and the changes we are currently observing with the reshoring of certain economic activities as well as the degree of dependence on foreign inputs for the production of critical technologies. The variety of contexts to which network tools can be applied and the insights that this modeling technique may provide make network science extremely relevant for policymaking. Policy makers and regulators face dynamic and interconnected socioeconomic systems. The ability to map and understand this complex web of technological, economic, and social relationships is therefore critical for taking policy decision and action—even more, in the next decades when policy makers will face societal and economic challenges such as inequality, population ageing, innovation challenges, and climate risk. Moreover, network analysis is a promising tool also for investigating the fastest-changing areas of non-traditional financial intermediation, such as peer-to-peer lending, decentralized trading, and adoption of new payment instruments.

This chapter introduces network analysis providing suggestions to model data as a network for beginners and describes the main network tools. It is organized as follows. Section 2 provides an overview of recent applications of network science in the area of economics and finance. Section 3 introduces formally the fundamental mathematical concepts of a network and some tools to perform a network analysis. Section 4 illustrates in detail the application of network analysis to firm ownership and Sect. 5 concludes.

In economics, a large body of literature using micro-data investigates the effects of social interactions. Social networks are important determinants to explain job opportunity [18], student school performance [20],¹ criminal behavior [18], risk sharing [36], investment decisions [51], CEO compensations of major corporations [52], corporate governance of firms [42], and investment decision of mutual fund managers [26]. However, social interaction effects are subject to significant identification challenges [61]. An empirical issue is to disentangle the network effect of each other’s behaviors from individual and group characteristics. An additional challenge is that the network itself cannot be always considered as exogenous but depends on unobservable characteristics of the individuals. To address these issues several strategies have been exploited: variations in the set of peers having different observable characteristics; instrumental variable approaches using as instrument, for example, the architecture of network itself; or modeling the network formation. See [14] and [15] for a deep discussion of the econometric framework and the identification conditions. Network models have been applied to the study of markets with results on trading outcomes and price formation corroborated by evidence obtained in laboratory [24]. Spreading of information is so important in some markets that networks are useful to better understand even market panics (see, e.g., [55]). Other applications are relevant to explain growth and economic outcome. For example, [3] find that past innovation network structures determine the process of future technological and scientific progress. Moreover, networks determine how technological advances generate positive externalities for related fields. Empirical evidences are relevant also for regional innovation policies [41]. In addition, network concepts have been adopted in the context of input–output tables, in which nodes represent the individual industries of different countries and links denote the monetary flows between industries [22], and the characterization of different sectors as suppliers to other sectors to explain aggregate fluctuations [1].²

In the area of finance,³ since the seminal work by [5] network models have been revealed suitable to address potential domino effects resulting from interconnected financial institutions. Besides the investigation of the network structure and its properties [28], this framework has been used to answer the question whether the failure of an institution may propagate additional losses in the banking system [75, 72, 65, 34]. Importantly it has been found that network topology influences contagion [43, 2]. In this stream of literature, financial institutions are usually modeled as the nodes, while direct exposures are represented by the linkages (in the case of banking institutions, linkages are the interbank loans). Some papers use detailed data containing the actual exposures and the counterparties involved in the transactions. However, those data are usually limited to the banking sector of a single country (as they are disclosed to supervisory authorities) or a specific market (e.g., overnight interbank lending [53]). Unfortunately, most of the time such a level of detail is not available, and thus various methods have been developed to estimate networks, which are nonetheless informative for micro- and macro-prudential analysis (see [8] for an evaluation of different estimation methodologies). The mapping of balance sheet exposures and the associated risks through networks is not limited to direct exposures but has been extended to several financial instruments and common asset holdings such as corporate default swaps (CDS) exposures [23], bail-inable securities [50], syndicated loans [48, 17], and inferred from market price data [62, 13]. Along this line, when different financial instruments are considered at the same time, financial institutions are then interconnected in different market segments by multiple layer networks [11, 60, 68]. Network techniques are not limited to model interlinkages across financial institutions at micro level. Some works consider as a node the overall banking sector of a country to investigate more aggregated effects [31] and the features of the global banking network [64]. Other papers have applied networks to cross-border linkages and interdependencies of the international financial system, such as the international trade flows [38, 39] and cross-border exposures by asset class (foreign direct investment, portfolio equity, debt, and foreign exchange reserves) [56]. Besides the different level of aggregation to which a node can be defined, also a heterogeneous set of agents can be modeled in a network framework. This is the approach undertaken in [13], where nodes are hedge funds, banks, broker/dealers, and insurance companies, and [21] that consider the institutional sectors of the economy (non-financial corporations, monetary financial institutions, other financial institutions, insurance corporations, government, households, and the rest of the world).

Both in economics and finance, the literature has modeled firms as nodes considering different types of relationships, such as production, supply, or ownership, which may create an intricate web of linkages. The network approach has brought significant insights in the organization of production and international investment. [12] exploit detailed data on production network in Japan, showing that geographic proximity is important to understand supplier–customer relationships. The authors, furthermore, document that while suppliers to well-connected firms have, on average, relatively few customers, suppliers to less connected firms have, on average, many customers (negative degree assortativity). [6] exploring the structure of national and multinational business groups find a positive relationship between a groups’ hierarchical complexity and productivity. [32] provide empirical evidence that parent companies and affiliates tend to be located in proximity over a supply chain. Starting with the influential contribution of [58], an extensive body of literature in corporate finance investigates the various types of firm control. Important driving forces are country legal origin and investor protection rights [58, 59]. In a recent contribution, [7] describe extensively corporate control for a large number of firms, documenting persistent differences across countries in corporate control and the importance of various institutional features. [74] investigate the network structure of transnational corporations and show that it can be represented as a bow-tie structure with a relatively small number of entities in the core. In a related paper, [73] study the community structure of the global corporate network identifying the importance of the geographic location of firms. A strong concentration of corporate power is documented also in [70]. Importantly, they show that parent companies choose indirect control when located in countries with better financial institutions and more transparent forms of corporate governance. Formal and informal networks may have even performance consequences [49], affect governance mechanisms [42], and lead to distortions in director selection [57]. For example, in [42] social connections between executives and directors undermine independent corporate governance, having a negative impact on firm value. In the application of this chapter, we focus on ownership linkages following [74, 70], and we provide an overview of the worldwide network structure and the main patterns of control.

This section does not provide a comprehensive literature review, but rather it aims to give an overview of the variety of applications using network analysis and the type of insights it may suggest. In this way we hope to help the reader to think about his own data as a network.

This section formally introduces graphs⁴ and provides an overview of standard network metrics, proceeding from local to more global measures.

A graph \(G = \left ( V, E \right )\) consists of a set of nodes V  and a set of edges E ⊆ V ² connecting the nodes. A graph G can conveniently be represented by a matrix \(W \in \mathbb {R}^{n \times n}\), where \(n \in \mathbb {N}\) denotes the number of nodes in G and the matrix element w _ij represents the edge from node i to node j. Usually w _ij = 0 is used to indicate a non-existing edge. Graphs are furthermore described by the following characteristics (see also Figs. 1 and 2):

While a visual inspection can be very helpful for small networks, this approach quickly becomes difficult as the number of nodes increases. Especially in the era of big data, more sophisticated techniques are required. Thus, various network metrics and measures have been developed to help describe and analyze complex networks. The most common ones are explained in the following.

Most networks do not exhibit self-loops, i.e., edges connecting a node with itself. For example, in social networks it makes no sense to model a person being friends with himself or in financial networks a bank lending money to itself. Therefore, in the following we consider networks without self-loops. It is however straightforward to adapt the presented network statistics to graphs containing self-loops. Moreover, we consider the usual case of networks comprising only positive weights, i.e., \(W \in \mathbb {R}_{\ge 0}^{n \times n}\). Adaptations to graphs with negative weights are however also possible. Throughout this section let W ^(dir) denote a directed graph and W ^(undir) an undirected graph.

The network density \(\rho \in \left [0, 1\right ]\) is defined as the ratio of the number of existing edges and the number of possible edges, i.e., for W ^(dir) and W ^(undir), the density is given by: The density of a network describes how tightly the nodes are connected. Regarding financial networks, the density can also serve as an indicator for diversification. The higher the density, the more edges, i.e., the more diversified the investments. For example, the graph pictured in Fig. 1 has a density of 0.5, indicating that half of all possible links, excluding self-loops, exist.

While the density summarizes the overall interconnectedness of the network, the degree sequence describes the connectivity of each node. The degree sequence \(d = \left ( d_1, \ldots , d_n \right ) \in \mathbb {N}_0^n\) of W ^(dir) and W ^(undir) is given for all i = 1, …, n by:

For a directed graph W ^(dir) we can differentiate between incoming and outgoing edges and thus define the in-degree sequence d ⁽ⁱⁿ⁾ and the out-degree sequence d ^(out) as: The degree sequence shows how homogeneously the edges are distributed among the nodes. Financial networks, for example, are well-known to include some well-connected big intermediaries and many small institutions and hence exhibit a heterogeneous degree sequence. For example, for the graph pictured in Fig. 1, we get the following in- and out-degree sequences, indicating that node 4 has the highest number of connections, 3 incoming edges, and 2 outgoing edges:

Similarly, for weighted graphs, the distribution of the weight among the nodes is described by the strength sequence \(s = \left ( s_1, \ldots , s_n \right ) \in \mathbb {R}_{ \ge 0}^n\) and is given for all i = 1, …, n by:

In addition, for the weighted and directed graph W ^(dir), we can differentiate between the weight that flows into a node and the weight that flows out of it. Thus, the in-strength sequence s ⁽ⁱⁿ⁾ and the out-strength sequence s ^(out) are defined for all i = 1, …, n as: For example, for the graph pictured in Fig. 1, we get the following in- and out-strength sequences: Node 2 is absorbing more weight than all other nodes with an in-strength of 32, while node 4 is distributing more weight than all other nodes with an out-strength of 38.

The homogeneity of a graph in terms of its edges or weights is measured by the assortativity. Degree (resp. strength) assortativity is defined as Pearson’s correlation coefficient of the degrees (resp. strengths) of connected nodes. Likewise, we can define the in- and out-degree assortativity and in- and out-strength assortativity. Negative assortativity, also called disassortativity, indicates that nodes with few edges (resp. low weight) tend to be connected with nodes with many edges (resp. high weight) and vice versa. This is, for example, the case for financial networks, where small banks and corporations maintain financial relationships (e.g., loans, derivatives) rather with big well-connected financial institutions than between themselves. Positive assortativity, on the other hand, indicates that nodes tend to be connected with nodes that have a similar degree (resp. similar weight). For example, the graph pictured in Fig. 1 has a degree disassortativity of − 0.26 and a strength disassortativity of − 0.24, indicating a slight heterogeneity of the connected nodes in terms of their degrees and strengths.

The importance of a node is assessed through centrality measures. The three most prominent centrality measures are betweenness, closeness, and eigenvector centrality and can likewise be defined for directed and undirected graphs. (Directed or undirected) betweenness centrality b _i of vertex i is defined as the sum of fractions of (resp. directed or undirected) shortest paths that pass through vertex i over all node pairs, i.e.: where \(s_{jh} \left ( i \right ) \) is the number of shortest paths between vertices j and h that pass through vertex i, s _jh is the number of shortest paths between vertices j and h, and with the convention that \(s_{jh} \left ( i \right ) / s_{jh} = 0\) if there is no path connecting vertices j and h. For example, the nodes of the graph pictured in Fig. 1 have betweenness centralities \(b = \left ( b_1, b_2, \ldots , b_5 \right ) = \left ( 5, 5, 1, 2, 1 \right ) \), i.e., nodes 1 and 2 are the most powerful nodes as they maintain the highest ratio of shortest paths passing through them.

(Directed or undirected) closeness centrality c _i of vertex i is defined as the inverse of the average shortest path (resp. directed or undirected) between vertex i and all other vertices, i.e.: where d _ij denotes the length of the shortest path from vertex i to vertex j. For example, the nodes of the graph pictured in Fig. 1 have closeness centralities \(c = \left ( c_1, c_2, \ldots , c_5 \right ) = \left ( 0.80, 0.50, 0.57, 0.57, 0.57 \right ) \). Note that in comparison to betweenness centrality, node 1 is closer to other nodes than node 2 as it has more outgoing edges.

Eigenvector centrality additionally accounts for the importance of a node’s neighbors. Let λ denote the largest eigenvalue of the adjacency matrix a and e the corresponding eigenvector, i.e., λa = ae holds. The eigenvector centrality of vertex i is given by: The closer a node is connected to other important nodes, the higher is its eigenvector centrality. For example, the nodes of the graph pictured in Fig. 2 (representing the undirected and unweighted version of the graph in Fig. 1) have eigenvector centralities e = (e ₁, e ₂, …, e ₅) = (0.19, 0.19, 0.19, 0.24, 0.19), i.e., node 4 has the highest eigenvector centrality. Taking a look at the visualization in Fig. 2, this result is no surprise. In fact node 4 is the only node that is directly connected to all other nodes, naturally rendering it the most central node.

Another interesting network statistic is the clustering coefficient, which indicates the tendency to form triangles, i.e., the tendency of a node’s neighbors to be also connected to each other. An intuitive example for a highly clustered network are friendship networks, as two people with a common friend are likely to be friends as well. Let a denote the adjacency matrix of an undirected graph. The clustering coefficient C _i of vertex i is defined as the ratio of realized to possible triangles formed by i: where d _i denotes the degree of node i. For example, the nodes of the graph pictured in Fig. 2 have clustering coefficients C = (C ₁, C ₂, …, C ₅) = (0.67, 0.67, 0.67, 0.67, 0.67). This can be easily verified via the visualization in Fig. 2. Nodes 1, 2, 3, and 5 form each part of 2 triangles and have 3 edges, which give rise to a maximum of 3 triangles (C ₁ = 2∕3). Node 4 forms part of 4 triangles and has 4 links, which would make 6 triangles possible (C ₄ = 4∕6). For an extension of the clustering coefficient to directed and weighted graphs, the reader is kindly referred to [37].

Furthermore, another important strand of literature works on community detection. Communities are broadly defined as groups of nodes that are densely connected within each group and sparsely between the groups. Identifying such groupings can provide valuable insight since nodes of the same community often have further features in common. For example, in social networks, communities are formed by families, sports clubs, and educationally or professionally linked colleagues; in biochemical networks, communities may constitute functional modules; and in citation networks, communities indicate a common research topic. Community detection is a difficult and often computationally intensive task. Many different approaches have been suggested, such as the minimum-cut method, modularity maximization, and the Girvan–Newman algorithm, which identifies communities by iteratively cutting the links with the highest betweenness centrality. Detailed information on community detection and comparison of different approaches are available in, e.g., [67] and to [30].

The concepts and measures we had presented are general and can be applied to any network.⁷ However, their interpretation depends on the specific context of application. A knowledge of the underlined economic/financial phenomenon is also necessary before starting to model the raw data as a network. When building a network, a preliminary exploration of the data helps to control or mitigate errors that arise from working with data collected in real-world settings (i.e., missing data, measurement errors…). Data quality, or at least awareness of data limitations, is important to perform an accurate network analysis and to draw credible inferences and conclusions. When performing a network analysis, it is also important to remind that the line of investigation depends on the network under consideration: in some cases it could be more relevant to study deeply centrality measures while in others to detect communities.

For data processing and implementation of network measures, several software are available. R, Python, and MATLAB include tools and packages that allow the computation of the most popular measures and network analysis, while Gephi and Pajek are open-source options for exploring visually a network.

In this section we present an application of network analysis to firm ownership, that is, the shareholders of firms. We first describe the data and how the network was built. Then we show the resulting network structure and comment the main results.