The availability of large-scale data sets about firms has boosted research about economic networks in the recent decade (Schweitzer et al., 2009). To construct such networks, different types of interactions between firms have been analyzed, for example, knowledge transfer (Reagans and McEvily, 2003; Vaccario et al., 2018), financial relations (Battiston et al., 2012; Nanumyan et al., 2015), supply chains and trade networks (Garlaschelli and Loffredo, 2005; Fagiolo et al., 2010; Mizgier et al., 2013; Burkholz and Schweitzer, 2019), or ownership (Glattfelder and Battiston, 2009; Vitali et al., 2011; Mani and Moody, 2014; Garcia-Bernardo et al., 2017).

In this article, we build on the latter, by reusing a data set about the global ownership relations among firms (Vitali et al., 2011; Glattfelder and Battiston, 2009; Zhang and Schweitzer, 2019) obtained from the Orbis database of 2007. ¹ This reports information about the share firm A holds on firm B, that is, links in the ownership network are directed and weighed. Further information about the operating revenue of each firm is also available in this database. This data set has been previously analyzed to quantify corporate control (Glattfelder and Battiston, 2009; Vitali et al., 2011).

Similar to the mentioned works, in the following, we focus on transnational companies (TNCs) which, according to definition by the Organization for Economic Co-operation and Development (OECD), operate in more than one country. They are known to form the backbone on the ownership network (Glattfelder and Battiston, 2009). These TNCs directly or indirectly participate in other firms, called participated companies (PCs) which are mostly direct or indirect subsidiaries of TNCs.

Starting from the list of TNCs, we recursively include all companies that are participated by TNCs, or companies that are shareholders of TNCs, directly or indirectly. With this procedure, we end up with a large network that contains 600,508 economic entities connected by 1,006,987 ownership relations.

Our analysis is focused on the very small, but densely, connected core, that is, a strongly connected component, of this network (Vitali et al., 2011), which is also visualized in Figure 1. It comprises 1,318 firms that are connected by 12,184 ownership relations; that is, on average, each firm is connected to 20 other firms, and there is at least one directed path from any firm to other firm in this core. The overall operating revenue for firms in this core accounts for 20% of the operating revenue by all firms in the global ownership network. So, we are looking here at the heart of the global economy.

We will use this ownership network to later explore the network controllability by identifying the set of driver nodes. For this, we also need to specify the dynamics that connect these firms.

The ownership relations between firms do not only determine corporate influence but they also influence reputation (Fombrun and Shanley, 1990; Brammer and Pavelin, 2006; Delgado-García et al., 2010). For example, with sample data of selected firms at the country level, Fombrun and Shanley (1990); Brammer and Pavelin (2006); Delgado-García et al. (2010) reported that features of the ownership structure, such as the concentration of ownership in institutional investors, are correlated with corporate reputation. Further, Kang (2008) found that because of independent and active monitoring, institutional shareholders can greatly reduce the likelihood of negative reputation spillover.

Following the previous line of research, in a recent article (Zhang and Schweitzer, 2019), we have distinguished two phases, which differ in the directionality for the reputation spillover. In an initial phase, the reputation of the owners, that is, the firms investing into a newly founded company, largely determines the reputation of this firm because with their reputation, early shareholders signal trust to invest in this yet unknown firm.

In the second phase, the reputation of the invested firm can feed back on the reputation of its stakeholders, both in positive and negative ways. We have seen many scandals that have shaken the business world because reputed stakeholders, who also represent a considerable corporate control, have been made responsible for the malfunction of their firms. For example, in Germany, the emission scandal of the car-building company Volkswagen led to a negative reputation spillover to its largest shareholder, Porsche SE, for neglecting its supervisory obligations. But investors also use the positive reputation of firms, for example, in the green energy sector, to brush up their own reputation—as the recent debate on ethical investments witnesses (Mallin et al., 1995; Michelson et al., 2004).

Thus, it is justified to discuss the reputation dynamics of firms by utilizing their ownership network. In the following paragraphs, we focus on the core of the ownership network, which represents a mature economy of established firms. This allows us to consider the second phase, where the directionality of the ownership links is opposite to the directionality of the reputation spillover, that is, reputation spills over from the invested firm to its shareholders.

To quantify reputation, we assign to each firm a scalar value, x _i (t), that changes with time according to the following dynamics (Zhang et al., 2019): x ˙ i ( t ) = ∑ j = 1 N a j i x j ( t ) − ϕ x i ( t ) . (1) The variables a j i = log ( c w j i ) capture the key assumption that the reputation spillover from firm j to firm i via the ownership link, where w _ij is the reported share firm i holds in firm j and c is a normalization constant such that a _ji is always equal or larger than 1. The second term in Eq. 1 captures the assumption that reputation fades out exponentially at a rate ϕ if it is not maintained (Lundgren, 2011).

This dynamics has been also applied to model the reputation dynamics in online social networks (Schweitzer et al., 2020). For application scenarios, it is more convenient to use relative reputation values r i = x i / x max , instead of absolute values x _i . But in this article, we are only interested in the reputation ranking of firms; therefore, we use x _i .

In matrix form, the previous linear dynamics can be conveniently expressed as follows: X ˙ ( t ) = A T X ( t ) − ϕ X ( t ) . (2) The matrix A ^T contains the information about the network topology, and the vector X ( t ) = [ x 1 ( t ) , x 2 ( t ) , … , x N ( t ) ] contains the reputation values of all firms.

We set ϕ as the largest eigenvalue of A. This allows the dynamics of Eq. 2 to converge to an equilibrium with only positive entries, which are then used for the ranking. Here, we emphasize that with this configuration, we do not look at the time scale of the model, and we only use the equilibrium values.

Once we have identified the set of driver nodes, as described in the following section, we have to consider a control signal, that is, an induced change that modifies the reputation of only the driver nodes. The resulting linear dynamics can be then expressed as follows: X ˙ ( t ) = A T X ( t ) − ϕ X ( t ) + BU ( t ) . (3) The vector U ( t ) ∈ ℝ N c contains N c control signals u k ( t ) ( k = 1 , … , N c ) , and the matrix B ∈ ℝ N × N c determines which firms are influenced directly by control signals, which means the elements b i j ≠ 0 if control signal u j ( t ) is applied to firm i. To apply the concept of network controllability, we are still left with determining the set of driver nodes.

The recent framework of structural controllability for complex networks (Liu et al., 2011) allows to identify minimum sets of driver nodes, that is, a small number of nodes that can be utilized to control the whole network. This method can be applied to directed networks. Because we cannot repeat all details of the method here, we summarize the respective steps and refer to the literature for subsequent information (Wang et al., 2012; Cornelius et al., 2013; Zhang et al., 2016; Zhang et al., 2019).

A complex network of N nodes can be controlled by different sets of driver nodes. MDS denotes the minimum set of drivers to control the whole network, and the size of this set is N d . It is computationally infeasible to enumerate all the possible MDSs. Therefore, in our article, we use two randomly chosen MDSs for the visualization, as shown in Figure 2, and calculate our control-related measures based on 10.000 random samples. These samples are generated using a sophisticated random sampling procedure, as described in Jia and Barabási (2013). Note that for a given network, all of its MDSs are of the same size N _d .

In different MDSs, we usually find different nodes, but some of them appear in every MDS. The probability P(D _i ) that a given node i appears in an MDS is also known as control capacity K i (Jia and Barabási, 2013). Further, each driver i controls a non-overlapping part of the network of size N _i . The probability that a given node is in the subnetwork controlled by node i is given by P(N _i ). We combine these two pieces of information in the conditional probability P ( N i | D i ) that a given node is part of the subnetwork controlled by i, given that i is a driver node. The upper bound of this probability is also known as control range, ℛ i (Wang et al., 2012).

To eventually combine the control range and control capacity, we have proposed a new measure, controzl contribution C i = K i ℛ i (Zhang et al., 2019). This node-based measure gives us the probability for any node in a network to be controlled by node i joint with the probability that i becomes a driver. Larger C i indicates that node i is more important in driving the whole network to a desired state. Concrete values for C i can only be obtained algorithmically. (For an illustrative calculation and an algorithm, we refer to Reference Zhang et al. (2019).) There, it was also demonstrated that control contribution is better suitable than the control range or control capacity to classify the importance of nodes in controlling a network.

Applying the methods described earlier, we now have three different types of information for each firm in the ownership network: (i) its operating revenue Ω i , (ii) its reputation x _i (which takes the weighted ownership relations w _ji into account), and (iii) its control capacity K i , control range ℛ i , and control contribution C i . These measures reflect different dimensions to describe the importance of firms in an economic network, namely, their economic activity, their dependence on other firms, and their influence on other firms. Therefore, we can now address research questions that link these different dimensions, for instance, are firms with a high operating revenue or firms with a high reputation also most influential in network control?

To quantify such relations, we perform an enrichment analysis, a statistical method which is commonly used to identify genes or proteins that are overrepresented (Wuchty, 2014). To illustrate the idea, suppose there are N balls characterized by colors s and types t. We have three colors, that is, s: (white, black, and grey) and two types t: (heavy and light). Enrichment analysis can, for example, tell whether heavy balls are more likely to be white balls or not. To do so, we need to compare the number of heavy balls whose color is also white, N l s , and the number of heavy balls N l R , if we randomly sample N/3 balls.

Here, we apply this analysis to the firms that are part of the driver set of size N _d . Our “colors,” or categories, are now reputation values, that is, s : (low, medium, and high) reputation. To define these groups, we first calculate the reputation x _i using Eqn. (1) and then rank firms according to their reputation values in equilibrium. Note that this reputation ranking is also produced in Zhang and Schweitzer (2019) with the same dynamics and assumptions. Here, we further split the ranked set into three groups of equal size N _d /3.

Second, we specify which types l we are interested in, for example, whether firms have a low, medium, or high control contribution C _i . N l s ≤ N d / 3 then denotes the number of firms which are in the reputation group s and have a type l regarding their control contribution, which means, instead of just looking into correlations across all firms, we define groups of firms with certain features and then address the question whether firms with these features appear more frequently than expected in each reputation group.

For this comparison, we need a random set R that has the same size N _d /3 but is sampled from all N firms with respect to the feature l. N l R is the number of firms in the random set with, for example, medium control contribution. The random sampling is performed 10.000 times, to obtain a distribution for the values N l R , from which we can calculate the mean μ ( N l R ) and the standard deviation σ ( N l R ) . For the comparison between the category s and the type L, we then use the z-score: z l s = N l s   − μ ( N l R ) σ ( N l R ) . (4)

Obviously, a positive z-score shows an enrichment of the given category s in the type l. Enrichment means that firms with a given type l appear more frequently in the category s than expected at random. Additionally, we report the probability p in which N l s is larger (smaller) than N l R when the z-score is positive (negative).

To classify firms as driver nodes, we first need to determine the size of the MDS. We find that from the 1,318 firms in the ownership network, we need to control a minimum number of N _d = 341 firms directly in order to control the whole network. Note that the size of the MDS is mainly determined by characteristics of the network topology, such as the degree distribution. Then, out of the large number of possible MDSs with the same size, we have to generate 10.000 random samples, on which our further analysis is based.

As an illustration, Figure 2 presents visualizations of two random MDSs embedded in the ownership network shown in Figure 1. We notice that both MDSs only have a few driver nodes in common. Further, the right MDS contains more TNCs with high operating revenue as driver nodes, whereas the left contains mostly PCs with lower operating revenue.

To further quantify these differences, we first investigate how many TNCs are present in a randomly sampled MDS. The distribution obtained from 10.000 MDS is shown in Figure 3A. We find that on average, about 26 TNCs are present in an MDS of size 341, that is, <10%. One could naively assume that because of their economic importance, TNCs would also be the most important driver nodes and thus should appear more often. Interestingly, this is not the case. Even more, the average of 26 TNCs, which corresponds to 8.7% of all TNCs in the core of the ownership network, is far below the expected number of TNCs obtained from a random sample of firms, which is 28.8%. This leads to the conclusion that TNCs are statistically underrepresented in the sets of driver nodes.

Further, the distribution is well-formed between a minimum of 17 and a maximum of 34 TNCs, which means we can find indeed MDSs in which the number of TNCs is only about 5%. Why is this of interest? These MDSs because of the different number of TNCs also represent a very different economic value, as proxied by the operating revenue Ω i of their TNCs. Figure 3B shows the distribution of Σ Ω i of all TNCs in the 10.000 sampled MDSs. On average, the TNCs in an MDS hold a total operating revenue of 720 trillion USD, which accounts for 9.6% of the amount held by all firms in the network. But these values can be as low as 350 or as high as 1,300 trillion USD. So, we have a remarkable number of “cheap” MDS available.

We remind that all MDS fulfill the same purpose, namely, to control the whole network. But a “cheap” MDS, as proxied by the total operating revenue, with a low number of TNCs would potentially be more easily accessible. Remember that network controllability requires us to apply a control signal to a firm, which means we need to consider some sort of access cost to utilize a given firm as a driver node. It is likely more expensive to access a TNCs of high operating revenue than a PC of low operating revenue. Because we have no way to directly quantify the access cost, in the following, we take the operating revenue Ω i as a proxy of this access cost.

One could still argue that firms from a “cheap” MDS are less likely to be chosen as driver nodes because they are more often PCs. Again, this reflects the underlying assumption that TNCs should be more important as driver nodes and therefore should also be more often present in different MDSs. To refute this argument, we have investigated the distribution of the control capacities K i , which give the probability that a firm is chosen as a driver node. The results are shown in Figure 4 both for TNCs and for PCs. We find that most firms, despite belonging to an MDS, only have a very low probability to be chosen as driver nodes. This holds for both TNCs and PCs. Then, there is a very broad distribution of K i values, which is largely dominated by PCs. Firms with a control capacity close to 1 are always present in any MDS. We find that these firms are PCs.

Thus, in conclusion, firms that are PCs are most often present as driver nodes. Second, their access cost should be considerably lower than for TNCs. Therefore, we can safely choose “cheap” MDSs with a high fraction of PCs, to reach an efficient control of the whole network. This is an important insight because it links network controllability to economic measures and allows for policy advice.

So far, we have mainly explored the economic and control properties of the firms that are part of the sets of driver nodes. Now, we focus on the different types l of nodes, specifically the roles of firms in (a) maintaining controllability and (b) controlling the network. We start from our reputation ranking of firms, which lead to the formation of groups of size N _d /3 with s: (low, medium, and high) reputation, as described in Section 2.3.

We first analyze how these groups to correlate with the roles of firms in maintaining control. This requires us to specify the node types l accordingly. Maintaining control means that the set of driver nodes is still able to fully control the network, if a respective node i would be isolated. Following Vinayagam et al. (2016), we can then distinguish three types l of driver nodes: nolistsep.

(a) node is indispensable if after its isolation, more driver nodes are needed to control the rest of the network;

Based on this classification, to identify the role l of firm i in maintaining controllability, we need to calculate the minimum number of driver nodes if i is isolated and compare it with the minimum number of driver nodes if i is not isolated. We have to keep in mind that isolating a node implies changing its local ownership relations, which definitely impacts the size of the minimum set of drivers. An MDS of size 341 only holds for the non-perturbed network. Also, different from Vinayagam et al. (2016) in which a protein can be knocked out, we cannot remove a firm from the ownership network even it is bankrupt. Instead, we can isolate a firm by removing all its ownership relations.

The results are shown in Figure 5A in terms of the z-score defined in Eq. 4. Firms in each group with low, medium, and high reputation do contain all three types of driver nodes, indispensable, redundant, and dispensable. But the z-score tells us whether such roles are enriched in a particular group. We see that indispensable nodes are most enriched (p = 100%) in the set of high-reputation firms; this is in accordance with our expectation that the most reputable firms channel control signals through the ownership network. Interestingly, indispensable nodes are mostly underrepresented (p = 99.8%) in the group with medium reputation, instead of low reputation. This can be partly explained from the fact that the ownership network forms a strongly connected component. Therefore, the isolation of a low-degree node, which is likely a firm of medium to low reputation, may leave some nodes with no incoming links, which have to be controlled directly with additional drivers. In conclusion, this analysis shows the importance of firms with high reputation in maintaining controllability.

Second, we analyze how the three reputation groups correlate with the role of firms in controlling the network. In this case, we have to specify the types l of nodes with respect to their control contribution C i , introduced in Section 2.3. We remind that C i captures the probability for a firm to become a driver node, joint with the probability for any firm to be controlled by this firm. Hence, firms with a high control contribution are top drivers. We use the values of C i to distinguish three groups of equal size N _d /3 with low, medium, and high control contribution.

The results are shown in Figure 5B in terms of z-score defined in Eq. 4. We find that the top driver nodes are mostly firms with medium reputation, not with high reputation, which is a very interesting result. Firms with high reputation are strongly embedded into the ownership network and connected to other firms with high reputation. Consequently, to utilize such firms as driver nodes would imply a considerable access cost. But this is not needed. Instead, network controllability can be best achieved with firms of medium reputation.

In Table 1, we also list the top 10 driver nodes with respect to their control contribution C _i and provide their reputation rank. We observe that none of these firms has a high reputation, and only one of them is a TNC. This confirms that the top drivers are likely not firms with high reputation in the ownership network.

A summary our findings from the two enrichment analyses is given as follows: (a) firms with high reputation maintain the controllability of the network but are unlikely to become top driver nodes and (b) firms with medium reputation are most likely to become top driver nodes, but they are also dispensable for maintaining controllability.