Quantitative analysis of trade networks: data and robustness

One of the fundamental mathematical structures in network theory is the adjacency matrix, which represents the connections between nodes in a network. Equation (1) shows an adjacency matrix for N countries:In this paper, the trade flows between countries are considered. Because trade flows between countries are not symmetric (i.e., import (export) of country i from (to) country j is not necessarily equal to import (export) of country j from (to) country i), and countries do not import from themselves, the network in this study is a weighted directed network without self-loops. Therefore, the adjacency matrix of this network is an asymmetric adjacency matrix with zero diagonal elements. In the import network, \(a_{ij}\) represents the import value of country j from country i and in the export network, the \(a_{ij}\) represents export of country i to country j.

In this paper, the 2016 trade flow data is studied. In 2016, only 154 countries provided their reports. After studying the trading partners of those countries, 234 trading units, including the 154 countries, were found to be the trading partners. So, a 234-by-234 adjacency matrix is reconstructed with 154 reporting countries labeling the first 154 columns without the loss of generality and the 234 trading partner countries labeling the rows.

Degree is the total number of links incident on a node, or the number of links a node is connected to. In weighted networks, it is the sum of the weights of the links which is called strength. In directed networks, the concept of the degree is subdivided to in- and out-degree which are defined aswhere \(d^{in}_i\) stands for in-degree of node i, \(d^{out}_i\) stands for out-degree of node i, and \(a_{ji}\) and \(a_{ij}\) are elements of the corresponding adjacency matrix with \(a_{jj}=0\) for all j [3, 31].

Degree distribution: The fraction of nodes that have degree k is given by the degree distribution, which is a characteristic of the network structure and can have different functional forms [2, 32‐34]. For example, random networks have binomial degree distribution which is approximately a Poisson distribution for large networks. In regular networks where all nodes have the same degree, their distribution is a delta function at that degree. Random networks and regular networks are two extreme graphs in network theory and small-world networks have properties between these two extremes [35]. Other types of networks that have a specific degree distribution include scale-free networks which have a power-law degree distribution [36‐38], with probability proportional to a negative power of the degree.

A central node can be described by either how connected that node is or how influential its neighbors are [3]. For instance, based on degree centrality, a central node is a node that has many connections. Also, in-degree indicates that a central node has many inflows and out-degree assigns central node status to a node that has many outflows.

In economic terms, for a weighted directed import network of a commodity, in-degree centrality indicates how dependent a country is on other countries from which it imports that specific commodity group. Particularly, a country with a high in-degree centrality measure is a highly dependent country on importing a specific commodity group and does not satisfy its own need for that commodity group. Conversely, the out-degree centrality states the extent to which a country is a source (although not necessarily a producer) of a specific commodity group for a group of other countries. Also, for a weighted directed export network of a commodity, out-degree centrality indicates the strength of a country to be a potential supplying market of that commodity. On the other hand, in-degree centrality in an export network represents the dependence of a country on the other supplying markets of a specific commodity. Here, for simplicity, in-degree and out-degree centrality measures are considered for the import and export networks, respectively.

The centrality of a node can also be determined by its neighbors’ characteristics. Therefore, a node that is central based on one type of centrality measure is not necessarily central under other such measures [3, 39]. In this subsection, the centrality measures of PageRank, hubness, and authority are introduced, which are widely used global centrality measures. Note that eigenvector centrality is one of the other measures that captures the centrality of a node based on its neighbors’ characteristics. However, the eigenvector centrality cannot be measured in directed networks [31], so the present study does not examine this quantity.

PageRank: In this measure the centrality of a node derives from its network neighbors’ and is proportional to their centrality divided by their out-degree. In this case, nodes that point to many others pass only a small amount of centrality on to each of those others, even if their own centrality is high [31]. In mathematical terms the PageRank centrality, \(x_i\), of node i is defined bywhere \(\alpha\) and \(\beta\) are positive constants and \(x_i\) and \(x_j\) are the centralities of nodes i and j, respectively. This centrality measure is defined by two endogenous and exogenous components. The endogenous component, \(\alpha \mathop {\sum _{j}} a_{ij}\cdot {{x_j}/{d^{out}_j}}\), is based on the network topology, and the exogenous component, \(\beta\), is independent of the network structure [40].

The economic significance of this centrality measure is that a country with a high PageRank centrality is connected to a central country (i.e., a country with high PageRank centrality). Also, central countries themselves have high PageRank centrality measures indicating they are core sources in that specific commodity.

Hub and authority: Page Rank centrality assigns a country high centrality if other trading units that have trade flows to this country, have high centrality. However, in the case of directed networks it is possible to accord a country high centrality if it points to (has flows to) other trading units with high centrality. Thus, there are really two types of central countries in such a network: authorities are countries that are important and central points of a specific commodity group in the import sector, and hubs are trading units that trade with important and central points of a specific commodity and export to other important and central countries (authorities) in the network. In other words, hub countries point to authorities. Besides, an authority may also be a hub and vice versa [31]. In this study, following [13], and for simplicity, hubness is considered for the export networks and authority is used for the import networks.

Note that two other types of centrality measures, betweenness, and closeness, exist, which determine the centrality of a node based on being on the path of other nodes [3, 31]. The issue with calculating these centrality measures is that the weights of the links must be the cost of reaching a node from the other [41, 42]. Since we could not find a suitable data for the cost of trade among countries, these centrality measures are not considered in this study.

In this case, there is lack of data on bilateral trade statistics between some of the countries and territories in the list of the trading partners’ countries. Therefore, based on Eq. (5), the full matrix A with \(N=234\) is not available because of the lack of availability of data and this case goes from matrix B to D.

Step 1: Available matrices in Eq. (5) in the first case In 2016, 154 countries reported their trade flows but 234 countries are reported on. So, a \(234^2\) matrix is available that contains missing data. Therefore, there would be 154 columns (reporting countries) and 234 rows (trading partners for those reporting countries) with no missing data. This adjacency matrix is similar to matrix B in Eq. (5) with \(N=234\) and \(P=154\). In order to study the trade relations between the reporting countries, it is necessary to eliminate the rest of the countries from the 234-country network. Therefore, the trade flows of the reporting countries with the rest of the world is set to zero; i.e., all entries in rows \(P+1\) to N are set to zero to yield a matrix like C in Eq. (5) with \(N=234\) and \(P=154\). Thereafter, a full adjacency matrix like matrix D in Eq. (5) with \(P=154\) is constructed based solely on trade between reporting countries.

Our aim here is to test the effect of inferring missing data on nonreporting countries from data provided by those countries that did report. Since we do not have access to the full 234-country data, we carry out this test for trade between the 154 countries that did report, which gives the largest available full matrix. We artificially render this matrix incomplete by setting to zero the same fraction of columns as are missing from the 234-country case due to nonreporting. We then use the remaining data to estimate as many as possible trade flows for the nonreporting countries and compare the resulting network structure with that of the full 154-country network. Note that for constructing the incomplete 154-country network, first, the 154 countries are ranked descendly based on their total trade (import/export) value. Second, the trade values of top 102 countries (\(\dfrac{154}{234}\simeq \dfrac{102}{154}\simeq 0.66\)) are kept and the trade value of the rest of the countries are set to zero. Therefore, the incomplete 154-country matrix with 102 nonzero columns is constructed. The logic behind keeping the top 102 countries is that the major trading countries always report their trade flows. Accordingly, without the loss of generality, the incomplete 154-country matrix is constructed this way.

Step 1: Available matrices in Eq. (5) in the second case The analysis is started with the largest available full matrix like matrix A in Eq. (5) in which \(N=154\). Then, the same proportion of columns of data as is missed from the original \(234^2\) matrix (\(\dfrac{154}{234}\) \(\simeq\) \(\dfrac{102}{154}\)) is intentionally deleted in the full matrix. This yields a matrix like B in Eq. (5) with \(N=154\) and \(P=102\). To see the trade relation between countries with no missing data, it is needed to eliminate the rest of the countries from the 154-country network. Therefore, the trade flows of the 102 representative countries with the rest of the 154 countries is set to zero. Hence, a matrix like matrix C with \(P=102\) is constructed and then the square part of this matrix is extracted and a matrix like D is constructed with \(P=102\).

Step 2: Checking the structural position of the countries In this case, studying the structural position of the countries is twofold. First, the rank of 102 countries in matrix D and B are compared and secondly, the rank of those countries are compared in matrix B and A. Simply put, centrality measures of 102 countries are calculated in import and export networks of various commodity groups; i.e., trade networks based solely on the bilateral trade between those 102 countries. Then, these countries are ranked rest on their centrality measures in import and export networks. This process is done anew in the trade network formed on the trade between reporting countries which is intentionally made incomplete; i.e., a matrix like B with \(N=154\) and \(P=102\) in Eq. (5). Further, rank of the 102 countries are compared in these two networks based on their centrality measures which contributes to a correlation coefficient like \(r_1\).

Additionally, the centrality measure of these countries are calculated in the complete trade networks; i.e., networks build exclusively on the bilateral trade between reporting countries. The adjacency matrix of these networks are similar to matrix A in Eq. (5) with \(N=154\). Next, these countries are ranked rest on the centrality measures and the ranks are compared to those in the intentionally incomplete networks, i.e. the trade networks with an adjacency matrix like B in Eq. (5). The rank comparison contributes to a correlation coefficient like \(r_2\). Lastly, the probability of similarity between \(r_1\) and \(r_2\) are tested in various commodity groups and import and export networks by using the Fisher Z transformation. That is to say, the similarity of the structural position of these countries is checked.

We argue that if the core countries in trading different commodity groups in reporting countries’ network is the same as the core countries in trading partners’ network, the structure of the trade network is not significantly affected by the missing data. The logic behind this is that the core countries are extracted by using a composite index of the network structural measures. The composite index out of the centrality measures are constructed using principal component analysis (PCA) method. In addition, entropy is applied on the composite indices to find the core countries.

Principal Component Analysis: PCA is the oldest and the most well-known multivariate statistical technique being used in dimension reduction in all science disciplines [49‐51]. One of its applications is the construction of mixed indices considering different features and variables [52].

PCA computes a linear combination of the original variables equal in number to the original ones. Suppose there is a \(X_{N\times K}\) matrix with K variables (indicators) and N observations. Define \(R_{K\times K}\) the correlation matrix between the K variables with \(\lambda _{i}\) (\(i=1,\ldots ,K\)) as the ith eigenvalue, and \(v_{K\times 1}^i\) (\(i=1,\ldots ,K\)) as the ith eigenvector of the \(R_{K\times K}\) correlation matrix. Assume, \(\lambda _{1}>\lambda _{2}>\ldots >\lambda _{K}\), then, the principal components are derived as Eq. (6).where \(PC_i\) stands for the ith principal component, X is the defined matrix with K variables and N observations, and \(v^i\) is the eigenvector corresponding to the ith eigenvalue being sorted in descending order [49, 53].

Note that \(\lambda _{i}\) is also defined as the variance of the ith principal component, i.e., \(\lambda _{i}\)=\(var(PC_{i})\). consequently, the first principal component is the linear combination of the initial indicators that has the biggest variance. The second principal component is the linear combination of the initial indicators that has the second biggest variance. Similarly, the Kth principal component is the linear combination of the initial indicators that has the smallest variance. The principal components are used as the basis of a single mixed variable construction out of the original variables [50, 53].

In this study, the in-degree, PageRank, and authority centrality measures are used to capture structural network characteristics of countries in the import sector. Also, the out-degree, PageRank, and hubness are considered to extract the structural network aspects of countries in the export sector. According to the notation, there are \(K=3\) variables for each network, and \(N=154\) and \(N=234\) observations in the reporting countries’ and trading partners’ networks, respectively. In an attempt to compare the core countries in the reporting countries’ and trading partners’ import and export networks, four composite indices are constructed applying PCA to the centrality measures.

As stated above, in PCA, the lowest order components describe the most variations in the original variables. Accordingly, a new index can be constructed using the lowest order components. Following Chen and Woo (2010), the new variable is calculated as Eq. (7).where \(CI_{n}\) is the composite index of the nth observation, \(x_{j}\) is the jth column of matrix \(X_{N\times K}\), and \(\omega _{j}\) is the final weight of the jth variable.

Equation 8 shows the composite index, \(CI_{n}\), is proportional to the principal components’ values.Hence, it is straightforward to construct the composite index, \(CI_{n}\) based on the PC values [53]. Accordingly, this study examines two 234-length series of \(CI_{n}\) for the trading partners’ import and export networks, and two 154-length series of \(CI_{n}\) for the reporting countries’ import and export networks.

The goal of this subsection is to compare the core countries in the reporting countries’ and trading partners’ networks. After constructing the composite index out of the network structural measures, namely the centrality measures, we need to find the countries that are the main contributors of the trade network structure. In an attempt to find the core countries in the trade networks, entropy concept is applied on the composite index. Consequently, we argue that if the core countries in the trading partners’ networks are the same as the core countries in the reporting countries’ networks, the structure of these two sets of trade networks would be similar.

Entropy: This concept is an appropriate measure that determines the most variations being captured in the observations [54]. Entropy was first introduced in the field of thermodynamics in the nineteenth century. Later, Shannon (1948) used entropy in the field of information theory [55]. Entropy is the main measure in the information theory, and it estimates how much information is produced in an information source [56].

Shannon (1948) presented a discrete information source as a Markoff process. Based upon this, supposed there is a set of possible events with \(p_1,p_2,\ldots ,p_n\) probabilities of occurrence. In order to find a measure that captures how much choice/uncertainty/disorder- or chiefly, how much information- is involved in the selection of the event, Shannon proved that there is a measure, say \(H(p_1,p_2,\ldots ,p_n)\), as a measure of information in information theory. This measure has a form like Eq. (9).where M is a positive constant. Accordingly, he called Eq. (9) as the entropy of probabilities \(p_1,p_2,\ldots ,p_n\). Besides, Shannon stated that \(p_i\) can be defined as the average frequency of event i, \(f_i\) [56].

Note that the information can be stored in variables which can have different values [55]. For the case of Shannon, the information is stored in strings of symbols, but it can be stored in any other kind of variables. Similarly, Skillicorn introduced the entropy concept in the context of dataset, and dimension reduction, particularly, singular value decomposition (SVD). Suppose the dataset properties are presented in a space of r dimensions and we want to represent the dataset in a space of dimension k where (\({k}\le {r}\)). Skillicorn [54] states that one way to find k, is to discuss the contribution of each singular value to the whole, which is computed by Eq. (10).where \(s_i\) is the ith singular value. By definition, the proposed contribution of each singular value is equivalent to the average frequency of each singular value. Skillicorn [54] states that “entropy measures the amount of disorder in a set of objects”, and calculates the entropy of the dataset as Eq. (11) which is equivalent to the entropy measure proposed by Shannon (1948) [54, 56].In the context of this study, we aimed at finding the main contributors to the trade networks, i.e., the core countries that capture the most information on structural characteristics of the trade networks. Like the messages in Shannon (1948), and the singular values in Skillicorn [54], the composite index of structural network measures associated with countries (CI) is considered as the variable storing the information on the structural characteristics of the trade networks. Accordingly, the contribution of the composite index of structural network measures associated with N countries is calculated as Eq. (12), which is similar to Eq. (10) proposed by Skillicorn [54].where \(CI_n\) is the composite index out of the centrality measures for the nth country, \(N=154\) for reporting countries’ and \(N=234\) for trading partners’ trade (import/export) networks. So, \(f_n\) denotes the contribution of the nth country in the considered trade network [54]. The next step is to calculate the entropy of the dataset as Eq. (13), which is similar to Eq. (11) proposed by Skillicorn [54].Entropy has a value between 0 (all variation in the dataset is captured in the first country) and 1 (all countries are equally important). The magnitude of the entropy infers how many (and which) counties are the main contributors of the network [54, 57]. In an attempt to find the main contributors, following Zekri et al. [57], first, the CI associated with N countries are sorted in descending order. Second, the entropy calculated by Eq. (13) is multiplied by N as follows.Finally, the main contributor countries of the trade network structure, or equivalently the core countries, are the top number of contributor countries [calculated by Eq. (14)] according to their CI being sorted in descending order.

The step-wise network approach to study the robustness of studying trading partners’ network adopted in this study is outlined in Fig. 1.