The global data on productive trade in services was sourced from the official OECD website database. This database includes the productive service trade between 46 countries and regions from 2005 to 2022. The productive service industries in OECD-STRI are road transportation, railway transportation, sea transportation, air transportation, logistics loading and unloading, logistics warehousing, freight forwarding, logistics customs clearance, broadcasting, video, audio, telecommunications, accounting, computer, commercial banking, insurance, law, architectural design, and engineering design. In combination with the continuity and integrity of the data obtained, 42 countries or regions were finally selected as the network nodes ^①

① Australia, Austria, Belgium, Canada, Chile, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Israel, Italy, Japan, South Korea, Luxembourg, Mexico, Netherlands, New Zealand, Norway, Poland, Portugal, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey, Britain, USA, Chinese Mainland, Columbia, Hong Kong (China), India, Latvia, Lithuania, Russia, Saudi Arabia.

, and they were analyzed in three stages: 2005-2010, 2011-2016, and 2017-2022. The total amount of productive service trade between the sample countries during 2022 was 114.70 trillion USD, accounting for 84.29% of the global productive service trade volume during that period (136.08 trillion USD). To investigate the topological characteristics of the global productive service trade network, matrix processing must be performed on the raw data of global productive service trade to transform it into a 42×42 unweighted matrix.

The global network of productive trade in services can be defined as:

In Equation (1), $a_{i j}^{t}$ represents the existence of productive service exports from country i to country j in year t; w_ij^t represents the number of patent applications from country i to country j in year t.

The complete network A^t is a social network that reflects the productive service trade relations among countries. Analyzing this network can not only clearly describe the topological characteristics of global productive service trade, but also reveal how changes in productive service trade among several node countries affect other node countries.

Since China's global productive service trade network has topological characteristics, and the topological properties and evolutionary laws of the network cannot be explained by the random graph paradigm, the widely used indicators in network feature analysis were selected for analysis. The main indicators are described in detail below.

Due to the spatial correlation of global productive service trade, examining its spatial correlation characteristics is crucial for understanding the forms of global productive service trade. There were 42 node countries (regions) in the global productive service trade network from 2002 to 2022 (Table 1), and the density and number of relationships in the network showed a significant upward trend. Among them, the number of relationships in the network increased from 973 in 2005-2010 to 1276 in 2017-2022, and the increasing number of relationships also increased the network density from 0.565 in 2005-2010 to 0.741 in 2017-2022. Correspondingly, the average degree of the network also increased from 32.476 in 2005-2010 to 37.810 in 2017-2022. However, as productive service trade relations between network node countries (regions) became increasingly close, the average path length of the network also significantly decreased, from 1.435 during the period from 2005 to 2010 to 1.259 during the period from 2017 to 2022. In addition, from the perspective of the maximum output of the network, the maximum values during the periods of 2005-2010, 2011-2016, and 2017-2022 were 30, 35, and 41, respectively, indicating that a certain node country in the network exported productive service products to 30, 35, and 41 other countries (regions) during the corresponding sample period. From the perspective of the maximum input, the maximum values during the periods of 2005-2010, 2011-2016, and 2017-2022 were 29, 35, and 36, respectively, indicating that a certain country in the network imported productive service products from 29, 35, and 36 other countries (regions) during the corresponding sample period. Thirdly, the scale of global productive service trade increased significantly from 2005 to 2022, with the average trade volume rising from 23.686 trillion USD in the period of 2005-2010 to 41.213 trillion USD in the period of 2017-2022. The maximum export and import intensities of node countries (regions) during the period of 2017-2022 were 59.102 trillion USD and 36.536 trillion USD, respectively. Overall, the scale of global productive service trade continues to expand, showing an increasingly interconnected and networked trend.

The number of areas (i.e., the number of network partitions where the node is located) in the global productive service trade network from 2005 to 2022 was 1 (Table 1), indicating good connectivity of the overall global productive service trade network. The network efficiencies during the periods of 2005-2010, 2011-2016, and 2017-2022 were 0.305, 0.411, and 0.563, respectively, indicating that the relationships between node countries (regions) in the global productive service trade network are interrelated and the network is relatively stable overall. The hierarchical degrees of the network during the periods of 2005-2010, 2011-2016, and 2017-2022 were relatively small, with values of 0.211, 0.183, and 0.132, respectively. This indicates that the hierarchical division of the global productive service trade network is not clear, and countries (regions) with different levels of development in their productive service industries also have spatial dependence relationships. In addition, the average path length of the global productive service trade network has continued to decline (from 1.435 during 2005-2010 to 1.259 during 2017-2022), but the clustering coefficient of the network increased from 0.576 during 2005-2010 to 0.741 during 2017-2022. This indicates that the global productive service trade network has “centralization” and “small world” characteristics. In other words, although the node correlation of the global productive service trade network is becoming increasingly close, there is a clear core periphery structure, and the monopolistic position of core countries (regions) in the productive service trade network is becoming more obvious.

Analyzing the centrality of network nodes is crucial for exploring their status and role in social networks. This study analyzed the influence of countries (regions) in the global productive service trade network from the perspectives of the absolute centrality and intermediary centrality of network nodes. Note that the total degree, export degree, and import degree of global productive service trade network nodes have significantly increased between 2005-2010 and 2017-2022 (Table 2).

In addition, based on the regional and national distribution of global productive service trade, traditional developed countries (regions) such as the United States, the Netherlands, South Korea, Spain, and Germany rank high in export centrality, indicating that these European and American countries have leading positions in productive service exports, while developing countries such as Mexico, India, Brazil, and China have high import centrality, indicating that these countries (regions) are the main importers of productive service trade. From the perspective of total degree centrality, countries such as the United States, the Netherlands, China, and South Korea have always been at the center of the global productive service trade network, with the highest number of countries (regions) involved in the import and export of productive service trade. However, developing countries such as Hungary and Chile rank relatively low. China's central position in the global productive service trade network has continued to rise, from 7th place during 2005-2010 to 2nd place during 2017-2022, and the number of countries (regions) engaged in productive service trade with China has increased from 33 to 39. Therefore, China's network position and influence in global productive service trade are constantly increasing.

Figure 1 shows the complete network diagrams of global productive service trade from 2005 to 2010 and from 2017 to 2022, with the node sizes reflecting node degrees. That figure shows a clear “center periphery” phenomenon in the global productive service trade network. Among the countries, developed countries such as the United States, the Netherlands, South Korea, Japan, and Canada have more relationships and occupy “central positions” in the network. However, except for a few developing countries such as China and Brazil, the node degrees of other developing countries are generally small, and they are at the “edge” of the network.

The intermediate centrality of nodes in the global productive service trade network was also calculated. From 2005 to 2010, the top 10 countries in terms of centrality were the United States, Sweden, Luxembourg, Lithuania, Saudi Arabia, Japan, Spain, Portugal, Italy, and Russia (Table 3). Among these 10 countries, two are developing countries and eight are developed countries. The top 10 countries in terms of centrality between 2017 and 2022 were the United States, Spain, Saudi Arabia, Portugal, Italy, Mexico, Colombia, France, Latvia, and Canada. Among these 10 countries, four are developing countries and six are developed countries. This indicates that the “bridging role” of developing countries in the global productive service trade network is becoming increasingly evident.

The community ownership and stability of the global producer services trade network were analyzed using formulas (2) and (3) above. Although there are differences in the community divisions of node countries during 2005-2010, 2011-2016, and 2017-2022 (indicating the community ownership of some marginal countries is unstable), the community ownership of most countries is relatively stable (Table 3). The first club included 12 countries, the United States, Australia, Belgium, Czech Republic, Denmark, Finland, France, Germany, Japan, Spain, Sweden, and Turkey, which are mainly European and American countries with developed service industries. The second club included 12 countries, Canada, Chile, Iceland, Israel, the Netherlands, Chinese Mainland, Colombia, Greece, Italy, India, Mexico, and South Korea, which are mainly countries with strong vitality in the development of the service industry. The third club included 11 countries, New Zealand, Austria, Portugal, Poland, Norway, Luxembourg, Switzerland, Saudi Arabia, Ireland, the United Kingdom, and Hong Kong (China), which are mainly countries with great potential for service industry development. The other seven countries (see Table 3) are scattered among different societies and are unstable members of the society. Overall, the modularity of the global productive service trade network has continued to decline, indicating that in the context of economic globalization, the degree of integration of the global productive service industry is increasing, and the spatial connectivity network composed of node countries is becoming increasingly loose.

Due to the large number of edges in the national productive service trade spatial correlation network, a top-level structural network was further constructed in order to reveal its network characteristics more clearly, and the status and influence of network nodes were analyzed (Zhou et al., 2018). It was found to be a Top 1 network, meaning that only country i ranks first in the export of productive services to country j, and these two countries establish connections. The top-level network structure diagrams for the periods of 2005-2010, 2011-2016, and 2017-2022 are shown in Figure 2.

Note in Figure 2 that the number of regions in the top-level networks during the three periods of 2005-2010, 2011-2016, and 2017-2022 remained constant at 1, and the structure became increasingly flat, indicating a higher degree of network integration in the global productive service trade. This suggests that the strategic position of service trade in the open economies of various countries is becoming increasingly important, and the trend of service-oriented linkage in the global economy is becoming stronger. Productive service trade between countries is becoming a new driving force for international trade growth. Table 4 shows the statistics of the top five regions in the global production service trade top-level network node export rankings from 2005-2010, 2011-2016, and 2017-2022. Clearly the top five countries in the top-level network are all developed countries, indicating that developed countries are still the main members of the production service industry exports. The role of core members in the top-level network of global productive service trade can be illustrated using Germany's Top1 as an example. From 2005 to 2010, Germany's node export degree was six, indicating that Germany was also the largest exporter among the six node countries in the global productive service trade network. By 2017-2022, Germany's node export in the top-level network had increased to 11, indicating that Germany has always been a major exporter in the productive service trade for most countries in the world. Overall, the control and influence of developed member countries on global productive service trade continue to increase, mainly due to the high added value of the productive service industry, which has a significant controlling influence on the global industrial chain. Therefore, developed countries such as European countries, the United States, Japan, and South Korea have taken the lead in the transformation of international economic and trade rules, further extending the modern service industry with competitive advantages to the world.

Based on the topological structural characteristics of the global productive service trade network discussed above, its spatial evolution presents a clear core edge pattern. Sun et al. (2018) stated that the central and peripheral characteristics of a network are closely related to the geographical locations of the nodes. From the block model analysis results mentioned above, the global productive service trade network structure has a significant neighborhood interaction effect, with stronger productive service trade relationships between countries in the same region. At the same time, the top-level structural network analysis also found that productive service trade relationships between countries in the same regional alliance are closer. Therefore, the proximity of geographical locations is an important factor affecting global productive service trade. The second factor is the relationship between national economic development, mainly the differences in the level and mode of economic development among countries. The block model analysis of the global productive service trade network found that countries with stable economic development levels and modes within the same community are relatively close. In fact, the basis of competition between countries lies in the competition for economic strength and market share. Under this mechanism, the governments of various countries focus more on strategic competition among countries with similar levels of economic development, economic regulatory capabilities, and economic development modes. Therefore, countries with similar levels of these three factors are more likely to engage in productive trade in services. The third factor is institutional. Sheng and Liao (2004) suggested that regional trade arrangements have a significant impact on trade, and the trade connections between countries that sign trade agreements may be closer, while the opposite is also true. The economic development level of each country can be measured by per capita GDP (Lin and Liu, 2000) and infrastructure level (Wang et al., 2007). The economic regulatory capacity of each country can be measured by the national economic fiscal burden rate, following the method of Liu (2009). In addition, the economic development model used here drew on the method of Li et al. (2014) to select indicators for measurement, such as industrial structure and economic openness. Regional trade arrangements used here drew on the method proposed by Sun et al. (2018) to select a matrix that reflects whether two countries have signed a trade agreement to represent this factor. Based on the above theoretical analysis, the following econometric model was constructed:

In Equation (9), PS represents the matrix data of the global productive service trade network among countries; and S represents the geographical adjacency matrix. If two countries are geographically adjacent, the value is 1, otherwise it is 0. Pgdpc, Roadc, Pfrc, Struc, and Openc respectively represent the difference matrix of per capita GDP, infrastructure level (mileage of roads per unit of land area), national economic fiscal burden rate (proportion of fiscal expenditure in regional GDP), industrial structure (proportion of manufacturing output in regional GDP) and economic openness (proportion of total imports and exports in regional GDP) of each country. The calculation of the difference matrix referred to the method of Li et al. (2014). First, the average value of each variable in each country was selected, then the observed value of the variable was subtracted from the average value, and the absolute value was taken to obtain the component difference matrix. The regional trade arrangement variable was represented by a matrix (F) that reflects whether the two countries have signed a trade agreement. If the two countries have signed a trade agreement, then it is 1, otherwise it is 0. The sample data were sourced from the GeoDist database and UNComtrade_SITC database of CEPII.

The results from using the R language to conduct a QAP correlation analysis on the factors influencing the global productive service trade network are shown in Table 5. The spatial correlation network and the factors influencing the development of China's regional knowledge intensive service industry were tested, and the correlation coefficient results are also shown in Table 5. During the three periods of 2005-2010, 2011-2016, and 2017-2022, the correlation coefficients between the topological correlation network of global productive service trade and the geographical adjacency matrix were all significantly positive, indicating that geographical factors are one of the important conditions affecting productive service trade between countries, and that productive service trade between geographically adjacent countries is more frequent.

Secondly, the spatial difference matrix coefficient of per capita GDP of each country is significantly negative in all three stages, and the absolute values of the coefficient during 2005-2010, 2011-2016, and 2017-2022 became increasingly large, indicating that producer services trade between countries with similar economic development levels has become more likely to occur in recent years. The spatial difference matrix of infrastructure level is negative but not significant, indicating that the internal correlation of infrastructure level with global productive service trade is not significant. This may be due to the time lag effect of positive externalities generated by global transportation infrastructure construction. The spatial difference matrix coefficient of the national economic fiscal burden rate is significantly not negative, indicating that the smaller the differences in fiscal expenditures among countries around the world, the closer their exchanges in productive service trade. In addition, in all three stages, the estimated coefficient of industrial structure (i.e., the proportion of manufacturing output value to the GDP of the region) is significantly negative, and its absolute value is the largest, indicating that the similarity of industrial structures among countries is a key factor for explaining the topological structure of global productive service trade. The similarity of unit manufacturing output value is an important factor for explaining the exchange of productive service trade. This may be because the productive service industry is a supporting service industry for manufacturing enterprises, so it runs through the upstream, midstream, and downstream links of the manufacturing industry chain and supply chain. Therefore, as an intermediate input, the productive service industry has a close relationship with the development of the manufacturing industry. The spatial difference matrix of economic openness (i.e., the proportion of total import and export volume to the GDP of the region) is significantly negative, indicating that productive service trade between countries with similar economic openness is more likely to occur. The correlation coefficient of the variable matrix of regional trade arrangements is significantly positive, indicating that the productive service trade links between countries that have signed trade agreements are closer. Following the approach of Zhou (2024), variables with insignificant correlation coefficients were excluded from the QAP regression analysis in the following discussion.

A further QAP correlation analysis among the six selected independent variables shows that the geographical adjacency matrix and the difference matrix of per capita GDP, national economic financial burden rate, industrial structure, economic openness, and regional trade arrangement matrix are correlated at the 5% significance level. Therefore, if the data for these six variables with associated attributes were used for a traditional least squares regression, there may be multiple collinearity, which would distort the regression results and further verifies the rationality of using the QAP method for this test.

QAP regression analysis was used to explore the effects of influencing factors on the spatial correlation network matrix of the productive service industry. Its basic principle is to first perform multiple regression on the influencing factors and the spatial correlation matrix of productive service trade, and then randomly replace the rows and columns of the spatial correlation matrix of productive service trade for re-estimation (5000 random replacements). The regression results in Table 6 show that the goodness of fit of the equations for the three periods of 2005-2010, 2011-2016 and 2017-2022 are 52.6%, 60.2% and 75.8%, respectively, which indicates that in each period the geographical adjacency matrix, the difference matrix of per capita GDP, financial burden rate of national economy, industrial structure and economic openness, as well as regional trade arrangements could jointly explain 52.6%, 60.2% and 75.8% of the global productive services trade relations. The unit test results in Table 6 are all less than 0.01, indicating that the goodness of fit of each regression equation is significant at the 1% significance level after the permutation of rows and columns in the correlation matrix of productive service trade.

The regression results in Table 7 show that the regression coefficient of the geographical adjacency matrix is significant at the 5% significance level, indicating that geographical adjacency plays an important role in the correlation of productive service trade between countries, and there is a “neighborhood interaction” pattern in global productive service trade. During 2005-2010 and 2011-2016, the regression coefficients of the matrix variable of per capita GDP difference among countries were significantly negative at the 10% significance level, and during 2017-2022, it was significantly negative at the 5% significance level, indicating that the topological connection of the global producer services trade has a “club group” effect based on economic development level. In other words, the more similar the economic development level, the more frequent the producer services trade. At the same time, the estimated coefficients of the difference matrix of the national economic fiscal burden rate are all significant at the 10% significance level, indicating that similar levels of fiscal expenditure promote the exchange of productive service trade. The estimated coefficients of the difference matrix between industrial structure and economic openness are significantly negative, indicating that countries with similar economic development modes have stronger correlations in productive service trade. The estimated coefficients of the regional trade arrangement matrix are significantly positive, indicating that the productive service trade links between countries that have signed trade agreements are closer.

An additional QAP regression analysis was conducted on the global productive service trade spatial correlation matrix (i.e., the weighted trade network constructed using the productive service export data of 42 sample countries) for 2005-2010, 2011-2016, and 2017-2022. The empirical results in Table 8 show that except for the insignificant geographic adjacency matrix, the regression results are consistent with the previous discussion. This indicates that the geographical adjacency matrix, per capita GDP of each country, financial burden rate of national economy, industrial structure, differences in economic openness, and signing of regional trade agreements all have significant impacts on the global network of producer services trade. However, the geographic adjacency matrix does not have a significant impact on the volume of productive service trade between countries, but it only has a significant impact on the formation of productive service trade relationships between countries.

This study used social network methods to construct a global productive service trade spatial correlation network with 42 countries as nodes and the import and export trade relations of productive services as edges. Furthermore, social network and QAP methods were used to analyze the evolutionary pattern and determining factors of the global productive service trade network. The results show four key features of this system. 1) From 2005 to 2022, the number of relationships in the global productive service trade spatial correlation network continued to increase, from 973 in the period of 2005-2010 to 1276 in the period of 2017-2022. Correspondingly, the network density of the global productive service trade spatial correlation network increased from 0.565 to 0.741. The tightness of the global productive service trade spatial network is relatively high, and the connectivity effect of the network is good, with a correlation degree of 1. Network efficiency and other factors have been constantly improving. 2) Developed countries such as the United States, Germany, Japan, South Korea, and the United Kingdom have always been at the core of the global productive service trade network, with more export relationships in their productive service trade, while developing countries have more import relationships in their productive service industry. Notably, the “bridging role” of BRICS countries in the global productive service trade network is more evident. 3) From 2005 to 2022, the “factions” of the spatial association network of global producer services trade are clearly divided, and there are three stable associations. The first association is composed of 12 countries: the United States, Australia, Belgium, Czech Republic, Denmark, Finland, France, Germany, Japan, Spain, Sweden, and Turkey. The 12 countries of Canada, Chile, Iceland, Israel, the Netherlands, China, Colombia, Greece, Italy, India, Mexico, and South Korea have formed the Second Society. The Third Society is composed of 11 countries (regions): New Zealand, Austria, Portugal, Poland, Norway, Luxembourg, Switzerland, Saudi Arabia, Ireland, the United Kingdom, and Hong Kong (China). Other countries that are not listed are separate from the different associations and are unstable members of the association. Overall, the modularity of the global productive service trade spatial correlation network has been continuously declining, and the “allocation” of members is becoming increasingly loose. The integration level of the Top1 network of global productive service trade spatial correlation is gradually increasing. The node regions of the United States, Japan, and Germany in the Top1 network are very stable, and they are becoming important benchmark regions that affect the development of productive service trade in other countries. 4) The QAP correlation analysis found that the differences in per capita GDP, national economic financial burden, industrial structure and economic openness, and the signing of regional trade agreements have significant impacts on the formation of the spatial linkage network of global producer services trade, but the matrix variable of infrastructure level differences has no significant impact on it. Regional adjacency plays an important role in the spatial correlation of global productive service trade, and global productive service trade has a “neighborhood interaction” effect. Further QAP regression test results indicated that during the three sample periods of 2005-2010, 2011-2016, and 2017-2022, geographic adjacency, economic development level, economic regulation ability, similarity in economic development mode, and the signing of regional trade agreements could jointly explain 52.6%, 60.2%, and 75.8% of the changes in the spatial correlation of global productive service trade.

The guiding significance of the research conclusions for improving the status and influence of China's productive service industry on a global scale lies in four main policy recommendations.

(1) Increase investment and consolidate the foundation of the productive service trade industry. In the global productive service trade network, some “peripheral” countries have relatively lagging infrastructure construction. To enhance their position in the global productive service trade, it is necessary to increase investment in their infrastructure, especially in the digital economy field. This will help to promote the coordinated development of goods trade and productive service trade, and further promote the sustained growth of demand for the productive service trade.

(2) Deepen opening up and enhance the level of openness in the productive service trade. Although the global productive service trade has formed relatively stable trade partnerships, to further optimize the business environment, it is still necessary to establish flexible market management mechanisms, gradually eliminate trade barriers, and accelerate the marketization process of productive service industries. At the same time, the entry threshold for foreign investment in knowledge-based fields of the productive service trade should be lowered to attract more foreign investment and inject new vitality into the development of the productive service trade.

(3) Innovation leads and enhances the international competitiveness of the productive service trade industry. Therefore, to enhance the international competitiveness of the productive service trade industry, productive service enterprises should be encouraged to increase their investments in capital, technology, and talent. By establishing high-level public service platforms and technology research and development centers for information services, creative design, product evaluation, etc., the research and development investment costs of enterprises can be reduced, and the supply level of the productive service trade can be improved. This will help companies to occupy a more advantageous position in the global market.

(4) Strengthen cooperation and enhance the density of the global productive service trade network. With the continuous increase in the global productive service trade network density, countries should further strengthen cooperation and jointly learn advanced technology and experience. Appropriately increasing the imports of production factors with strong technology spillover effects, optimizing allocation with domestic production factors, and achieving resource sharing and complementary advantages will help to improve the overall efficiency and development level of global productive service trade.