*3.3. Methods*

#### 3.3.1. Social Network Analysis

A social network is a collection of social actors as nodes and their relationships. For international transboundary water cooperation, it can be abstracted as a network collection with state actors as nodes and cooperative linkages as social ties. Based on the research needs, this article eventually established the undirected weighted network:

$$\mathbb{C} = (N, \mathbb{R}), \tag{1}$$

where *C* is the global TWC network; *N* is the nodes of state actors; and *R* is the water cooperation linkages weighted by connection frequency or connection intensity.

The centrality of a node reflects its influence in the network. According to the theory of social network analysis, the degree, weighted degree centrality, and weighted betweenness centrality [84] are introduced to analyze the individual network characteristics of the TWC network, so as to quantify the importance and connectivity of the nodes (Table 2).

**Table 2.** Main analysis indicators of network characteristics.


Notes: Where *n* is the number of nodes in the network, *bjk* (*i*) is the probability that node *i* is on the shortest path between node *j* and node *k*, *wij* is the connection strength between node *i* and *j*.

#### 3.3.2. QAP Analysis

The traditional multiple regression model is based on the ordinary least squares (OLS) method, and its basic assumption is that there is no correlation between the independent variables. In the real world, however, "relationships" are usually not independent. To investigate the determinants of TWC, quadratic assignment procedure (QAP) is used.

QAP analysis is a nonparametric test applied to the "relationship-relationship" level. Its purpose is to examine the regression relationship between a matrix and other multiple matrices, as well as to evaluate the influence and significance of each independent variable on the dependent variable. Different from the OLS regression model, QAP regression does not require assumptions on the mutual independence between variables. In the QAP procedure for network analysis, the standard errors are estimated using repeated permutations of the data set [85]. The calculating logic of QAP is consistent with the analytical logic of multidimensional proximity, and the essence is to explore the degree of "proximity" between actors from the perspective of the relationship. Therefore, the QAP regression model is established as follows:

$$Y = \beta\_0 + \beta\_1 X\_1 + \beta\_2 X\_2 + \dots + \beta\_n X\_n + \mu\_\prime \tag{2}$$

where *Y* is the dependent variable matrix, depicting the intensity of TWC between countries; and *X*1, *X*2, ... , *Xn* as the independent variable matrices, which are specific indicators of multidimensional proximity between countries.

#### **4. Results**

#### *4.1. Time Series of TWC Events*

The latest update allows us to analyze the TWC trends more precisely. To capture their dynamics, the article counted the number of global TWC events by year, as shown in Figure 3. The latest TWC time series shows that there were 1423 water cooperation events around the world from 1948 to 2013. The maximum number of events appeared in 1992, which was 114; the minimum appeared in 1948, which was only seven events. The overall scale of events also increased significantly, from 33 in the Cold War era to 64.7 in the Post-Cold War era on the annual average level. In a certain period of time, the changes in the number of events were often not linear, mostly fluctuating. It can be found that sharp changes occurred around 1991, the number of events rise from 13 in 1986 to 114 in 1992.

**Figure 3.** Annual variations of the number of global TWC events.

The key reasons for these characteristics are the disintegration of the bipolar system and the development of the multi-polarization trend, namely, the change of the international system. From 1989 to 1991, major geopolitical events occurred in succession within three years. In particular, on 25 December 1991, the Soviet Union formally collapsed into 15 countries, resulting in an increase in the number of transboundary rivers and basins, as well as TWC events. After that, in the Post-Cold War era, peace and development became the themes of the times. With the ease in international political tension, the scale of TWC in this stage is higher than that of the Cold War era, and the interaction between countries has shown steady fluctuations.

#### *4.2. Spatial Differentiation of TWC Linkages Based on Frequency*

Taking state actors as nodes, TWC linkages in the Post-Cold War era as edges, and applying the connection frequency to give weight, a global TWC frequency network *C*<sup>1</sup> is constructed. The weighted degree centrality and weighted betweenness centrality of nodes in the network are calculated and Table 3 shows the top countries ranked by them. In terms of weighted degree centrality, the top 20 countries are all from Europe-North America, Africa, and Asia, and their distribution is relatively balanced, with eight, seven, and five countries, respectively. Compared with the former, the ranking of weighted betweenness centrality differs more among regions. Europe-North America, Africa, and Asia have five, five, and 10 countries, respectively, and more than half of countries come from Asia. Specifically, China, Egypt, Germany, the United States, and Russia have always occupied the top five in the two indicators, with China always occupying the first place. On the basis that they have the cooperative ability, this result is mainly related to the geographic and environmental factors of these countries. These countries have longer border lengths or a larger number of neighboring countries, which naturally determines their needs and willingness for TWC. However, it can also be found that for some countries with short borders and few neighboring countries, their status in the network is also prominent. The reasonable explanation is that this is related to their own specific interest demands, which include both water-related and non-water-related interests. Some countries have high water security needs, so they would actively take TWC to meet their water-related interests, such as Israel. Other countries are more expected to meet other interests through TWC, such as questing for their international status or enhancing their national image. A typical case is Japan. After World War II, Japan has long carried out economic diplomacy with ODA (Official Development Assistance) as the main means and provided assistance to many countries, and TWC affairs are one of its priorities. Therefore, while exporting its own successful water management experience, Japan continuously expands its political and economic interests as well as enhances its international image.


**Table 3.** Countries' hierarchies based on weighted centrality indicators.

To make better sense of the network structure, the spatial pattern of TWC between countries is illustrated. As shown in Figure 4, the frequency network of TWC in the Post-Cold War era has obvious topological and spatial structure heterogeneity. First, Asian countries participate in TWC much more frequently than others, and the local structure of the network in Asia is also denser and more complex. The highest frequency of cooperation has occurred between China and Russia, up to 78 times. Among the top 20 partnerships, there are 14.5 pairs of Asian countries. Second, the network structure consists of triangular or quadrilateral structures within the continent, which are commonly found in the Nile, Zambezi River Basin in Africa, Danube River Basin in Europe, Mekong River Basin in Asia, etc. Clearly this shows that TWC is sensitive to geographical distance, and its geographical proximity is prominent. Third, extra-regional powers are widely involved in TWC. On the one hand, intercontinental interaction among countries is obvious, for example, the United States and Canada are widely involved in TWC in Asia, while European countries maintain a high level of interaction with African and South American countries. On the other hand, some island countries actively participate in TWC among continental countries. For example, Japan has extensive cooperation with countries in East, Southeast, and West Asia. Additionally, the UK has extensively established cooperative relations with countries in East and West Africa.

**Figure 4.** Spatial pattern of transboundary water cooperation linkages based on frequency weighting.

#### *4.3. Spatial Differentiation of TWC Linkages Based on Intensity*

Cooperation frequency can reflect the scale of cooperation, but cooperation intensity can more effectively reflect the quality of cooperation. Taking state actors as nodes, TWC linkages in the Post-Cold War era as edges, and applying the connection intensity to give weight, a global TWC intensity network *C*<sup>2</sup> is constructed. Calculating the weighted degree centrality and weighted betweenness centrality of state actors (Table 3), on the one hand, it can be found that among the top 20 countries compared with network *C*1, the proportion of Asian countries has remained stable, and the proportion of European countries has increased. Most of the countries with high centrality are located in the surrounding areas of China, as well as Eastern and Southern Europe. On the other hand, the status of extraregional countries, such as the United States and Japan, has declined.

For the former, its cause is inseparable from the constraints of the geographical environment and the relatively successful mechanism construction of the areas. In Asia, as Asia's water tower, the Tibetan Plateau closely connects China and neighboring countries through transboundary rivers, making the region have a lot of water cooperation needs and practices. In Europe, due to the high level of regional integration and the relatively complete construction of cooperation mechanisms, countries usually carry out high-intensity water cooperation.

For the latter, the cause may be that the cooperative willingness of countries outside the region is weaker than that inside the region. Although countries such as the United States, Japan, and South Korea have a prominent centrality in the frequency network, they are not located in the hot spot basins, and their participation in TWC is mostly in the form of economic and technical assistance. Therefore, they are less likely to achieve in-depth and decisive cooperation results with relevant countries than local participants. It is worth noting that China's two centrality indicators both rank first in both frequency and intensity networks, reflecting that China occupies an extremely important position in the network and is a very important participant in global TWC.

In terms of network linkages, it can be found that the topological and spatial structure heterogeneity of the TWC intensity network has become more obvious compared with the frequency network (Figure 5). First, the network hierarchy is obvious and there are far more low strength linkages than high strength linkages. Linkages with a strength higher than 10 accounted for only 36.1% of the total. Second, Asia is the continent with the most complex TWC spatial pattern and the highest concentration of hot spots. Highintensity water cooperation runs through the Eurasian and African continents. High strength linkages only exist between geographically neighboring countries within a certain geographic area. Countries located in the Amur, Mekong, Ganges, Indian, Aral Sea, Jordan, and the Nile River Basin have carried out high-intensity water cooperation.

**Figure 5.** Spatial pattern of transboundary water cooperation linkages based on intensity weighting.

#### *4.4. QAP Multiple Regression Results*

By importing the multidimensional proximity variable matrices of global TWC into the QAP regression model, and then having performed 2000 times matrix random permutations to estimate the standard errors, the regression results were obtained. Table 4 reports the results of the QAP regression. The goodness of model fit is 0.272, indicating that the variables can explain the difference in TWC intensity between countries. The regression results show that geographical proximity, economic proximity, organizational proximity, and colonial proximity have significant effects on TWC. This also allows our theoretical framework to be quantitatively verified.


**Table 4.** QAP multiple regression results.

First, the significant impact of geographical proximity on TWC has been fully verified. On the one hand, the capital distance between countries is negatively correlated with the intensity of TWC, and the result is significant at the 0.1% level, indicating that the closer the countries are, the greater the likelihood and intensity of TWC. On the other hand, whether countries belong to the same transboundary basin is positively correlated with the intensity of TWC. The result is significant at the 0.1% level, indicating that high-intensity TWC is more likely to occur between countries with spatial connections at the transboundary basins. Additionally, this result is also clearly reflected in the spatial pattern of the TWC intensity network (Figure 5). Second, economic proximity has a significant positive effect on TWC. The bilateral trade volume, as its specific indicator, is significant at the 0.1% level, indicating that closer trade between countries is more conducive to the development and deepening of TWC. With the development of globalization, the dependence of economy and trade have increasingly become the anchor of political relations between countries [86]. The higher the degree of trade dependence between countries, the more it leads to shared benefits, which in turn will affect political relations between countries and promote mutual cooperation. Third, organizational proximity has a significant positive effect on TWC, and the result is significant at the 0.1% level, which means that the more water organizations exist among countries, the greater the intensity of TWC. For example, a variety of cooperation regimes have been formed in the Mekong River Basin [87], including the GMS (Greater Mekong Subregion Economic Cooperation), the MRC (Mekong River Commission), the AMBDC (ASEAN-Mekong Basin Development Cooperation), the LMI (Lower Mekong Initiative), the MGCI (Mekong-Ganga Cooperation Initiative), and the LMC (Lancang-Mekong Cooperation). These regimes provide various dialogue platforms for the basin countries and play an important role in promoting transboundary water governance and economic cooperation in the region. Fourth, colonial proximity is significantly positively correlated with the intensity of TWC, and the results of each indicator are significant at the 0.1% level. This suggests that former colonies prefer to maintain a high level of cooperation in transboundary water matters with former colonizers as well as other former colonies. For example, in 2007, Uganda and the Congo-Kinshasa had to refer to the agreements and maps reached in Europe in the past to resolve the dispute over the lake islands between the two countries. Another example is when the UK established a new close bond by transferring

power to the regime that was most beneficial to its own interests while recognizing the independence of the colonies, and at the same time incorporated the newly independent country into the Commonwealth, thus establishing a new kind of close ties. Thus, in Figures 4 and 5, we can see that the UK is widely involved in African water affairs.
