The multi-regional input-output (MRIO) model has been universally chosen for revealing the virtual water flows between regions [39]. In a MRIO framework, regions are linked together by trade. The MRIO table used in this study contains 30 provincial-level administrative regions, including 23 provinces, three autonomous regions and four municipalities at the year 2007, 2010, 2012, and 2015. The MRIO data is obtained from CEADs database ¹ , published papers and books [40–42]. The data on water consumption and population at the year 2007, 2010, 2012, and 2015 are taken from China Statistical Yearbook [43–46] and China Urban-Rural Construction Statistical Yearbook [47–50]. For the sake of brevity, we use particular codes in figures to abbreviate the name of different provinces, which are shown in Table A1 in the Appendix.

Multi-regional input-output table provides a useful approach that can be used to reveal the virtual water flows among sectors or regions [42]. As shown in Table 1, Chinese economic system consists of N regions. The goods or services imported from Region i to Region j can serve either as intermediate use (denoted by x _ij ) or final use (denoted by f _ij ). Thus, the total output in Region i, denoted by x _i , is the sum of intermediate inputs and the final demand, which is shown in Eq. 1. x i = ∑ j = 1 N x i j + ∑ j = 1 N f i j . (1)

The direct consumption coefficient a _ij reflects the required quantity of imports from Region i per unit output in Region j, which is expressed as: a i j = x i j x j . (2)

By substituting Eq. 2 into Eq. 1, the matrix expression of the basic form of MRIO model is obtained as following: X = A X + F . (3)

It follows from Eq. 3 that X = ( I − A ) − 1 F , (4) where I is a N-by-N identity matrix. (I − A)⁻¹ is Leontief inverse matrix, which contains both direct and indirect inputs required to meet one unit of final demand in monetary value [51].

In multi-regional input-output analysis of China, the direct water intensity coefficient WI _i of province i is defined as: W I i = Y i x i , (5) where Y _i and x _i respectively represent the water consumption and total output of province i. By multiplying the direct water intensity coefficient by Leontief inverse matrix, the total virtual water coefficient ɛ can be expressed as following: ε = W I ( I − A ) − 1 . (6)

Combined with the total virtual water coefficient matrix and the final demand matrix, the inter-provincial virtual water transfer amount T can be calculated, which is expressed as: T = ε F . (7)

In the inter-provincial VWT network, the roles of provinces on water consumption are different and change over time. On the one hand, due to regional heterogeneity, different provinces play different roles. On the other hand, the role of provinces in the network evolves dynamically over time. Therefore, the panel regression model can be used to analyze the relationship between the roles of provinces and water consumption in China from the spatial and temporal dimensions: Y i t = α + β 1 i t X i t + β 2 i t c o n t r o n l i t + γ i t ( i = 1,2 , … , N ; t = 1,2 , … , T ) , (17)

X _it , Y _it and control _it separately represent the set of core explanatory variables, explained variable and control variables. β1 _it and β2 _it are the coefficients of explanatory variables and control variables. γ _it is the residual.

1) Explained variable

This paper aims to use panel regression model to evaluate the drivers of water consumption in China. The water consumption of each province is selected as explained variable.

2) Core explanatory variables

Indicators reflecting the structural centrality of the inter-provincial VWT network are chosen as core explanatory variables, including: in-degree (D ⁱⁿ ), out-degree (D ^out ), in-strength (S ⁱⁿ ), out-strength (S ^out ), in-closeness (C ⁱⁿ ) and out-closeness (C ^out ).

3) Control variables

Control variables are used to eliminate some important common factors affecting virtual water consumption in the province. The increase in the urbanization rate and the improvement of the water-saving system will drive the reduction of water consumption. The final demand represents the final use or consumption of the social total products by consumers. If a province has a high consumption capacity, this drive the province’s water use from production to consumption to a certain extent. In addition, there are differences in production technology in different provinces. Advanced technology can promote the reduction of water consumption. Water intensity reflects the difference in production technology, and thus can effectively describe the difference in water consumption between different provinces. Therefore, in this paper, urbanization rate (U) ² , final demand (F) and water intensity (WI) are selected as control variables. In order to eliminate potential multicollinearity between variables, each variable in the panel regression model is estimated separately.

The initial inter-provincial VWT network based on MRIO table in China is a fully connected network. In complex network theory, it is difficult to reveal the essential characteristics for a fully connected network [36]. To better understand the basic structural characteristics of virtual water transfer flows between provinces, it is essential to set a threshold to eliminate the disturbance of edges with negligible weights. Firstly, the edges in the initial VWT network from 2007 to 2015 are sorted by weight. Then, we find that at most 40% of the edges have transferred more than 80% of the total virtual water flow. The remaining more than 60% edges are very weak connections, which are relatively trivial for structural analysis. Figure 1 shows the distribution of the ratio of edge weight to total weights in the initial VWT network during the period 2007–2015, arranged in ascending order from left to right. The yellow dashed line represents the critical threshold for filtering out those unimportant edges, which is set to guarantee the total weight of the removed edges accounts for 20%. Thus, only those edges that accounted for 80% of the total volume of the virtual water transfer are retained in the final networks. The inter-provincial VWT networks in 2007, 2010, 2012, and 2015 are shown in Figure 2. During 2007–2015, virtual water flow played an increasingly important role in water usage of China. The virtual water flow was 233.6, 228.6, 239.2, and 251.6 billion cubic meters for 2007, 2010, 2012, and 2015 respectively. In other words, the virtual water flow has increased by more than 7% in the past few years. Figure 2A shows the virtual water in Xinjiang, the largest exporting province at that time, was mainly transferred to the Central region in 2007, then to the South Coast region through the East Coast in 2010 (shown in Figure 2B) and 2012 (shown in Figure 2C). By 2015, Jiangsu has replaced Xinjiang as the largest exporter, the virtual water in which were mainly transported to the South Coast region (shown in Figure 2D).

It can be found from Table 2 that the network structure has characteristics of evolution with time. From 2007 to 2015, the density of inter-provincial virtual water transfer networks showed an increasing trend, indicating that network connections have become closer. The growing network density reflected the increasing frequency of virtual water transfer between provinces, which was related to the rapid development of inter-provincial trade in China. The average clustering coefficient was roughly around 0.6, implying that the probability of virtual water transfer between neighboring provinces was relatively high. The average shortest path length of the network experienced a slight fluctuation around 1.6, which showed a quite fast transfer speed of virtual water between provinces. The small average shortest path length indicated that the virtual water in one province could flow either directly or indirectly through at most one intermediate province to the destination province. Large average clustering coefficient and small average shortest path length made the inter-provincial virtual water transfer network exhibit the characteristics of a small world network, where most nodes were non-adjacent and could reach every other node in relative small steps [57].

The structural roles of each province in the inter-provincial VWT network were identified with the aid of complex network analysis. It is widely known that the node degree characterizes the position of the node in a network. It can be seen from Figure 3 that several provinces have sustained high out-degrees over these years, which indicated that those provinces had many exporting partners. The changes in the number of exporting partners differed greatly among those provinces with high out-degrees. The number for exporting partners has shown a slight upward trend for Jiangsu, Anhui, Henan and Hunan provinces, while a slight decline for Guangxi province. The changes in the number of exporting partners differed greatly among these provinces with high out-degrees. The number for exporting partners has shown a slight upward trend for Jiangsu, Anhui, Henan and Hunan provinces, while a slight decline for Guangxi province. Compared with the case of out-degree, provinces with relatively high in-degree have many import partners. Over these years, the number of importing partners has grown slightly for Guangdong, Zhejiang and Henan provinces, while declined slightly for Shanghai. In particular, Jiangsu and Guangdong with both high in-degree and out-degree values had a close virtual water transfer relationship with other provinces, which was consistent with their economic status in the entire supply chain of China.

Another important role character is the node strength. The in-strength and out-strength of each province in the VWT network were shown in Figure 4. Jiangsu, Xinjiang, Guangdong, Anhui and Heilongjiang with high out-strengths, were the main virtual water suppliers. On the contrary, provinces including Jiangsu, Guangdong, Zhejiang, Henan and Shanghai had high in-strengths and were the main virtual water consumption markets. In addition, water stress index (WSI) was employed to evaluate the scarcity of water resources in a region. It is calculated by the ratio of water withdrawn to available water. The range of WSI values is between 0.01 and 1. A WSI of 0.5 is usually set as a threshold of medium and high water stress [58, 59]. The shade of the color on the map in Figure 2 represents the WSI value of each province. Combined with the results in Figures 2, 4, it can be found that the positions among these provinces are different. In 2007, 2010, 2012, and 2015, Beijing, Shanghai, Tianjin and Jiangsu are provinces with severe water shortage due to extremely high WSI values. Beijing, Shanghai and Tianjin are the main NVW importers. However, Jiangsu was the main NVW exporter, ranking second in net exports. Xinjiang, Heilongjiang, Guangxi, Anhui, Guangdong and Zhejiang were rich in water resources. Especially in Guangxi, Anhui, Guangdong and Zhejiang, WSI was lower than 0.1. However, the situation between them was quite different. Xinjiang, Heilongjiang, Guangxi, and Anhui were the main NVW exporters during 2007–2015, while Guangdong and Zhejiang were the main NVW importers, ranking the top two in terms of net imports.

The closeness of a node accounts for the importance of the node from another perspective. Figure 5 shows that the values of in-closeness and out-closeness were relatively large, indicating that the virtual water transfer speed was fast. In particular, the out-closeness values of Jiangsu, Guangdong and Xinjiang were all above 0.8, which imply that virtual water in these provinces could flow to other provinces through relative short paths. Moreover, the values of in-closeness and out-closeness both showed a slight upward trend. This trend demonstrated that changes in province would soon spread to other provinces due to closer connections between provinces.

The stability of a complex network depends largely on its structure. Therefore, in the inter-provincial VWT network, the structural role of provinces has a significant impact on their water consumption. To test this assumption, panel regression analysis was applied to examine the impact of structural roles of provinces on their water consumption. Here the fixed effects model was selected as the panel regression model through Hausman test. In this paper, water consumption is chosen as the explained variable, D ⁱⁿ , D ^out , S ⁱⁿ , S ^out , C ⁱⁿ and C ^out are the core explanatory variables, and urbanization rate, final demand and water intensity were the control variables.

Table 3 lists the descriptive statistics of the variables considered. Significant differences between some variables can be found from this table. Before substituting into the panel regression model, all variables need to be normalized to eliminate the impact of different dimensions between variables, so that equivalent comparisons can be made between the effects of variables. The results of unit root test and co-integration test showed that all normalized variables were stable, and there was a co-integration relationship between variables, as shown in Tables 4, 5.

The correlation between variables was shown in Table 6. The results showed that there was a strong correlation between some variables. For instance, the correlation between in-degree and in-strength was 0.906 and the significance level was less than 1%. Thus, this paper constructed six panel regression models to evaluate the influence of the structural characteristics of inter-provincial VWT network on water consumption, and the other two models were used to evaluate the influence of control variables and all variables on water consumption. Table 7 showed the results of the panel regression model. Model 1 measured the impact of control variables on water consumption. The results showed that both water intensity and final demand had a significant positive impacts on water consumption. Specifically, water intensity characterized by technical effects was an important driving force for changes in water consumption [42]. Technological progress and improvement of technical efficiency have led to a reduction in water intensity and greatly reduced water consumption. Final demand was the continuous driving force of economic growth, which in turn would drive water consumption. The level of urbanization had a weak but significantly negative effect on water consumption. The urbanization process accompanied by industrial agglomeration and technological progress has promoted the improvement of water use efficiency and the reduction of water consumption [60]. In addition, the high population density brought about by urbanization made infrastructure sharing inevitable, which also contributed to the reduction of water consumption to a certain extent.

In models 2–7, the effects of the six structural roles (in-degree, out-degree, in-strength, out-strength, in-closeness and out-closeness) of provinces on water consumption in China were evaluated. The results of panel regression analysis showed that out-degree and out-strength had a significant positive effect, while in-closeness had a significant negative effect. In-degree, in-strength and out-closeness had no significant impact on water consumption. The impacts of out-degree indicated that the provinces would consume more water if they had more exporting partners. Similar situation occured when the impact of out-strength was investigated. Provinces exporting more virtual water tended to consume more water. The reason lies in the products with plenty of virtual water embodied in, produced by these provinces are not only to meet the needs of the province, but also to be exported to downstream provinces. This demonstrated that water consumption was mainly affected by direct production demand from downstream provinces. Therefore, by controlling the direct production demand from downstream provinces, water consumption could be significantly reduced. Conversely, the significant negative impact of in-closeness indicated that the transferring speed of the virtual water played a key role in water consumption. One province would consume less water resource if products would be imported from other province in a very short time. Thus, this province was often considered as a direct or indirect downstream market for many provinces in the production chain. In other words, if the production demand of a downstream province would be quickly filled from the upstream provinces, resulting in its less consumption of water resources. The impact of in-closeness indicated that indirect production demand would also significantly drive water consumption. In conclusion, the reduction of water resources should be implemented in both direct and indirect production activities.