This study uses network density, hierarchy and efficiency to measure the characteristics of the overall network.

Network density is defined as Equation 4: D e n s i t y = T / N N − 1 (4) where D e n s i t y is the network density, T denotes the total number of existing export and import connections between provinces, and N is defined as the total number of provinces.

Network hierarchy assesses asymmetrical reachability across nodes in a graph. Based on Equation 5, it can be calculated as: H = 1 − T / ⁡ max ⁡ T (5) where T denotes the number of symmetrically accessible node pairs, and max ⁡ T denotes the maximum possible quantity of symmetrically accessible node pairs.

Network efficiency assesses the level of redundant lines within a graph. Based on Equation 6, it can be calculated as: E = 1 − Q / ⁡ max ⁡ Q (6) where Q denotes the number of redundant lines, and max ⁡ Q denotes the maximum number of redundant lines.

This study computes outdegree, betweenness and closeness centrality to characterize the individual network structure.

Outdegree centrality. To compare the changes in the degree centrality of different provinces across various periods, this study selects relative degree centrality for analysis. Equation 7 for Outdegree centrality is as follows:

To partition network communities, the modularity index is calculated. Equation 10 for the modularity index is: Q = 1 2 m ∑ i j w i j − s i s j 2 m δ v i , v j (10) Q represents the modularity index, which is measured by the expected difference between the proportion of edges connecting nodes within a community in the actual network and in a random network, and s i denotes the total trade flows associated with node i in the network, s j denotes the total trade flows associated with node j in the network, m is the total sum of all trade volume. δ v i , v j is defined as whether countries i and j belong to the same community in the vegetable trade network. If they belong to the same community, δ v i , v j = 1; otherwise, it is 0.

Existing research suggests that the evolution of network relationships are influenced by both exogenous and endogenous mechanisms (Wu et al., 2024). Among them, exogenous mechanisms refer to actors’ attribute selection preferences and external situational factors, while endogenous mechanisms mainly refer to the self-organizing effects of the network (Liu et al., 2024).

Exogenous mechanisms refer to the influence of node and edge attributes outside the internal relationships of the network on network evolution. This study divides them into attribute-related effects and spatial externalities. Network actor attributes include receiver effects, sender effects, and homophily effects (Robins et al., 2007), which refer to different network relationship structures formed due to differences in actor attribute characteristics. Specifically, in the spatial association network of agricultural water use efficiency of this study, receiver effects refer to the probability that provinces with certain specific attributes become recipients of agricultural water use efficiency, whereas sender effects refer to the probability that a certain type of province is more inclined to radiate efficiency outward (Pan et al., 2022). In addition, homophily effects refer to the likelihood of establishing efficiency spillover relationships among provinces with the same attributes (Cai et al., 2022). Spatial externalities refer to the impact of exogenous networks on efficiency-related spatial networks. This study uses a geographic distance matrix between provinces to explore the effect of spatial externalities on network evolution (Htwe et al., 2020). Generally, efficiency spillover effects exhibit spatial proximity. The farther the provinces are from each other, the higher the cost of efficiency interaction (Pan et al., 2022), making it difficult to establish efficiency spillover connections.

Endogenous mechanisms refer to the self-organizing effects of networks. Network relationships can undergo self-organizing processes, where existing connections influence the formation of other relationships, ultimately forming certain network patterns. This process is defined as the self-organizing effect of networks, including reciprocity effects, preferential attachment effects, and transitivity effects (Robins et al., 2007). This study examines the impact of reciprocity and transitivity effects on network evolution. Reciprocity effect refers to the bidirectional interactive relationships of efficiency spillovers among provinces (Lu and Qin, 2025). According to complex network theory, reciprocity helps enhance trust and stability among entities (Shi et al., 2024), thereby facilitating information dissemination and resource flow between provinces. Therefore, an increase in reciprocity within the network may promote the diffusion of efficiency. Transitivity effects include both transitive triples effect and 3-cycles effects. The transitive triples effect refers to the tendency of Province A to establish efficiency spillover links with Province B, while also forming efficiency spillover connections with B’s partner provinces. It is used to measure the likelihood of provinces with shared partners establishing efficiency spillover relationships (Pan et al., 2022). The 3-cycles effect places greater emphasis on the direction of efficiency spillover relationships, reflecting the tendency for several provinces to form a ‘closed-loop’ relationship (Cai et al., 2022). These two types of endogenous structures can test whether regions prefer to cooperate in a ‘group’ pattern during the process of agricultural water resources efficiency diffusion (Liu et al., 2024).

SAOM includes a rate function and an objective function. The rate function, denoted as ω i x , α , has a value that is fixed as a constant during model fitting, and the objective function is denoted as f i x , β .

The rate function determines the waiting time for changes in the efficiency association relationships between provinces in the network. Let the initial network state of province i be x 0 . The time variable t is a random variable that follows an exponential distribution with a parameter of ∑ n = 1 31 ω n x , α . Let the network state after a time increment Δt be x 1 . The probability distribution function for province i to gain an opportunity to change its relationship status is expressed as Equation 11: δ i = ω i x , α ∑ n = 1 31 ω n x , α (11)

For all provinces j ≠ i in the network, province i will select the next network x 1 with a probability ε i j . And it is shown in Equation 12. ε i j = exp f i x 1 , β − f i x 0 , β ∑ n = 1 31 exp f i x 1 , β − f i x 0 , β (12)

This paper assumes that the SAOM rate functions ω i x , α all select province nodes with the same probability ε i . The model’s objective function reflects the changes in efficiency association relationships that a province will make after receiving an opportunity to change its relationship status. Specifically, it is expressed as a linear combination of the effects of endogenously generated network structural variables, one-way exogenous control variables, and two-way exogenous control variables (see Equation 13). f i x , β = ∑ β k S k i x (13) where: x is the given observed values of the longitudinal network, S k i x represents the endogenous structural effects and exogenous influencing factors, β = (β0, …, βl) is the parameter values corresponding to the various effects in the model’s objective function, reflecting the degree to which these effects influence the probability of a change in the efficiency relationship status for province i in the network. A positive (negative) β k indicates that the corresponding network effect S k i x will increase (decrease) the probability of province i forming a corresponding relationship.

This study evaluates the overall structural characteristics of the spatial association network of agricultural water use efficiency from the perspectives of network density, network hierarchy, and network efficiency (Figure 4). The results show that network density exhibits a fluctuating upward trend, stabilizing at 0.26–0.27 after 2010, which is consistent with the findings of Han et al. (2024) and Lai et al. (2025). This indicates that the network structure has become relatively stable. However, the average network density during the sample period is 0.26, suggesting that spatial linkages among provinces remain relatively loose, and further efforts are needed to strengthen interregional links and cooperation. From the perspective of network hierarchy, the index shows a fluctuating downward trend but consistently remains above 0.1, suggesting that the spatial association network has maintained a relatively high level of hierarchy with a clear ordered structure. This is due to the huge differences in resource endowment and socio-economic development levels among provinces and cities (Zhang et al., 2024). Nonetheless, with increasingly close interprovincial linkages, the degree of hierarchy has gradually declined. Network efficiency remains relatively stable, fluctuating around 0.67, indicating that redundant relationships among nodes have been reduced and the efficiency of interprovincial communication and cooperation is relatively high. Overall, the findings suggest that while the spatial spillover effects of agricultural water use efficiency across provinces in China are relatively strong, there remains considerable room for improving the density and stability of the spatial association network.

To reflect the position and role of individual nodes, it is necessary to measure the individual network characteristics of the spatial association network. This study presents the outdegree centrality, betweenness centrality, and closeness centrality of each province in China’s agricultural water use efficiency spatial association network in 2005 and 2023, as shown in the Table 4.

Outdegree centrality of each province reflects its efficiency spillover capacity within the network. Over the sample period, outdegree centrality of most provinces have increased, reflecting a growing spillover effect of agricultural water use efficiency. Among them, Guangdong, Shanghai, and Fujian, which have high levels of economic development and relatively convenient transportation, have consistently played a strong efficiency spillover role in the network. As these regions possess higher technological levels and economic strength, they should be encouraged to promote cooperation with their grain trade partner provinces to maximize their efficiency-driving effect. Meanwhile, the rankings of Qinghai, Neimenggu and Xizang increased significantly, indicating an enhanced efficiency spillover effect in the western provinces. With the support of policies such as the Western Development Strategy, the communication between the western provinces and other areas has become increasingly close. However, given the fragile ecological environment, their agricultural water use efficiency should be further improved by enhancing vegetation coverage, strengthening infrastructure development and promoting water-saving technologies. That will ensure the sustainability of their efficiency spillover effect.

Betweenness centrality reflects the capacity of provinces to act as a bridge controlling information flow within the network. The level of betweenness centrality also increased in 2023, indicating that more provinces are beginning to act as “intermediaries” and “bridges” within the network. In 2005, the provinces acting as “bridges” were mainly developed coastal provinces such as Beijing, Shanghai, and Guangdong. They controlled the transmission of information within the network by virtue of their relatively convenient transportation conditions and frequent economic exchanges with surrounding provinces. By 2023, provinces like Neimenggu, Jiangsu, and Fujian joined this group. Most of the provinces with high betweenness are economically developed and geographically important. And they play an important role in controlling the spillover and reception channels related to agricultural water use efficiency (Lai et al., 2025).

Closeness centrality measures the average degree of spatial proximity between a province and all other provinces in the network. The average closeness centrality of all provinces rose from 0.013 in 2005 to 0.015 in 2023. This indicates that the entire spatial association network for agricultural water use efficiency has become more efficient. A higher individual closeness centrality means that the province is more likely to receive efficiency spillovers from other regions. In 2023, 13 provinces had a closeness centrality above the average. Most of these regions are located in the central and western parts of the country. While their economic levels may be lower, their advantageous geographical location along key corridors connecting the coastal and inland areas gives them a potential advantage for improving water resource efficiency.

To characterize the role transformation of each province within the network and clarify their future strategies for improving agricultural water use efficiency, this study performs a classification of the provinces. Based on agricultural water use efficiency levels, outdegree centrality, betweenness centrality and closeness centrality, provinces can be categorized into four types: global-core, regional-core, potential and peripheral provinces. Figure 5 shows that, compared with 2005, the roles of most provinces had shifted in 2023. The number of regional core provinces has increased significantly compared to 2005, reflecting the further expansion of the network and the growing spillover effects brought about by improvements in agricultural water use efficiency.

The global core provinces, including Shanghai, Guangdong, Fujian, and Chongqing, should promote multi-channel cooperation with other regions to fully leverage their radiative and driving effects. Moreover, Provinces such as Jilin, Zhejiang, Henan, Guizhou, and Tianjin have transitioned from potential provinces to regional core status, while Qinghai, Shaanxi, Xinjiang, and Guangxi have moved from marginal status to regional core status. Some provinces, including Hebei, Yunnan, Guizhou, and Liaoning remain in a stage of potential growth. In the future, they need to strengthen exchanges and cooperation with neighboring provinces and fully utilize the network connectivity to enhance more efficient use of agricultural water resources.

Additionally, it is worth noting that although the total of peripheral provinces decreased in comparison with 2005, 66.67% of the peripheral provinces are major grain-producing areas in 2023. Such regions still face significant challenges in achieving green agricultural water resource production. This highlights the urgent need to strengthen production-marketing linkages and technological collaboration with more advanced regions, thereby promoting a comprehensive green transformation of agriculture in these grain-producing provinces.

The dynamics of communities in the spatial association network has undergone significant changes (Figure 6). In 2005, the network could be divided into three communities, while it evolved into four communities in 2023. Moreover, composition of these communities also changed.

Spatial association of agricultural water use efficiency among provinces was relatively unbalanced. The largest community within the network consisted of 18 provinces located in northern and western China, exhibiting extensive interregional and inter-basin linkages. The community’s internal edge density was 0.23, indicating relatively loose interprovincial connections. The second community includes 10 provinces in central and southern China, such as Anhui, Hubei, and Guangdong. Shanghai, Fujian, and Guangdong were the core provinces of this community. The provinces in this community are geographically close, have frequent economic exchanges, and exhibit relatively close ties within the community. In addition, Shandong, Henan, and Gansu formed a smaller community that was isolated within the overall network and had weaker connections with other regions.

The community structure in 2023 exhibits characteristics of multi-centralization and deep integration between the northern and southern regions. Nine provinces in central and western China, together with Shanghai, Fujian, and Hebei, formed the largest community in the network. The community’s density increased to 0.30, indicating that collaboration and connections among the central and western provinces were strengthened. Benefiting from the state’s investment in ecological protection and water resource management, as well as the development of high-value-added agriculture in the central and western areas, the efficiency spillover capacity of western provinces such as Inner Mongolia, Qinghai, and Tibet significantly increased. And they have attracted beneficiaries such as Shanghai and Fujian. Meanwhile, Heilongjiang, Beijing, and Jiangsu joined a community dominated by Anhui and southern provinces, forming the second-largest community in the network. Compared with 2005, the scope of influence of this community has expanded. Among them, Guangdong strengthened its position and absorbed provinces like Heilongjiang, Beijing, and Jiangsu. Guangdong as a major grain consumption area and economic center, has driven the efficiency improvement of the grain-producing regions in the North and Northeast through its demand. Shandong, Shanxi, Henan, and Zhejiang form a small community, with relatively close internal connections within the group. In addition, this community also has close connections with Community 2. This indicates that grain trade and inter-regional water rights trading strengthened the spatial association. Meanwhile, Liaoning and Jilin constitute the smallest community in the network, which suggests that these two provinces are relatively isolated within the network. Although its internal connections were strong, its external linkages with other regions were limited, indicating that this region remained relatively isolated within the overall network.

The evolutionary trend of community structures reveals that, as interprovincial linkages have become increasingly close, agricultural water use efficiency spillovers are gradually breaking through geographical boundaries to form new spatial diffusion patterns. In future regional water management policies, the coordination among provinces should be planned from a holistic network perspective. Furthermore, the central and western provinces are no longer on the network periphery but have become important sources of efficiency spillover through the improvement of water resource efficiency and cross-regional economic ties. This suggests a significant improvement in the regional fairness and synergy of China’s agricultural water resource management.