Towards Inclusive Growth: Perspective of Regional Spatial Correlation Network in China

: China’s economic growth has been impressive, but the persistent income inequality poses a threat to its sustainability. To address this issue, we use the complex network analysis method for the ﬁrst time to explore the structural characteristics of the regional spatial correlation network of inclusive growth (RSCN) of 26 provinces (autonomous regions and municipalities) in China from 1999 to 2020. We use exponential random graph models to explore the internal mechanisms and driving factors that shape this network. Our results show that inclusive growth dependencies between regions are widespread and increasing. Beijing, Shanghai, Jiangsu, and Zhejiang serve as benchmark regions, while provinces in the middle reach of the Yangtze River play an increasingly important bridging role. The northwestern region mainly acts as a receiving region. Our study identiﬁes transitivity, reciprocity, and high interaction tendency as critical microstructures. Furthermore, we ﬁnd that infrastructure, digital economy development, ﬁnancial marketization, ﬁscal expenditure linkages, and inter-provincial trade linkages are crucial factors in shaping network relationships. Our study provides theoretical support for the development of China’s regional coordinated development strategy and sustainable economic growth policies.


Introduction
China's efforts toward building a well-off society have been remarkable, yet the persistence of income inequality poses a significant threat to the sustainability of economic growth and macroeconomic stability.Despite China's rapid economic development and increasing international influence after its accession to the World Trade Organization (WTO), the country continues to face challenges of unequal development opportunities and regional disparities.Inclusive growth, a new strategic vision for achieving sustainable economic growth and addressing these issues, has gained increasing importance.The World Bank defines inclusive growth as economic growth compatible with sustainable development that reduces inequality of opportunity and improves social stability.International conferences, such as the 5th APEC Ministerial Conference on Human Resource Development and the European 2020 strategy, have highlighted the importance of achieving inclusive growth and addressing the social problems that arise from economic development.Inclusive growth has thus become a global concept that plays a crucial role in promoting sustainable economic growth.
In order to better understand the regional characteristics of China's inclusive growth and promote inclusive development, we go beyond traditional spatial econometric methods that only analyze neighboring relationships and use complex network analysis to explore the structural characteristics of China's regional spatial correlation network of inclusive growth (RSCN).We also employ exponential random graph models to examine the internal mechanisms and influencing factors of network formation.By doing so, we provide theoretical support for the implementation of China's regional coordinated development strategy and the formulation of sustainable economic growth policies.
Over the past few years, the importance of inclusive growth in achieving sustainable economic development has been increasingly recognized by scholars, leading to a surge of research interest in this area.Inclusive growth can be broadly categorized into three areas of inquiry: first, the definition of inclusive growth.The Asian Development Bank (ADB) defined inclusive growth as achieving fair opportunities and narrowing the income gap while developing the economy [1].Ali et al. [2] viewed inclusive growth as the growth of social opportunities and fairness and emphasized that inclusive growth entails poor people having easier access to benefits and opportunities.Meanwhile, Berg et al. [3] defined inclusive growth in terms of income growth and distribution.
Secondly, researchers have attempted to measure inclusive growth using a variety of measures.For instance, Ali et al. [4] used the social opportunity function to measure inclusive growth, and Silber and Son [5] proposed the Bonferroni index to measure the level of social inclusion based on this function.The Organization for Economic Cooperation and Development (OECD) [6] used household income, job opportunities, and health status as proxy variables for inclusive growth.Xu et al. [7] constructed a generalized Bonferroni index based on the Bonferroni index to measure inclusive growth, while Li et al. [8] measured inclusive growth using the Lorenz curve and Gini coefficient curve.
Thirdly, empirical research on inclusive growth has focused on various aspects, including the impact of inclusive growth on regional balanced development and economic development (Lin [9]; Chakrabarty [10]; Farhana et al. [11]; Parolin et al. [12]).Inclusive growth is necessary for sustainable economic growth (Cichowicz and Sadowska [13]; Corrado and Corrado [14]; Gupta and Vegelin [15]).Tan and Lv [16] showed through the panel data model that China's inability to achieve inclusive growth in various regions will hinder economic development.Rytova et al. [17] evaluated Russia's regional inclusive growth and found that there is a significant correlation between various dimensions of sustainable growth and that inclusive growth should not be ignored in the process of regional sustainable development.Surya et al. [18] proposed to promote balanced economic development through technological innovation and community enterprise productivity.Jagódka and Snarska [19] found that failure to achieve inclusive growth is an obstacle to sustainable development and that finding ways to promote inclusive growth is crucial.Rauniyar et al. [20] argued that increasing support for infrastructure and human capital can promote inclusive growth.Surya et al. [21] studied the spatial dynamics of communities and sustainable economic growth.Corrado et al. [14] analyzed the role of inclusive finance in promoting inclusive growth, while Kapoor et al. [22], Sugiawan [23], and Zou et al. [24] studied the mechanisms by which digital finance and financial development promote inclusive growth.Masduki et al. [25] analyzed the impact of quality government spending on inclusive growth, while Gupta et al. [26] studied strategies for inclusive development during COVID-19.
However, these studies on inclusive growth relatively ignored the degree of correlation at the regional level.This is especially important in China, where there are large regional differences (Wu et al. [27]).Sun et al. [28] have employed spatial lag models and spatial error models to study the spatial correlation characteristics of economic growth, and Li et al. [29] stated that maintaining an income gap within a reasonable range is an inevitable requirement for China to achieve common prosperity.Therefore, the challenge China faces is how to narrow the income gap between urban and rural areas; Lu et al. [30] explored the impact of China's transportation infrastructure on the income gap between urban and rural areas; Yuan et al. [31] investigated the inclusive growth effect of urban and rural financial development, and found that improving the scale and efficiency of rural financial development will improve inclusive growth; and Cui et al. [32,32] used the spatial Dubin model to find that both financial inclusion and renewable energy consumption have a positive contribution to inclusive growth.These studies were limited to the spatial correlation of adjacent regions, and the conclusions that were drawn for increasingly complex economic systems were likely to have been biased.The purpose of this paper is to construct the correlation of inclusive growth at the regional level through complex network analysis methods and find ways to promote inclusive growth more effectively according to local conditions to achieve high-quality economic growth and sustainable development.
The use of the complex network method is to describe the relationship of various parts of the real system in the form of a network, which can better understand the nature of the real system (Zhang et al. [33]).Driven by the spatial spillover effect, the economic system gradually presents an intricate network structure, and the complex network method has a strong adaptability to the current economic system (Li [34]; Yang [35]; Yu et al. [36]; Zhang et al. [33]).Its application in economics is mostly in the form of constructing a regional network to study the relationship between the development of various regions.The regional spatial association network constructed by the complex network overcomes the limitation suffered by the traditional spatial measurement method, which can only analyze adjacent areas, and thus obtains a more comprehensive and complex spatial association relationship.At the same time, the variables used to study regional correlations are themselves relational data, and it is difficult to avoid multicollinearity problems between observations.Using the traditional least squares method for regression may lead to biased parameter estimates, while models based on complex networks, such as Quadratic Allocation Procedures (QAP) and Exponential Random Graph Models (ERGM), can incorporate relational data into research (Lv and Chen [37]; Jiang [38]; Wang [39]; Zha [40]).
Li et al. [41] have constructed a spatial correlation network of regional economic growth in China and analyzed its structural characteristics and influencing factors.Xu et al. [42] constructed a regional association network through the method of a complex network to study the method of regional coordinated and sustainable development.Tian and Wang [43] conducted research on regional innovation and sustainable development through the complex network method.Fan and Xiao [44] constructed a regional spatial network to study China's green innovation, providing a new network perspective for the formulation of China's green innovation policy.Huang [45] studied how the endogenous evolution of a production network maintains economic growth by studying the network structure.Yu et al. [36] studied the factors affecting China's economic growth through a regional spatial network.Wu and Hui [46] found close economic linkages among provinces in eastern China through a complex network approach.Liu and Du [47] showed that a reasonable and orderly spatial network structure is an important support for the coordinated and sustainable development of regional economies.
To summarize, some of the existing research results explore the measurement methods and growth mechanisms of inclusive growth, and the other part shows that the spatial correlation and spatial agglomeration characteristics between regions are becoming more and more significant and that the spatial spillover effect cannot be ignored in formulating economic development strategies.There are still limitations in the research on inclusive growth.No scholars have used complex network analysis methods to study the spatial correlation network structure and influencing factors of China's inclusive growth.Most of the existing research results use traditional spatial econometric methods, which only explain a small amount of correlation between inclusive growth, and most of the research on the influencing factors of inclusive growth was based on the macroeconomic data of various provinces.Based on this, we measured China's inclusive growth from 1999 to 2020, and used complex network analysis methods to incorporate the relationship between non-adjacent regions into the study to reveal the nature of its complex network.Moreover, for the first time, we explored the formation mechanism of China's inclusive growth spatial correlation through spatial network data and ERGM.
The structure of this paper is as follows: Section 2 introduces the methodology of this paper.Section 3 introduces the structural characteristics of the spatial correlation of inclusive growth.Section 4 provides the analyzes of influencing factors of spatial correlation of China's inclusive growth.Section 5 presents the conclusions and insights.

Measuring Inclusive Growth
We use the generalized Bonferroni curve to measure inclusive growth.The most significant advantage of the generalized Bonferroni curve is that it is more sensitive to changes in the welfare of low-income groups, which not only reflects the idea of inclusiveness, but also fits more closely with the current situation of China's national income distribution.The specific measurement method is as follows.
The Bonferroni curve is defined as the ratio of the partial mean of the cumulative population in the income distribution to the overall mean under the corresponding cumulative population proportion, that is, where p is the percentile, µ p is the mean income of the accumulated population at the p percentile of the income distribution, and µ is the overall mean income.Based on the Bonferroni curve, the Bonferroni index is defined as B, as follows: Considering the limited value range of the Bonferroni curve and the possibility of cross-sections, the method of constructing the generalized Lorenz curve is used for reference, and the average income is introduced into the Bonferroni curve to obtain the generalized Bonferroni curve.The corresponding generalized Bonferroni index is defined as B G , as follows: Based on the generalized Bonferroni curve, the amount of inclusive growth is defined as the income level that everyone in the society can obtain, and finally: where y p represents the cumulative average income of the cumulative population at the pth percentile of the income distribution.It can be seen that the inclusive growth y * is the graphic area enclosed by the generalized Bonferroni curve, the straight line p = 1 and the coordinate axes.

Construction of RSCN
We use the modified gravity model to construct the inclusive growth network for two reasons: firstly, using the gravity model to construct an RSCN can not only comprehensively consider economic and geographical factors but also better reveal the dynamic evolution trend of the network; secondly, the VAR model is too sensitive to the choice of the lag order, and the VAR model is not accurate enough to describe the characteristics of the network structure.The revised gravity model is shown in Equation (5).
Within Equation ( 5), i and j both represent provinces (districts, cities); Y ij is the gravitational force between the inclusive growth of province i and province j; B i and B j are the inclusive growth of province i and province j; k ij = B i /(B i + B j ), P i and P j are the end-of-year permanent population of cities and towns in province i and province j; G i and G j are the actual gross regional products of province i and province j; and k ij represents the contribution rate of province i in the inclusive growth linkage between province i and province j.
The gravitational matrix of inter-regional inclusive growth is obtained by using Equation (5).The average value of each column of the matrix is taken as the threshold, and the gravity higher than the threshold is recorded as 1, indicating that the province in this row has a spatial spillover effect on the province in the column; on the contrary, it is recorded as 0, indicating that the row of provinces has no spatial spillover effect on the inclusive growth of the row of provinces.A directed and unweighted network is constructed from the asymmetric binary matrix.

Analysis Method of Structural Characteristics of RSCN
We use two methods of network hierarchy analysis and micro-pattern analysis to study the structural characteristics of the RSCN, in which the method of hierarchy analysis involves the global level, intermediate level, and individual level.

Hierarchy Analysis of RSCN
In this section, the following methods are used to analyze the global level, intermediate level, and individual level of the RSCN.
(1) Global level.Network density, average geodesic distance, spectral radius, and network clustering coefficient.The average geodesic distance is the mean value of the geodesic distances of each node; the smaller the value, the better the cohesion of the network.The spectral radius is the largest non-zero eigenvalue of the network adjacency matrix.The larger the spectral radius, the higher the overall connection degree of the network.The calculation formula for the network clustering coefficient is: where m is the number of relationships existing between nodes connected to node i, and d i represents the point degree of province i.
(2) Intermediate level.First, the Dyad census.Dyad refers to a subgroup that contains only two nodes in the RSCN, and it has three states: mutual relationship, asymmetric relationship, and no association.From the Dyad census, we can obtain the correlation mode of inclusive growth among provinces in the network.Second, the network block model.The output of the network block model can be used to classify nodes.The model uses the integration classification likelihood criterion to determine the most appropriate classification of the network and uses the maximum a posteriori criterion to determine which category each node belongs to.We find it most reasonable to divide the nodes of the RSCN into five categories.Through the analysis of the middle level of the RSCN, we can understand the linkage mode among Chinese provinces and obtain the categories to which each province belongs and the function each category plays.
(3) Individual level.First, out-degree and in-degree represent the overflow relationship and receiving relationship of each province (district, city) in the network.Second, the eigenvector centrality (EC) can measure the importance of each province (district, city) in the network.Through the analysis of the individual level of the RSCN, the status and role of each province can be clarified.

Micro-Pattern Analysis of the RSCN
Motifs show the basic connection patterns between nodes in the network and can link the microstructure of the network with its overall characteristics [48].They are called "primitives" which constitute the network.We use the Rand-ESU algorithm to detect motifs to explore the micro-interaction patterns in the RSCN.In the process of subgraph mining, this algorithm mines subgraphs by continuously expanding adjacent points at random, which greatly reduces the time needed for detecting motifs and can detect more types of motifs [49].

Analysis Method of Influencing Factors of RSCN
The above only conducts descriptive statistical analysis on the RSCN and does not analyze the internal mechanism of the formation of the correlation relationship.However, traditional spatial econometric methods need to assume that the research objects are not related or independent and that the linkage of inclusive growth in various regions is becoming closer and closer, which obviously cannot satisfy the independence assumption.Therefore, we will further construct a probability model centered on the network structure, incorporate the spatial correlation of inclusive growth into the empirical analysis, and explore the influencing factors and mechanisms of the spatial correlation of inclusive growth.

Analysis of the Structural Characteristics of the Spatial Correlation of Inclusive Growth in China
We collected and sorted out the grouped per capita disposable income data of 26 provinces (districts, cities) in China from 1999 to 2020, and calculated the inclusive growth.The data comes from the grouped per capita disposable income in the statistical yearbooks of each province.In view of the missing years in some provinces, we do the following processing and use the China Family Panel Studies (CFPS) data of Peking University to fill in the provinces with fewer missing years.If a province has many missing years or its sample size in CFPS is small, we delete the province.Finally, 26 provinces (districts, cities) were selected from 2000 to 2020.In 1999, Gansu was missing, and there were 25 provinces.Since the number of relationships generated in Gansu is not large, comparisons can still be made.In order to eliminate the impact of price changes, GDP, per capita GDP, and per capita disposable income were deflated with 1999 as the base period.

Construction of China's RSCN
The RSCN diagram of inclusive growth in Figure 1 is a directed network.From Figure 1, it can be clearly seen that the network has become increasingly complex and dense over time.The number of network edges has increased from 86 in 1999 to 165 in 2020, and the relationship and central cities are increasing yearly.It can be seen from Figure 1 that the formation of spatial correlations is not limited to geographical proximity.(1) Global level analysis.The spatial correlation of inclusive growth in China has formed a huge network covering all provinces (districts, cities) in China.It can be seen from Table 1 that as time progresses, the network density of the RSCN rises.The spectral radius has nearly doubled from 1999 to 2020.This shows that in the past 21 years, the coordination and linkage relationship of inclusive growth among regions has continued

Hierarchy Analysis of the RSCN in China
(1) Global level analysis.The spatial correlation of inclusive growth in China has formed a huge network covering all provinces (districts, cities) in China.It can be seen from Table 1 that as time progresses, the network density of the RSCN rises.The spectral radius has nearly doubled from 1999 to 2020.This shows that in the past 21 years, the coordination and linkage relationship of inclusive growth among regions has continued to deepen.While the network clustering coefficient is gradually increasing over time, the average geodesic distance is continuously decreasing, indicating that the RSCN gradually shows the characteristics of a small-world network, and the network cohesion continues to increase.From the global level indicators of the network, it can be found that the mutual influence of inclusive growth among provinces in the RSCN is deepening day by day, showing the spatial network structure of the interaction between coastal areas and inland areas, and the joint development of eastern, central, and western regions.
(2) Intermediate level analysis.It can be found from Table 2 that the mutual relationship (Mut) in the network increases with the number of network relationships.However, the asymmetric relationship (Asym) has always been the main relationship in the network, indicating that two-way spillovers are increasingly common in the network.However, one-way spillovers, where the other party only receives the impact, are more prominent.On the one hand, this reflects that the development of China's inclusive growth is not coordinated enough.On the other hand, it reflects the leading role of China's economically stronger provinces in leading and supporting the weaker provinces.Next, the network block model is used to divide the nodes (that is, provinces) in the network and to study the changes in the functions in each category and the members in each category.Finally, the nodes are divided into five categories.The circles in Figure 2 correspond to five categories, and the size of the circle represents the number of members of each category.The thickness of the arrow represents the in-degree or out-degree in each category.From the size of the nodes in Table 3 and Figure 2, it can be found that the members of each category are constantly moving to the previous category.
network and to study the changes in the functions in each category and the members in each category.Finally, the nodes are divided into five categories.The circles in Figure 2 correspond to five categories, and the size of the circle represents the number of members of each category.The thickness of the arrow represents the in-degree or out-degree in each category.From the size of the nodes in Table 3 and Figure 2, it can be found that the members of each category are constantly moving to the previous category.The first type of members has the largest total out-degree (that is, the overflow relationship), and the total in-degree (that is, the receiving relationship) is also larger.The members of this category are the benchmark areas for inclusive growth.The number of members of the first category is increasing.In the early days, only Beijing and Shanghai could cover the whole country.Now, it has grown to four regions.The members of the second type are the central provinces of each region.It can be found that the number of  The first type of members has the largest total out-degree (that is, the overflow relationship), and the total in-degree (that is, the receiving relationship) is also larger.The members of this category are the benchmark areas for inclusive growth.The number of members of the first category is increasing.In the early days, only Beijing and Shanghai could cover the whole country.Now, it has grown to four regions.The members of the second type are the central provinces of each region.It can be found that the number of members of the second type has also increased from two to four.Although their scope of influence is not as wide as that of the first type, they also lead local inclusive growth.The members of the third category are typical bridge regions, mainly the provinces in the middle reaches of the Yangtze River, which transmit the spatial correlation of inclusive growth.The number of members in this category increased from one to six.
Members of the fourth category are mainly in the northern, central, and western provinces, with low out-degree and high in-degree, playing the role of beneficiaries.The number of members in this category kept rising at first, and some members originally in the fifth category stepped into the fourth category.It reached its peak in 2015 and dropped to five in 2020.Some members entered the third category from the fourth category.This shows that these provinces have continued to improve after accepting the inclusive growth space spillover effects of other provinces and that their economic strength has been qualitatively improved.Members of the fifth category are mainly border areas, and their out-degree and in-degree are very low.These areas are at the edge of the RSCN.As time goes on, the number of members in the fifth category continuously declines (see Table 4).
(3) Individual level analysis.Table 5 lists the node degree of each province (district, city) in China's RSCN.Furthermore, in order to make the analysis clearer, we divide these provinces into eight economic zones, corresponding to the Beijing-Tianjin-Hebei economic zone, the Yangtze River Delta economic zone, the Southern Coastal economic zone, the Northeast economic zone, the middle reaches of the Yellow River economic zone, the middle reaches of the Yangtze River economic zone, the Southwest economic zone, and the Northwest economic zone.First, it can be found that there are differences in the spatial correlation of inclusive growth in various economic regions.In the eastern region, the out-degree (Out) is relatively large, and the in-degree (In) is small.The linkage between the western region and other regions is less.The out-degree and in-degree in the central region are the most balanced.Furthermore, the development within each economic zone is uneven.In the Beijing-Tianjin-Hebei economic zone, there is a large gap between Hebei and Beijing in terms of node degree.In addition to the gap in their own economic strength, this may be because the linkage relationship between Hebei and Beijing is much greater than that of other provinces.The most coordinated development is the Yangtze River Delta region.The nodes of Shanghai, Zhejiang, and Jiangsu rank among the top in the country, and their influence also covers the whole country.Among the southern coastal economic regions, Guangdong's node degree has always been high, and Fujian's node degree has increased rapidly since 2010, gradually becoming the center of the network.Hainan mainly plays the role of receiver, and its EC is constantly improving.With the development of the Hainan Free Trade Zone, Hainan's economic status in China is constantly improving.
Both the Northeast and the middle reaches of the Yellow River economic zone have a higher in-degree and a lower out-degree.Due to their lack of economic strength, they mainly play the role of recipients.The economic zone in the middle reaches of the Yangtze River is located between several major economic center provinces.Their in-degree and out-degree are at an intermediate level and are equivalent.Their EC is around 0.5, which acts as a bridge in the network.Due to the relatively remote geographical location and weak economic strength of the Northwest economic zone, the out-degree of Tibet, Gansu, and Xinjiang is very low.However, their in-degree and EC are increasing year by year, and more and more regions are helping them.

Micro-Pattern Analysis of China's RSCN
In order to explore the micro-association patterns of inclusive growth among different provinces, we identify the motifs in the network.Table 6 shows the results of the motif analysis.A total of 12 motifs were found in the network in 2020 and 11 in the rest of the years.There are five notable motifs in 2020 and three in other years.According to the analysis results, the motifs with codes 14, 36, and 164 have always been significant microstructure configurations for the past 21 years.Motif 36 is in a twostar structure, which indicates that the unbalanced in-degree distribution in the network; that is to say, the phenomenon that one area in the network receives spillover effects from multiple areas at the same time is significant.Motifs 14 and 164 have transitive and reciprocal relations between nodes, indicating that transitivity and reciprocity are important features of the RSCN.The measurement results show that the in-degree distribution imbalance, transitivity, and reciprocity of the relationship have a significant impact on the formation of the overall network.According to the analysis results, the motifs with codes 14, 36, and 164 have always been significant microstructure configurations for the past 21 years.Motif 36 is in a twostar structure, which indicates that the unbalanced in-degree distribution in the network; that is to say, the phenomenon that one area in the network receives spillover effects from multiple areas at the same time is significant.Motifs 14 and 164 have transitive and reciprocal relations between nodes, indicating that transitivity and reciprocity are important features of the RSCN.The measurement results show that the in-degree distribution imbalance, transitivity, and reciprocity of the relationship have a significant impact on the formation of the overall network.According to the analysis results, the motifs with codes 14, 36, and 164 have always been significant microstructure configurations for the past 21 years.Motif 36 is in a twostar structure, which indicates that the unbalanced in-degree distribution in the network; that is to say, the phenomenon that one area in the network receives spillover effects from multiple areas at the same time is significant.Motifs 14 and 164 have transitive and reciprocal relations between nodes, indicating that transitivity and reciprocity are important features of the RSCN.The measurement results show that the in-degree distribution imbalance, transitivity, and reciprocity of the relationship have a significant  According to the analysis results, the motifs with codes 14, 36, and 164 have always been significant microstructure configurations for the past 21 years.Motif 36 is in a twostar structure, which indicates that the unbalanced in-degree distribution in the network; that is to say, the phenomenon that one area in the network receives spillover effects from multiple areas at the same time is significant.Motifs 14 and 164 have transitive and reciprocal relations between nodes, indicating that transitivity and reciprocity are important features of the RSCN.The measurement results show that the in-degree distribution imbalance, transitivity, and reciprocity of the relationship have a significant  According to the analysis results, the motifs with codes 14, 36, and 164 have always been significant microstructure configurations for the past 21 years.Motif 36 is in a twostar structure, which indicates that the unbalanced in-degree distribution in the network; that is to say, the phenomenon that one area in the network receives spillover effects from multiple areas at the same time is significant.Motifs 14 and 164 have transitive and reciprocal relations between nodes, indicating that transitivity and reciprocity are important features of the RSCN.The measurement results show that the in-degree distribution imbalance, transitivity, and reciprocity of the relationship have a significant  According to the analysis results, the motifs with codes 14, 36, and 164 have always been significant microstructure configurations for the past 21 years.Motif 36 is in a twostar structure, which indicates that the unbalanced in-degree distribution in the network; that is to say, the phenomenon that one area in the network receives spillover effects from multiple areas at the same time is significant.Motifs 14 and 164 have transitive and reciprocal relations between nodes, indicating that transitivity and reciprocity are important features of the RSCN.The measurement results show that the in-degree distribution imbalance, transitivity, and reciprocity of the relationship have a significant  According to the analysis results, the motifs with codes 14, 36, and 164 have always been significant microstructure configurations for the past 21 years.Motif 36 is in a twostar structure, which indicates that the unbalanced in-degree distribution in the network; that is to say, the phenomenon that one area in the network receives spillover effects from multiple areas at the same time is significant.Motifs 14 and 164 have transitive and reciprocal relations between nodes, indicating that transitivity and reciprocity are important features of the RSCN.The measurement results show that the in-degree  According to the analysis results, the motifs with codes 14, 36, and 164 have always been significant microstructure configurations for the past 21 years.Motif 36 is in a twostar structure, which indicates that the unbalanced in-degree distribution in the network; that is to say, the phenomenon that one area in the network receives spillover effects from multiple areas at the same time is significant.Motifs 14 and 164 have transitive and reciprocal relations between nodes, indicating that transitivity and reciprocity are important features of the RSCN.The measurement results show that the in-degree  According to the analysis results, the motifs with codes 14, 36, and 164 have always been significant microstructure configurations for the past 21 years.Motif 36 is in a twostar structure, which indicates that the unbalanced in-degree distribution in the network; that is to say, the phenomenon that one area in the network receives spillover effects from multiple areas at the same time is significant.Motifs 14 and 164 have transitive and reciprocal relations between nodes, indicating that transitivity and reciprocity are important features of the RSCN.The measurement results show that the in-degree  According to the analysis results, the motifs with codes 14, 36, and 164 have always been significant microstructure configurations for the past 21 years.Motif 36 is in a twostar structure, which indicates that the unbalanced in-degree distribution in the network; that is to say, the phenomenon that one area in the network receives spillover effects from multiple areas at the same time is significant.Motifs 14 and 164 have transitive and reciprocal relations between nodes, indicating that transitivity and reciprocity are important features of the RSCN.The measurement results show that the in-degree  According to the analysis results, the motifs with codes 14, 36, and 164 have always been significant microstructure configurations for the past 21 years.Motif 36 is in a twostar structure, which indicates that the unbalanced in-degree distribution in the network; that is to say, the phenomenon that one area in the network receives spillover effects from multiple areas at the same time is significant.Motifs 14 and 164 have transitive and reciprocal relations between nodes, indicating that transitivity and reciprocity are  According to the analysis results, the motifs with codes 14, 36, and 164 have always been significant microstructure configurations for the past 21 years.Motif 36 is in a twostar structure, which indicates that the unbalanced in-degree distribution in the network; that is to say, the phenomenon that one area in the network receives spillover effects from multiple areas at the same time is significant.Motifs 14 and 164 have transitive and reciprocal relations between nodes, indicating that transitivity and reciprocity are According to the analysis results, the motifs with codes 14, 36, and 164 have always been significant microstructure configurations for the past 21 years.Motif 36 is in a two-star structure, which indicates that the unbalanced in-degree distribution in the network; that is to say, the phenomenon that one area in the network receives spillover effects from multiple areas at the same time is significant.Motifs 14 and 164 have transitive and reciprocal relations between nodes, indicating that transitivity and reciprocity are important features of the RSCN.The measurement results show that the in-degree distribution imbalance, transitivity, and reciprocity of the relationship have a significant impact on the formation of the overall network.
In 2020, there are the most notable microstructures, with a total of five.Motif 238 represents a high degree of interaction tendency, and it has gradually become a significant microstructure in the network, which represents that the hierarchical degree of China's inclusive growth is becoming lower and lower.Motif 12 is a typical one-way relational transfer structure.To summarize, it can be seen that transitivity, a highly interactive tendency, and reciprocity have gradually become important characteristics of China's RSCN, which also shows that the relationship between regions is becoming more balanced and common.

Analysis of Influencing Factors of Spatial Correlation of China's Inclusive Growth
Above, we studied the structural characteristics of the RSCN.Next, we use ERGM to analyze the impact of different influencing factors on the spatial correlation structure of China's inclusive growth.

Network Endogenous Structural Variables
We incorporate the number of edges and reciprocity into the network.The number of edges is closely related to network density, and reciprocity examines the tendency of a symmetric relationship formation in a network.Incorporating triangular structures into the model may lead to problems, such as approximate degeneracy or non-convergence of estimation results when the network is highly clustered.In the process of empirical research, it is found that the inclusion of transitivity and alternating k-triangle into the model will indeed lead to unsatisfactory estimation results, so the final transitivity and interactive k-triangle are not included in the model.
In order to account for the complex dependencies that exist in the network, geometric weighting terms are incorporated into the model.We found that only the model with GWDSP has the best fitting effect.The decay parameters of the geometric weighting items were selected a priori and a model estimation method, respectively, and we finally found that the decay parameters obtained by the model estimation method had the best fitting effect, so we adopted the Curved Exponential Family Models (CEF) determine its parameter settings.

Node Covariates
First, we examine the role played by the level of inclusive growth in various regions in the formation of the overall network.We test whether the homogeneity of inclusive growth plays a significant role in the formation of the overall network.This statistical item is denoted as Inclu.In addition, we divide the inclusive growth of each region into the three categories of high, middle, and low, which are recorded as IncluHigh, IncluMid, and IncluLow, to test whether regions with higher levels of inclusive growth are more likely to have spatial correlations with inclusive growth in other regions.Finally, it Is found that the homogeneity tendency of inclusive growth is more significant, and the fitting effect is better, so the statistical items of homogeneity of inclusive growth are included in the model.
In terms of node covariates, the individual-level analysis of local inclusive growth in China shows that the degree of nodes has a greater correlation with the level of economic development in provinces.Thus, the level of economic development may have an important impact on the formation of networks.The level of infrastructure can reflect the level of economic development.We use the number of highway mileage per unit of land area to measure the level of infrastructure and divide them into the three categories of high, middle, and low, and incorporate them into the model.Neoclassical growth theory and endogenous growth theory show that technology is an important factor for output improvement and economic development.Inclusive growth is closely related to economic growth, so technological progress (measured by total factor productivity) is included in the model and divided into three categories: high, middle, and low.
In addition, the Internet revolution has had a huge impact on China, and China's digital economy has developed rapidly.The Peking University Digital Financial Inclusion Index of China (PUD) measures the development of the digital economy and incorporates the development of the digital economy into the model to study the impact of the digital economy on the coordinated development of inclusive growth.Financial marketization can increase budget constraints and strengthen the level of supervision, thereby affecting productivity and the level of economic and social inclusiveness.The financial marketization level is measured by the regional financial marketization index, and the financial marketization level is included in the model.

Network Covariates
To test the impact of other spatial correlations on the spatial correlation of inclusive growth, we introduce the spatial correlation of government fiscal expenditure and interprovincial trade as network covariates into the ERGM.Among them, the spatial correlation of government fiscal expenditure constructs a correlation network through VAR Granger causality.In order to obtain the structural characteristics of the fiscal correlation network more deeply, the TOP2 network is constructed by retaining the top two regions with spillover effects.The inter-provincial trade matrix uses the inter-provincial railway freight volume data of each year to construct a matrix and retains the data exceeding the column average to construct a correlation network.

The Impact of Endogenous Structural Effects on China's RSCN
Firstly, the number of edges (Edges) and mutuality (Mutual) are included in the model.From the network endogenous structure effect in Table 7, it can be found that both are significant at the 1% level, and the coefficient of mutuality is 1.0023.The above results show that the spatial correlation network of inclusive growth shows more reciprocal relationships, and the levels of inclusive growth among many regions are interdependent, which is consistent with the results of the motif analysis.

The Influence of Individual Attribute Effect and Exogenous Network Effect on RSCN
Incorporate the homogeneity tendency of inclusive growth (Inclu), the link-in relationship of infrastructure level (Infra), the link-in relationship of technological progress (Tech), the development of digital economy (Finan), and the level of financial marketization (Finan) into the model.According to the individual attribute effects in Table 7, the homogeneity tendency (Inclu) of inclusive growth is significantly negative, indicating that similar inclusive growth levels between regions will reduce the formation probability of spatial correlation.Both the low-level (Infra-Mid) and high-level (Infra-High) link-in coefficients of infrastructure are significantly negative at the 1% level, and the probability of receiving spatial spillover effects in low-level infrastructure areas is seven times that of middle-level areas and 7.9 times that of areas with high levels of infrastructure.The probability of link-in effects in areas with middle-level technological development is 32% of that in areas with low-level technological development, and areas with high-level technological development have not passed the significance test.The development of the digital economy is significant at the 1% level.Moreover, for every unit increase in the development of the digital economy, the probability of association formation increases by 1.32 times.It shows that the development of the digital economy has played a role in promoting the coordinated and linked development of China's inclusive growth.The coefficient of financial marketization is also significantly positive.Next, we will further study the influence of individual attributes and exogenous network on inclusive growth network.Since the model is difficult to converge when the endogenous pure network structural effect is controlled, the exogenous individual attribute effect and exogenous network effect in the inclusive growth space network cannot be well displayed.Therefore, in the following model, only Edges, an endogenous structural variable, is controlled, and node covariates and network covariates are added for analysis.
Furthermore, consider the impact of exogenous network effects on RSCN.We introduce the spatial correlation network of government fiscal expenditure (FicalNet) and interprovincial trade correlation network (TradeNet) as network covariates into the ERGM.From the exogenous network effects in Table 7, it can be seen that the spatial correlation of fiscal expenditure and the spatial correlation of inter-provincial trade are significant at the significance level of 5% and 1%, respectively, and the estimated coefficients are both positive.It shows that when there is a spatial correlation of fiscal expenditure between two provinces, the probability of the formation of the spatial correlation of inclusive growth will increase by 2.3 times.Additionally, when there is an inter-provincial trade correlation between provinces, the probability of the formation of the spatial correlation of inclusive growth will increase by three times.

Analysis of Influencing Factors Based on CEF
Finally, in order to avoid high statistical inference errors in the results, the ERGM is extended, and the geometric weighting item GWDSP is incorporated into the ERGM and uses the CEF.
The analysis results of CEF in Table 7 show that the estimated coefficient of GWDSP is significantly negative, indicating that if two regions have more common partners, the formation probability of inclusive growth spatial correlation will be reduced.Moreover, the introduction of GWDSP significantly improves the overall fitting degree of the model.The curve index family model is the model that fits the real network best among the four models.Therefore, we focus on using CEF to investigate the formation mechanism of China's inclusive growth spatial correlation network.
In the CEF, the homogeneity tendency (Inclu) of inclusive growth is significantly negative.From the odds ratio in Table 8, it can be seen that the probability of correlation between regions with the same level of inclusive growth is 21% of the probability of regions with different levels, indicating that a similar level of inclusive growth among regions will reduce the formation probability of spatial correlation and that the spatial correlation of inclusive growth mainly occurs between regions with different levels of inclusive growth.The link-in relationship coefficients of Infra-Mid and Infra-High are significantly negative at the 1% level.Furthermore, it can be seen from Table 8 that the probability of receiving spatial spillover effects in low-level infrastructure areas is 3.34 times that of middle-level areas and 3.53 times that of high-level infrastructure areas.This shows the importance of improving the basic public service system and improving the social governance system of joint construction, joint governance, and shared benefits.The probability of link-in effects in areas with a middle level of technological development (Tech-Mid) is 40% of the probability of areas with a low level of technological development and areas with a high level of technological development (Tech-High) have not passed the significance test.The model results show that, for categorical variables, the spatial correlation of inclusive growth is more formed in the form of high-level spillover to low-level.Regions with lower levels will receive more assistance, which reflects the pattern of regional cooperation and mutual assistance in China and promotes the common development of developed and underdeveloped regions.
The development of the digital economy is significant at the 1% level, and for every unit increase in the development level of the digital economy, the probability of association formation increases by 1.28 times.The coefficient of the level of financial marketization is also significantly positive, and it can be seen from Table 8 that every time the level of financial marketization increases by one unit, the probability of spatial correlation of inclusive growth will increase by 1.15 times.
The spatial correlation of fiscal expenditure and inter-provincial trade is significant at 5% and 1% significance levels, respectively, and the estimated coefficients are both positive.
From the odds ratio, it can be seen that if there is a correlation in fiscal expenditure between two provinces, the probability of forming a spatial correlation of inclusive growth will increase by 2.34 times.However, if there is a trade relationship between two provinces, the probability of forming an inclusive growth spatial relationship between the two provinces will increase by 3.07 times.
Next, in order to examine the robustness of each influencing factor, we analyze other network usage CEF from 1999 to 2020.Since the PUD does not have years prior to 2011, we use the trend extrapolation methods to obtain the development of the digital economy in 2010.In 2005 and 1999, the variable of digital economy development was not included.The results are shown in Table 9.It can be seen from Table 9 that the coefficient of inclusive growth homogeneity is stable and negative, indicating that the spatial correlation of inclusive growth is mainly formed among regions with different growth levels.The level of infrastructure is also significantly negative, indicating that the effect of this variable on the formation of inclusive growth networks is stable.Significant changes have taken place in the significance and sign of the technology level coefficient, which needs to be further tested in follow-up research.The development of the digital economy, the level of financial marketization, fiscal linkages, and trade linkages all have positive and stable effects on the formation of fiscal linkages, and trade linkages have positive and stable effects on the formation of RSCN.

Goodness-of-Fit of the CEF
The fitting results of ERGM can be obtained not only through AIC and BIC, but also through model simulation.A large number of random graphs are simulated with the fitted model, and the differences in network characteristics between these simulated networks and the real observed network are compared and visualized with boxplots (see Figure 3 for details).

Conclusions and Insights
Taking 26 provinces (districts, cities) in China as nodes, we used the complex network analysis method to investigate the spatial correlation of inclusive growth in China from 1999 to 2020 for the first time and analyzed the influencing factors of the formation of the RSCN.We find that: (1) The coverage of China's RSCN is becoming wider and wider, and it shows the characteristics of a small world.(2) The members of each category in the block model are constantly moving to the previous category.As of 2020, the distribution of the number of members in each category is balanced.Beijing, Shanghai, Jiangsu, and Zhejiang play a benchmark role in the network, the central region mainly plays the role of a bridge, and the northern, central, and western regions mainly play the role of beneficiaries.(3) Significant small-scale connected subgraphs in China's RSCN continue to increase.
At the same time, the highly interactive tendency and transitivity play an

Conclusions and Insights
Taking 26 provinces (districts, cities) in China as nodes, we used the complex network analysis method to investigate the spatial correlation of inclusive growth in China from 1999 to 2020 for the first time and analyzed the influencing factors of the formation of the RSCN.We find that: (1) The coverage of China's RSCN is becoming wider and wider, and it shows the characteristics of a small world.(2) The members of each category in the block model are constantly moving to the previous category.As of 2020, the distribution of the number of members in each category is balanced.Beijing, Shanghai, Jiangsu, and Zhejiang play a benchmark role in the network, the central region mainly plays the role of a bridge, and the northern, central, and western regions mainly play the role of beneficiaries.(3) Significant small-scale connected subgraphs in China's RSCN continue to increase.At the same time, the highly interactive tendency and transitivity play an increasingly significant role in the network, indicating that the correlation of inclusive growth between regions is becoming more balanced and common.(4) The formation of inclusive growth spatial correlation by network endogenous structural variables, the homogeneous tendency of inclusive growth, infrastructure construction level, technological progress, digital economy development, financial marketization, fiscal expenditure spatial correlation, and inter-provincial trade correlation had a noticeable impact.
This study provides insights into the spatial correlation characteristics of inclusive growth in China and highlights the importance of considering regional differences and spillover effects when formulating policies to promote inclusive growth.The findings suggest that policies should be tailored to the unique characteristics of each region and should take into account their roles in promoting spatial spillover effects.Specifically, policies should focus on enhancing spatial conduction in regions that act as bridges, creating a better -receiving platform for spatial spillover effects in regions at the edge of the network, and further stimulating spatial spillover effects in regions in the core position.
Moreover, this study highlights the important role of the digital economy and financial marketization in promoting inclusive growth.The development and popularization of the digital economy can help promote economic growth and improve social inclusiveness, while the expansion of financial marketization can help provide more opportunities for underprivileged groups.In order to promote the balanced development of inclusive growth in various regions of China, a cross-regional coordinated governance pattern of fiscal and taxation systems should be built, and the coverage of inter-provincial trade networks should be further increased.
In conclusion, sustainable economic growth requires policies that prioritize inclusive growth and account for the unique characteristics and roles of different regions.By promoting the development of the digital economy, financial marketization, and cross-regional cooperation, China can achieve a more balanced and sustainable path to economic development.While this study provides important insights, further research is needed to explore the spatial correlation of inclusive growth and its implications for sustainable economic development in China.

Figure 2 .
Figure 2. Block model diagram of inclusive growth spatial correlation network.

Figure 2 .
Figure 2. Block model diagram of inclusive growth spatial correlation network.

Figure 3
Figure 3 reports the difference between the original RSCN and the Monte Carlo simulation results of the CEF.The thick black line represents the actual measurement results of China's RSCN, and the gray box line represents the measurement results of the simulated network of the CEF at the 95% confidence interval.It can be seen that the real network fits well in the model statistics, the geometrically weighted dyads sharing partners, and the minimum geodesic distance.It is further confirmed that it is reasonable to use the CEF to analyze the influencing factors of the RSCN.

Figure 3
Figure 3 reports the difference between the original RSCN and the Monte Carlo simulation results of the CEF.The thick black line represents the actual measurement results of China's RSCN, and the gray box line represents the measurement results of the simulated network of the CEF at the 95% confidence interval.It can be seen that the real network fits well in the model statistics, the geometrically weighted dyads sharing partners, and the minimum geodesic distance.It is further confirmed that it is reasonable to use the CEF to analyze the influencing factors of the RSCN.

Table 1 .
The global level index of China's RSCN.

Table 3 .
The number of members of each category in the network block model.

Table 3 .
The number of members of each category in the network block model.

Table 4 .
Member composition of each category in the RSCN.: Those who have been in the same category for more than 15 years are recorded as stable members, and the category of floating members is subject to the final category. Note

Table 5 .
Node degree of each region.
Z -Value p -Value Frequency (%) Z -Value p -Value
Z -Value p -Value Frequency (%) Z -Value p -Value
Z -Value p -Value Frequency (%) Z -Value p -Value
Z -Value p -Value Frequency (%) Z -Value p -Value
Z -Value p -Value Frequency (%) Z -Value p -Value
Z -Value p -Value Frequency (%) Z -Value p -Value

Table 7 .
Regression analysis results of ERGM.

Table 9 .
Regression analysis results of the CEF in other years.