The IWCNM, first proposed by Taylor in 2001, is a quantitative method that approximates intercity networks based on APS data. To understand network linkages, Taylor (2001) assumed that there are m producer services firms located in n cities. The service value of city a can be defined as the importance of the firm’s local office in one city within all its overall office networks. This can be expressed by V_aj, which represents the service value of firm j in city a. The n×m matrix consists of the service values of all the producer services firms in different cities. According to Neal (2012), Liu et al. (2012), and Hennemann et al. (2014), the compiled database of the producer services matrix is a two-mode network that comprises cities and firms. This database should be transposed into a one-mode network in order to project it as intercity relationships. Hence, the essentiality of the IWCNM algorithm is a transformation from a two-mode city-by-firm network matrix to a one-mode city-by-city network matrix. The fundamental transformation of service value matrix V can be expressed as follows:

where V_aj and V_bj are the service values of firm j in cities a and b, respectively, and C_{ab, j} indicates the linkages between city a and b based on firm j. Then, the total connections in cities a and b can be expressed as follows:

For each city, it has (n-1) similar links at most. Furthermore, any city’s node degree Ca can be expressed as follows:

However, although the IWCNM algorithm introduces an effective way to convert a two-mode network into a one-mode network, it results in numerous invalid linkages. Specifically, because this conversion overlooks the spatial characteristics of cities and hierarchical nature of firms, it leads to inevitable information losses and flattens the nodality of city networks (Neal, 2012; Liu et al., 2012; Hennemann et al., 2013). Moreover, technically, the IWCNM algorithm does not sufficiently reflect the degrees of closeness, betweenness, in-degree, and out-degree nor other network statistical indices because of the limitations of the model’s assumptions, which lead to a technical deficiency when analyzing intercity APS networks.

Based on the limitations of the IWCNM described above, Hennemann et al. (2014) proposed a new model algorithm characterized by hierarchical and geographic features in comparison with the IWCNM. Based on these characteristics, two modifications are mainly performed by the RCCM algorithm. First, firms’ worldwide hierarchical distribution information is incorporated into the calculation. By allowing for spatiality in the organization of APS firms’ office networks, and based on the APS firms’ distribution within the global office organization, a city with the maximum geographic service value is selected as the external linkage portal in each region. Moreover, the manufacturing services firms with lower geographic values connect to high-level cities through portal cities. This approach reflects the importance of geographical adjacency on network linkage and is closer to the network linkage of the producer services sector in real life. More importantly, this approach overcomes the flat city nodes produced by the IWCNM.

Second, a baseline model is established for the network linkage in order to preserve the basic parameter distribution properties of the network structure (e.g., degree distribution). For this purpose, the shuffling approach is applied to randomly shuffle the several iterative swapping steps necessary for node linkages (i.e., permutation or bootstrapping in the social network). After this random upper-level directed change, the linkage routes and directions among network nodes are retained for the intermediary calculation and analysis of nodes.

The basic process of the RCCM algorithm can be briefly described as follows: for any firm j, find its maximum service value in region k where the firm is registered by considering the following two cases. Firstly, if both the regional service values of firm j in cities a and b of the same region k are larger than 0, but not equal, then mark the intercity linkage of firm j’s network between a and b as 1, and 0 otherwise. Similarly, if cities a and b are from different regions, but each is the largest regional service value city in their own region, then the intercity linkage is marked as 1, and 0 otherwise. On this basis, we calculate the unidirectional C_ab,j’ as follows:

This formula, which reflects the directional and multiple relations between cities, can be used to calculate the degrees (in-degree and out-degree) of every city in the network. In-degree can be understood as all the relations a branch in a certain city has with its headquarters, while out-degree indicates all the relations that the headquarters has with its branches. Furthermore, the vector feature of (C′_ab+C′_ba) can be used to represent all relation linkages between cities a and b. In other words, out-degree reflects the power of a city in which the firms’ headquarters is located, while in-degree reflects a city’s prestige and ability to attract investments (Alderson et al., 2004).

The city network linkage is a crucial way to investigate the regional organization of cities. Given the feasibility of data processing, empirical cases refer to polycentric urban region network studies in Europe. Similarly, based on the producer services data on multinational corporations compiled by the GaWC, Hall and Pain (2006) adopted the interlocking model to measure spatial linkages and analyze the organizational process of APS networking in Europe’s megacity-regions.

Inspired by the past research of western scholars, two typical high-level development city-regions on the east coast of China, namely YRD and PRD, is selected as the study cases. In the YRD region, there are 16 prefecture-level cities: Shanghai, Nanjing, Zhenjiang, Suzhou, Nantong, Yangzhou, Changzhou, Wuxi, and Taizhou in Jiangsu Province and Hangzhou, Huzhou, Jiaxing, Ningbo, Shaoxing, Taizhou, and Zhoushan in Zhejiang Province. In the PRD region, there are nine cities, namely Guangzhou, Shenzhen, Foshan, Zhuhai, Dongguan, Jiangmen, Huizhou, Zhongshan, and Zhaoqing, all in Guangdong Province.

To collect data on producer services firms, we accessed the websites of APS firms that have branches in more than one city (data were collected in May, 2010 and checked in August, 2012) based on a related ranking of Chinese firms. In total, 290 firms were included in our sample (48 banks, 38 insurance companies, 30 law firms, 33 accounting firms, 31 consulting and architectural design firms, 25 advertising agencies, and 85 securities companies). All firms were assigned to one of the following six quantitative levels (from 0 to 5) based on the reference APS values provided by the GaWC. When a firm is assigned 0, it means no office or a branch in that city, 5 refers to a city in which the headquarters is located, and 2 refers to a standard office or branch. Moreover, 1 and 3 refer to an office that is one grade below and above the standard level, respectively, while 4 refers to a city in which the regional headquarters is located. Of the 290 producer services firms, 189 have branches in at least two cities. Hence, 25 cities and 189 producer services firms are available; eventually, we take 189×25 among the matrices as the database for the present research.

4.2.1 The spatial distribution of intercity network connections

Given its authority in the analysis of producer services networks, we use the IWCNM algorithm to calculate the intercity network connectivities between these 25 cities (Figure 2). Further, we focus on the network characteristics of the top 10 cities ranked by node degree (Table 1). We find that the IWCNM algorithm provides a symmetrical matrix, with the linkages of Shanghai-Guangzhou and Shanghai-Shenzhen being more than 1,000, much higher than Shanghai-Hangzhou (677) and Shanghai-Nanjing (576), even though Hangzhou and Nanjing are the two other provincial capital cities in the YRD. Moreover, the results of the IWCNM algorithm show significant spatial connections between non-core cities in different regions, which ignores hierarchical features and thus could result in anomalies. For instance, Zhoushan in Zhejiang Province is the weakest city in the economy among all YRD and PRD cities. Yet, it has obvious intercity network linkages between Jiangsu and even some cities in Guangdong, which even exceed those cities in its own province.

Then, we use the RCCM algorithm to analyze the intercity connectivities in the YRD and PRD (Table 2) and derive an asymmetrical table, which is fundamentally different from that provided by the IWCNM. The linkage assumption of the RCCM algorithm states that a subordinate firm must report to a higher one. First, we see that the top five intercity linkages in order are Shanghai→Hangzhou (25), Shanghai→Guangzhou (23), Shanghai→Shenzhen (23), Shenzhen→Shanghai (22), and Shanghai→Nanjing (20), indicating that Shanghai, Shenzhen, Nanjing, Guangzhou, and Hangzhou dominate regional producer services networks in the YRD and PRD.

Second, the asymmetry of the RCCM table reflects the disequilibrium distribution of producer services. By observing each row in the table, we can see that the number of headquarters or regional headquarters of producer services firms in Shanghai and Shenzhen is larger than 0, whereas the number of headquarters in Foshan is 0. This finding displays the characteristic of headquarters aggregation espoused in the world city hypothesis by Hall (1966) and Friedmann (1986) and confirms Sassen’s (2001) global city theory that the aggregation of management and control functions strengthens even though production activities dispersed regionally.

Third, the method of Taylor’s (2001) ignored the geographical restrictions branch business, while the RCCM emphasized the importance of the regional headquarters of producer services. The difference that confirmed in the last two rows, which indicates that cities such as Dongguan and Foshan, which rely on their manufacturing industries, have weaker control of their export-oriented producer services.

Figure 2 shows that the results of the RCCM algorithm somewhat “erased” the connections between some cities established by the IWCNM, clarifying the spatial pattern of intercity connections and mitigating the shortcoming that resulted from its calculation process (i.e., that it ignored geographical characteristics). In terms of the linking process within a firm’s internal network, a lower-level office will usually communicate with a local higher-level administrative office before contacting an office located outside the region. This process is dictated by the fact that the local higher-level office often has access to more company information and can guide the lower-level office towards which target local office(s) to contact, making it easier for the lower-level office to operate. In this way, the higher-level offices of a firm are more likely to act as network bridging nodes for the interregional connections between their lower-level offices.

Taylor et al. (2010) proposed two terms to explain the internal and external relationships of city connections: “town-ness” and “city-ness.” The former is characterized by connections to the hinterland and is closely related to traditional central place theory, whereas the latter focuses on intercity connections. Economic globalization is an example of such interregional and external connections. Taylor also quoted Jacobs’ (1969) viewpoint to illustrate the importance of the external connections of cities and regional hinterlands. A city cannot support the economic development of other regions if it relies solely on its connections within regional hinterlands. Since the RCCM begins by dividing regions, a city’s connectivities to other cities within and outside the region need to be investigated. The node sets within each region were therefore divided according to a geographical scale. The average linkage level k_si was then used to measure the links between node i and the other city nodes within the region, while k_ti was also used to express the average linkage level of node i in the whole network. Thus, the ratio of the regional connection of node i to the regional connections of all networks can be calculated as follows:

In equation (8), n_si and n_ti refer to the number of node i within the region and the whole network, respectively. If r_i is larger than 1, the connections of node i are considered to be intra-regional. If r_i is smaller than 1, this indicates that the external connections run outside the region.

By applying equation (8), we can thus calculate the intercity regional connectitives in the YRD and PRD for the two network model algorithms (Table 3). It can be seen that Shanghai, Nanjing, Hangzhou, and Ningbo have lower internal connection ratios based on the IWCNM in the YRD, which demonstrates the trend towards the delocalization of the functional connections between the four cities in these two city-regions. However, the internal connection ratios between all cities, based on the RCCM, are larger than 1. The internal connections of Nanjing are lower than Hangzhou, implying that the external connections of the former are more significant compared with the latter. In general, the contrasting results of these two algorithms are the natural outcomes of the RCCM emphasizing the regional headquarters of producer services firms.

Similarly, based on the IWCNM, Guangzhou, Shenzhen, Foshan, and Zhongshan of the PRD have internal connection ratios that are smaller than1. However, Shenzhen’s external connections are more apparent than Guangzhou when the RCCM is used. This finding demonstrates the regional focus of this algorithm: the function of Guangzhou is closer to that of a provincial capital city; hence, its internal connections ratio under the RCCM is larger than that of Shenzhen. This result also reflects Shenzhen’s stronger external linkages within the intercity producer service network.

Another noteworthy finding is that cities in the YRD have lower internal connection ratios, which is not as obvious under the IWCNM algorithm. We thus analyzed the t-test results of the independent samples (Table 4), with the significance level of the two-tailed test based on the Levene test value. The test showed that the significance level under the RCCM reaches p<0.01, whereas that under the IWCNM does not pass the t-test. This result means that intercity producer services connections have more prominent interregional link characteristics in the YRD under the RCCM algorithm. Hence, the level of external connections is higher in comparison with the cities in the PRD.

4.2.2 Node degree

Although node degree is an important issue in network research, there is no consensus on its numerical processing. A standardized method was thus adopted in our study, in which the maximum value of each type was set as 1, followed by the ratio conversion of the array distribution from 0 to 1. Next, we compared the degree distribution under the two algorithms. For both algorithms, Shanghai’s degree was the leading one, which was set as 1.

Figure 3 shows that node degree under the IWCNM is decreasing gradually. Only Zhaoqing and Jiangmen in the PRD and Zhoushan in the YRD have a degree lower than 0.2. However, according to the RCCM’s calculation results, a node degree below 0.2 is found for Yangzhou, Taizhou (Jiangsu), Changzhou, Nantong, Huzhou, and Jiaxing in the YRD as well as for Zhongshan, Zhaoqing, Jiangmen, Zhuhai, Dongguan, Huizhou, and Foshan in the PRD.

We further adapt the degree in both algorithms to P in order to analyze the rank-size distribution. According to Newman (2003), Clauset et al. (2008), and Boccaletti et al. (2006), the degree distribution curve of the observable nodes in the geospatial network matches the scale-free characteristics, thereby presenting a power law distribution. The power law distribution of the nodes in the network can therefore be expressed as follows:

The rank-size distribution curve of node degree is shown in Figure 4 (Zhoushan is excluded because of its low degree). For both the RCCM and the IWCNM, the distribution degree shows a clear power law distribution, with the coefficients of determination of the regression equations being greater than 0.9 (Table 5). The RCCM calculation displays a steeper slope for the distribution curve of the regression equation than the IWCNM (Figure 4). The α value in the rank-size regression equation is 0.859, which is higher than that under the IWCNM algorithm (0.609). The prominent hierarchical characteristic of the RCCM can also be observed in Figure 4. Hence, given the geospatial polarization of world cities, we can conclude that the RCCM shows the dominance and controlling position of several cities in the YRD and PRD.

Based on the polycentric measurement of megacity-regions proposed by Hall et al. (2006) and Meijers and Burger et al. (2010), we can adopt the degree rank-size equation in order to measure the spatial organization in the YRD and PRD. Both the degrees calculated by these two algorithms present a better fitting scale-free distribution and show clear differences in the spatial aggregation of producer services in the YRD and PRD. Specifically, the intercity network in the YRD shows a flatter organizational characteristic, while spatial polarization still exists in the PRD to some extent. This difference is confirmed by the fitting degree of the RCCM algorithm, where the corresponding coefficient reaches 1.176 and weakly conforms to the typical primate city distribution.

Further, we investigated the correlation of in-degree, out-degree, closeness, and betweenness in two algorithms respectively (Figure 5). We found the correlation coefficients between the node degree of IWCNM and out-degree/closeness/betweenness of RCCM are above 0.7, with a significant correlation of determination (R²) of 0.881, 0.870, and 0.752,respectively, while there is poorer correlation between node degree of IWCNM and in-degree of RCCM (R²=0.097). This result suggests that although the RCCM fits the IWCNM well, the out-degree of the RCCM better reflects the actual hierarchical characteristics (Table 4). Since the headquarters and other high-level offices of producer services firms are centralized in a few core cities (i.e., the out-degree of one city), an asymmetrical functional relation between cities is shown when using the RCCM. This relation is practically inevitable given the unequal regional distribution of headquarters and branches and it results in the spatial aggregation of the control functions of world and global cities, as proposed by Friedman (1986) and Sassen (2001). Similarly, after studying the inequality of the production networks of multinational corporations, Dicken (2006) also found a difference between the definition of control and being controlled among the various sections of different companies’ production systems. Moreover, Massey (1995) proposed that social relations are determined by ownership relations, namely the relation of spatial ownership portrayed as a geography of power relations—that of control versus being controlled and influence versus being influenced. Hence, the in-degree and out-degree of the RCCM show the asymmetry of the economic connections in the YRD and PRD and provide a geographical projection of the value chains in megacity-regions.

Table 6 shows the absolute dominance of Shanghai and Shenzhen in closeness and betweenness. However, Nanjing ranks third instead of Guangzhou for betweenness, which indicates the former plays the broker role more effectively, whereas the latter may better connect the closeness of the other cities. Lastly, 13 cities have a score of 0 for betweenness, reflecting the hierarchical phenomenon of the urban network linkages in the two metropolitan regions examined in this study.

We further classify the sampled cities into five types based on node degree, betweenness, and closeness and under the criterion of a standard deviation equal to 0.5. Type A cities are Shanghai, Shenzhen, Guangzhou, Nanjing, and Hangzhou, which have outstanding results for all three indices, that is good nodality, high connectivity, and key positions compared with the other nodes. Shanghai is the leading Type A city. Type B cities are Suzhou, Nantong, Wuxi, Changzhou, Ningbo, Shaoxing, and Dongguan, which have moderate nodality performance in producer services networks. Type C includes Zhuhai and Foshan, which have lower nodality degree and moderate values for the other indices. Zhenjiang, Yangzhou, Huzhou, Jiaxing, and Taizhou in Jiangsu and Taizhou in Zhejiang that have low closeness are classified in Type D. Lastly, Type E cities are Zhaoqing, Zhongshan, Jiangmen, Huizhou, and Zhoushan, which have lower-than-average nodality and closeness (Table 7).

In general, we conclude that the RCCM algorithm effectively improved the social network analysis results, showing that it more comprehensively characterizes the intercity APS networks than the IWCNM. Moreover, since the IWCNM algorithm assumes that all non-local branches are connected (i.e., an orthogonal network without the weights), it is inadequate for social network analysis, explaining why western researchers of world city networks rarely perform this analysis based on such an algorithm.