How to determine city hierarchies and spatial structure of a megaregion?

ABSTRACT Megaregion has emerged as a global urban form, typically based on the polycentric strategy to enhance regional development. How to measure megaregional spatial structure and discriminate different roles of cities has become increasingly important to enrich the knowledge of the formation of a megaregion. Meanwhile, various indices have been used to identify vital nodes in the field of complex network. Which indices, however, are suitable for megaregion analysis remain unsolved. To address this requirement, this study first reviewed the typical indices for identifying vital nodes in the complex network theory, and pointed out that in a weighted city network scenario, weighted degree centrality, hub & authority score, and S-core decomposition (which represent network centrality, connectivity, and structures, respectively) are suitable for analyzing megaregional spatial structures. Then, we explored the city hierarchies and spatial structure in Guangdong Province, China, using the three indices. The hierarchical structure of the weighted city network in Guangdong Province had been identified using S-core decomposition. From the perspective of polycentric structure, Guangzhou and Shenzhen have the strongest node degrees and strength of mobility flows, while the Guangzhou-Dongguan-Shenzhen corridor has been identified via the hub & authority score which is designed to evaluate the connectivity in a weighted network. Moreover, we conducted a comparison analysis of three indices. The findings of this study not only enrich the understanding of city hierarchies and the structure of a megaregion, but also highlight that although various indices are available, they should be applied selectively in accordance with the study context.


Introduction
"Megaregions/megapolitan region" in the USA (Foote and Walter 2017) or "Mega-city Regions" of Europe (Harrison and Hoyler 2015) are recognized as the trend of metropolitan expansion and changed urban form and structure (Lang and Knox 2009), such as the Washington-Baltimore-Richmond metropolitan area, the Great Lakes megaregion in the United States, and the Greater London in Britain.Although the definitions of "megaregion" vary among different counties and regions, most agree that a megaregion is a vast network of metropolitan regions linked by transportation infrastructure systems (Domingo, Thibaud, and Claramunt 2019), economic flows (Taylor 2001), information flows (Wheeler and Mitchelson 1989), or shared history and culture (Yang, Song, and Lin 2015).
In the megaregion context, a central focus is the hierarchy of cities and their influences on the megaregional structure, which are critical for assessing regional integration and growth (Harrison and Gu 2021).Following the notion of "Space of Flow" (Castells 1989), cities within a megaregion compose a connected and dynamic system represented via various activity flows, particularly the distribution of living/working spaces and the mobility flows that link them to each other (Green 2007;Angel and Blei 2016).This notion transformed the megaregion into complex network models, and therefore, identifying vital nodes has become an approach for city hierarchies and measuring their relations in a megaregion (Taylor, Hoyler, and Verbruggen 2010;Zhang and Thill 2019).However, there are numerous indices for identifying vital nodes in the field of complex network studies.Some of them emphasize neighborhood influence between nodes, such as degree centrality, local rank, coreness, and H-index (Gu and Wang 2022;Chen et al. 2012).Others consider paths between nodes, including closeness centrality, information index, and betweenness centrality (Lü et al. 2016).These indices can be further modified for weighted networks (Garas, Schweitzer, and Havlin 2012).However, it will cause conflicting results if different indices are used indiscriminately.More specifically, which indices are appropriate for measuring the megaregional structure?And how to interpret the results obtained from different indices, i.e. how to select appropriate indices?
To answer these questions, this study first reviewed and clarified vital node indices used in complex network studies, and three network indices (weighted degree, hub & authority score, and S-core decomposition) were selected for flow-based megaregion spatial analysis.We further discussed the dissimilarities and applicability of the selected three indices based on Guangdong Province, China.The contributions of this study are twofold: (1) By critically reviewed the typical indices for identifying vital nodes in complex network studies, this study identified three indices which are suitable for determining city hierarchies and structure in megaregion-related analyses.
(2) From perspectives of centrality, connectivity, and structure, this study provided a comparison analysis of the three indices for the weighted city network to generate city hierarchies and suggested a strategy for producing more scientifically results of city hierarchies in a megaregion.

From cities to megaregion
The concept of megaregion as a polycentric urban system is often traced to Jean Gottmann's work on "megalopolis" in 1961, which was developed to explain large-scale urbanization.Recently, a megaregion, often referred to as polycentric urban agglomerations, is functionally linked and understood as large, spatially linked clusters of urban agglomerations that maintain daily transactional movements of economic linkages, people, services, culture, and materials (Shiliang et al. 2017).In contrast to concepts such as an urban area, a megaregion is made up of critical cities that serve as both hubs and hinges in connecting cities, rather than a single urban system centered on a dominant city (Ma, Li, and Huang 2021).As Harrison and Hoyler (2015) pointed out, megaregions have emerged as a new nexus of globalization and localization.
Based on the flow-based notion, Taylor, Hoyler, and Verbruggen (2010) proposed an interlocking network model in which flows come to center stage and are emphasized in intercity relations as a complement to the classical central place theory.They argued that the centrality of cities lies in the directional flow between cities and measured connectivity and flow strength.The relations of both intracity and intercity in a megaregion can be represented by various connections among them, such as enterprise data for economic flow (Yeh, Yang, and Wang 2015), transportation schedules (Guimerà et al. 2005), travel record data (Roth et al. 2011), social media data (Yang et al. 2019;Gong et al. 2021), and mobile phone location data (Zhang et al. 2020;Liu et al. 2021).Flow intensity and cost are the two commonly used indicators to measure the relations (Liu et al. 2021).Flow intensity represents the strength of spatial interaction between cities by measuring the flows of people or freight through the transport system.Flow cost is usually measured based on travel time or distance, which has been investigated to measure the regional job-housing performance and commuting efficiency (Sánchez-Mateos et al. 2014).In summary, the weighted network is more proper than binary entities to model intercity relations (Fang et al. 2020;Lü et al. 2016).Therefore, weighted city network characteristics such as size, structure, and patterns should be taken into consideration when selecting appropriate network indices to analyze city hierarchies and spatial structures of a megaregion.

Network indices for identifying vital cities
In a broader view of the literature, intercity relations have been widely explored in the field of complex network analysis through network indices such as centrality, connectivity, and modularity (Wang et al. 2020).The idea of centrality comes from complex network theory.Freeman (1978) measured centrality in social networks by considering the number of connected nodes, closeness between nodes, and the number of shortest paths.The five centrality indices: degree centrality, closeness centrality, betweenness centrality, straightness centrality, and information centrality, have been applied to study both intracity and intercity flows, for instance, identifying the world vital cities by the number of producer service firms (Taylor 2001) and evaluating urban street networks (Crucitti, Latora, and Porta 2006).Meanwhile, a series of other complex network indices, such as vital node identification, connection intensity, and modularity, were also explored to unveil intercity structures and city hierarchies (Limtanakool, Dijst, and Schwanen 2007).Table 1 shows the most commonly recognized network indices for identifying vital nodes used in complex network theory.
Although indices for identifying vital nodes have been used in many applications, we do not know yet which indices best deliver on their promise for analyzing the megaregional structure.More specifically, what differences exist between these indices in terms of city hierarchies and structures in a megaregion?

Appropriate indices for analyzing megaregional structures
In many modern applications, exploring and analyzing a weighted network involves tasks of uncovering the network structure and detecting "central" nodes that could be proven to be significant.Since a megaregion can be represented by a weighted city network under the notion of space of flows, aiming at computational efficiency and high scalability, the following indices will be considered: Firstly, node importance is directly measured by counting the number of neighbors (known as degree centrality) or shortest paths passing through a node (known as betweenness centrality).Most megaregion networks are completed weighted networks (different edges have different weights, and cities are fully connected to each other in the network), so the weighted degree centrality is a more efficient and suitable way of quantifying nodes' significance.
Secondly, another widely used measurement for node importance is the probability that nodes will be visited by a random walker, such as PageRank, LeaderRank, and Hub & authority score.In a weighted city network, since no new nodes or edges come into being in the process (known as nonspreading), the hub & authority score is more appropriate than the other similar indices.
Thirdly, the city network of a megaregion is weighted and directed (inflows and outflows differ in value).Recent studies used k-core decomposition to uncover groups of nodes that play important roles in the unweighted network due to its simplicity and fast computational algorithms (Malliaros et al. 2020).While the links in the weighted city network are heterogeneous, the S-core decomposition (which is a generalization of k-core decomposition to weighted networks) is a suitable metric for identifying the city hierarchies in the weighted city network.
In summary, the three indices: weighted degree centrality, hub & authority score, and S-core decomposition, were selected for determining city hierarchies and the megaregional structure because they were suitable for the weighted city network with the above features.

Weighted degree centrality
In the unweighted network, the strength of a node vi is defined as the summation of edges associated to vi (Lü et al. 2016).In the weighted network, if the in-strength and out-strength are defined as s in i and s out i , the formula of Weighted Degree Centrality (WDC) is: A scale-free network of data packets The BC distribution follows a power law for both directed and undirected cases.
H-index (Hirsch 2005) Co-author networks The H-index measures scholars' academic achievements with number of publications and citations.PageRank (Page et al. 1998)

Real networks of web pages
PageRank is a global ranking of all web pages and search results can be ordered so that more important and central web pages are given preference.LeaderRank (Lü et al. 2011) Online leadership networks LeaderRank outperforms PageRank in ranking networks of people in terms of effectiveness, robustness against manipulations and noisy data.Mixed Degree Decomposition (MDD) (Zeng and Zhang 2013) 15 real networks from social (friendship), nonsocial system (power station), etc.
The MDD outperform k-shell and degree methods in ranking spreaders.
Semi-local centrality (Chen et al. 2012) Real networks of blogs, net-science, router level of Internet, and Email The proposed index performs better in identifying influential nodes than degree and betweenness centrality indices, and almost as good as closeness centrality index with much lower computational complexity.Information index (Altmann 1993)

Theoretical networks
This index can be reinterpreted as the electrical conductance in an electrical network; An improved calculation algorithm is proposed.K-shell decomposition (Kitsak et al. 2010) Real networks of website friendship, email contacts, etc.
The index proves successful in identifying most efficient spreaders in a spreading network.

S-core decomposition (Eidsaa and Almaas 2013)
Real networks of corporate ownership The index considers both weight and degree of network and it ranks the nodes more accurately, by placing nodes with higher spreading potential into shells closer to the core.
Hub & authority score (Kleinberg 1999) The World Wide Web (WWW) networks The hub & authority score originates from the WWW network, where authoritative websites are the more reliable ones of a topic, while hub websites are linked to many authoritative websites in that area.BiHITS (Liao et al. 2014) Artificial generated and real networks of users and papers in websites from tectonophysics forum The algorithm is robust against persistent authors (spammers), which makes the method readily applicable to the existing online scientific communities.
Group-based ranking (Gao et al. 2015) Bipartite networks of users and ratings from Netflix and Amazon The method is effective on evaluating users' reputations for those with high rating errors.
where WDC i is the weighted degree centrality and s denotes node strength.This index is simple and of low computational complexity.It is a direct description of a node without considering the neighboring environment.For a weighted city network, the weighted degree centrality focuses on the intercity flows because the degrees of all nodes are identical.

Hub & authority score
The hub & authority score originates from the WWW network, where authoritative websites are the more reliable ones of a topic, while hub websites are linked to many authoritative websites in that area.In a weighted network, the hub & authority score of one node equals the summation of the hub scores of all nodes pointing to it.Using the Hyperlink Induced Topic Search (HITS) algorithm (Kleinberg 1999), the hub & authority score is calculated through an iterative process (Xu et al. 2020): (1) Initiate all nodes with a hub score of 1.
(2) At time t, the authority & hub score of node i are: where a i , h i are two temporary notations, and A i , H i are the final hub & authority score after normalization.
(3) The end condition of iteration is when the normalized scores of all nodes reach a steady state.
The hub & authority score sometimes exhibits the "topic drift" phenomenon in the hypertext link structure, including tightly interconnected clusters (Xu et al. 2020).In other words, the most highly scored authority nodes tend to be off the original topic.However, "topic drift" will not influence the result of complete networks.In the weighted city network, the hub & authority score is represented by intercity human mobility flows, which emphasizes the flows considering neighboring node influence.

S-core decomposition
In complex network analysis, core decomposition is a fundamental and efficient method for determining the relative importance of nodes and identifying a distinct group of core subgraphs in which all the nodes in the k-core subgraphs link to each other internally.The core decomposition has been widely employed in a variety of application contexts, including analyzing user activity in social networks (Leskovec, Huttenlocher, and Kleinberg 2010;Rombach et al. 2014), visualizing complex network (Christos et al. 2014), and performing static analysis on large system software projects (Malliaros et al. 2020).In light of its multiple applications in real-world challenges, S-core decomposition has garnered substantial attention (Malliaros et al. 2020;Soldano et al. 2017).Figure 1 depicts an example of a graph G and the hierarchical logical structure of the subgraphs based on S-core decomposition.The Layer-n is the subgraph of G defined by the nodes that belong to the n-core but not to the (n-1)-core.As we observe, an attempt to create a higher-order core subgraph (i.e. the n-core of the graph, Layer-n) would result in an empty subgraph, since the removal of one of the nodes belonging to the n-core will force the removal of the remaining nodes.
The nested structure of subgraphs is indicated by the dashed lines shown in Figure 1.
To uncover groups of vital nodes, Eidsaa and Almaas (2013) proposed the S-core network decomposition approach, a generalization of k-core analysis to the weighted network.The S-core decomposition approach considers both node degree and strength, and the procedures are as follows: (1) The threshold value of S-core is: (2) Remove all nodes whose strengths � s nÀ 1 , and those removed nodes belong to S-core n network.
(3) Recalculate the node strength of remaining nodes and the threshold value, then remove nodes with strengths smaller than threshold.(4) Continue the process until all nodes are removed.
The S-core decomposition is not computationally complicated.Indeed, it is more of a categorization of nodes than a continuous evaluation value.In largescale weighted networks, the S-core decomposition analysis is a useful method for ranking nodes and selecting groupings of nodes.Therefore, this index can help us gain a better understanding of how the weighted city network is structured and determine city hierarchies by integrating the link strengths of intercity The S-core decomposition is not computationally complicated.Indeed, it is more of a categorization of nodes than a continuous evaluation value.In largescale weighted networks, the S-core decomposition analysis is a useful method for ranking nodes and selecting groupings of nodes.Therefore, this index can help us gain a better understanding of how the weighted city network is structured and determine city hierarchies by integrating the link strengths of intercity relations.

The Guangdong Province
To test the three indices, we used the Guangdong Province, China as the study area, which has one of the largest urbanization areas benefiting from the opening up policy (Shiliang et al. 2017).The Outline Development Plan ("Deepening Cooperation between Guangdong, Hong Kong, and Macao-Greater Bay Area (GBA) and Promoting the GBA Construction Framework Agreement") for the Guangdong Province marks China's effort to transform its manufacturing industry into an ecological civilization (Zhang et al. 2020).Hence, the GBA has been known as a continuing effort to transform the Guangdong Province into a Bay Area, driven by the high-level intervention plan and high-tech industrial systems.More details on China's megaregion policy and development can be referred to in Shiliang et al. (2017).Integration-oriented institutions and cooperation networks have been restructured to foster regional competitiveness and coordinated growth among cities in the Guangdong Province.For example, in multiple iterations of the Guangdong regional plan and the construction of cross-border industrial parks (Zhang et al. 2021).For example, Zhao, Derudder, and Huang (2017) compared the corporate network interactions of the Pearl River Delta (PRD) between 2001, 2008, and 2013 to analysis the shifting spatial organization of cities in the Guangdong Province.While integrationoriented strategies (e.g. the transportation corridors) have substantially benefited spatial restructuring, their success remains ambiguous, but it is important in terms of the institutional implications of the GBA's cityregionalization.Therefore, studies on functional structures and city hierarchies in the Guangdong Province are critical.
The study area comprises 21 cities in Guangdong Province, located in southern China.Together with Hong Kong and Macao, the study area has become the engine for economic development in southern China.Among these cities, 9 of them are included in the GBA.The 9 cities in the GBA cover an area of about 56,000 square kilometers, and their GDP accounts for about 80% of that in Guangdong Province, which is an important pole of China's regional economic growth (Luo and Zhu 2018).While the other cities are less economically developed.The GDP and population of each city in 2019 are shown in Figure 2. The objectives of developing Guangdong are to facilitate in-depth integration within the region and promote coordinated regional economic development.Therefore, this study examined the performance of network indices for identifying city hierarchies and analyzes their feasibility and suitability.

Mobile phone data
The mobile phone data used in this study was obtained from one of three mobile communication companies in China, which included each mobile phone user's origin and destination information between the 21 cities from March 22nd to March 28th, 2019.The daily inflows (e.g.mobile phone users traveling to Guangzhou from other cities) and outflows among the cities were presented in Figures 3 and 4.
Intercity flows could be generated from different means, such as metro/railway, public buses, private cars, water transport, freight, and all other means of transportation within cities.Although the mobile phone dataset only represented around 20% of the total population, it can cover all the transport modes, and the 20% sampling rate is enough for urban-related studies (Zhang et al. 2020).The limitation is that mobile phone data could only represent human flows, while it may not suitable to model information flows or capital flows.Figure 5 illustrates the workflow of this study.

Results
In this study, we built the weighted city network G = (N, E, W) using the mobile phone data, including N = 21 nodes (21 is the number of cities).The weight w ij of inflows/outflows edges e ij between nodes i and j had been inferred from the intercity human mobility data.The scores of three indices, weighted degree centrality, hub & authority score, and S-core decomposition, were calculated.The results were classified by the Jenks Natural Breaks classification method.Class 1-5 represented the degree of each metric, with 5 being the highest and 1 being the lowest.

Centrality in city hierarchies: weighted degree centrality
To measure the control and influence of cities in the Guangdong Province, the weighted degrees of the 21 cities were calculated and shown in Figure 6.The degree of each node in the weighted city network was affected by inflows/outflows of human mobility in Guangzhou and Shenzhen, which have the highest weighted degree centrality scores.Guangzhou and Shenzhen are two core cities in the GBA: Guangzhou is the capital of Guangdong and one of the five "national central cities".After 30 years of economic growth and urban expansion, Shenzhen has become another vital mega city in Guangdong.Cities that are not in the GBA, such as Maoming, Meizhou, Shantou, and Zhanjiang, had relatively lower scores than cities in the GBA, which implied that cities in the GBA had higher attractiveness in this period.This finding was  confirmed by Table 2, which showed that the percentages of daily inflows and outflows of human mobility of 9 cities in the GBA were 78.2% and 86.0%, respectively.Therefore, in the weighted city network, the weighted degree centrality was a simple but effective way to evaluate the centrality of different cities.The observable relative differences in scores could be further analyzed regarding the city's population and economy.These observed results are extremely consistent with our knowledge that Guangdong Province has major urban attention-catchers in the form of Guangzhou and Shenzhen (Zhao, Derudder, and Huang 2017;Yeh, Yang, and Wang 2015).

Connectivity in city hierarchies: hub & authority score
In general, the hub & authority score is designed to evaluate the importance of web pages.In the weighted city network of this study, the hub & authority score evaluated inflows/outflows of human mobility among the 21 cities in Guangdong Province (Figure 7).Firstly, the intercity connections among cities mainly radiated outward from the Guangzhou, Shenzhen, and Dongguan, with the highest hub & authority scores.Foshan, Maoming, Jieyang, and Huizhou formed secondary cities in the weighted city network, which can be explained by the fact that these three cities had close relationships with neighboring cities due to their geographical location.City hierarchies at different levels   of connectivity may have different roles in the future development of the megaregional structure.The Guangzhou-Dongguan-Shenzhen region had the highest level of connectivity, which played a guiding role in the development of the network structure in the Guangdong Province.To promote their own economic development, these cities were in charge of higher-level functions and services.Secondary cities, such as Foshan and Huizhou, may play the role of connecting "transit stations".Their neighboring and peripheral cities generated material exchanges at both the production and consumption levels, which led to the formation of a network system that affects the overall structure of the Guangdong Province.

Structures: S-core decomposition
The city hierarchies obtained by the S-core decomposition are shown in Figure 8. Different from the hub authority score, which emphasizes neighboring nodes, the S-core considers the interactions from one node to another in all paths, and flows that travel shorter distances have a higher weight than those that travel longer distances.This makes the S-core a suitable metric for identifying the influential nodes in the weighted network.Using this index, Guangzhou, Shenzhen, Dongguan, Huizhou, and Foshan had the highest S-core decomposition scores, which means these four cities were connected as the core subgraphs in the weighted city network of the whole megaregion.
Taking two prefecture-level cities, Guangzhou and Foshan as an example, cross-city commuting flows between two cities had become increasingly prevalent after the operation of the Guang-Fo (Guangzhou-Foshan) intercity metro line.According to the mobile phone signaling data of operators, in addition to the resident population brought by the integration of Guangzhou and Foshan, Foshan had attracted many working populations in recent years with the promotion of the GBA and the stationing of innovative industries.Therefore, Guangzhou and Foshan were functionally and physically integrated with each other.Due to the high cost of property in Shenzhen, a rising number of Shenzhen workers were relocated to Dongguan and Huizhou.Additionally, some manufacturing companies had  also moved from Shenzhen to Dongguan and Huizhou for more affordable land.In this context, strong intercity human mobility flows were observed in Shenzhen-Dongguan-Huizhou.
Table 3 compares results of the three indices in determining city hierarchies and structures.can be seen that the city hierarchies were not consistent when different indices were applied.For example, Foshan and Dongguan ranked in the 2-level by weighted degree centrality, while they ranked in the 1-level based on hub & authority score and S-core decomposition.Therefore, if the indices of the weighted city network were used indiscriminately, it would cause conflicting results in city hierarchies.In the next session, we will further discuss the three indices.

Reflections on megaregional structures
The megaregional structure could be analyzed from multiple perspectives.This study identified three indices that could be applied through centrality, connectivity, and structure perspectives to examine city hierarchies and spatial structures in a megaregion.

Centrality: polycentric structure
To identify the most influential and interconnected cities, centrality indices could be applied, such as weighted degree centrality, PageRank, and H-index, by measuring node influence or significance.In this study, we focused on the mobility flows by measuring the weight of directed links between nodes, and Guangzhou and Shenzhen have the strongest mobility flows.We were aware that the weighted degree of a node did not consider the neighboring environment in the weighted city network.Recent studies showed that degree centrality, as a local property, is less robust to identify globally central clusters of nodes than S-core decomposition (Eidsaa and Almaas 2013).This polycentric structure with Guangzhou and Shenzhen as the core can also be observed through the inter-firm producer service linkages among cities (Yeh, Yang, and Wang 2015), intercity knowledge flows (Ma, Li, and Huang 2021), functional and institutional networks (Zhang et al. 2021), etc.

Connectivity: connected cities and corridor
To measure cities with higher connectivity, the hub & authority score is an option.A node is considered to have more connectivity if it has more communication nodes that organize flow exchanges.Moreover, the hub & authority score also consider node topology, i.e. spatial relations in the network.Therefore, the hub & authority score can identify not only interconnected cities but also their spatial relationships.The Guangzhou-Dongguan-Shenzhen corridor has also been observed by other studies: (1) using the spatial functional patterns of financial networks (Feng, Growe, and Shen 2020); (2) an urban network pattern consisting of an absolute core circle composed of Guangzhou-Shenzhen-Dongguan (Zhou, Liu, and Li 2021); (3) the PRD megaregion manifests as a spatial connection network structure centered on Guangzhou, Shenzhen, and Dongguan (Zhou, Liu, and Li 2021), etc.

Structures: functional urban areas
To identify megaregional spatial structures, the S-core decomposition can be used, which is able to uncover the core subgraphs (maximum connected subsets of nodes) in a weighted city network considering the nonuniform strength of links.For example, a set of influential cities were found in our study area, which is consistent with the two largest functional urban areas, namely, Shenzhen-Dongguan-Huizhou and Guangzhou-Foshan, identified using intercity commuting flows (Chen and Gar-On Yeh 2022).Two functional urban regions known as "Polynuclear Urban Region", including Shenzhen-Dongguan-Huizhou and Guangzhou-Foshan-Zhaoqing, can also be evaluated through intercity cooperation network analysis (Zhang et al. 2018).For unweighted networks, the k-core method could be applied, in which a set of nodes with a high degree are tightly connected to other nodes.
In summary, weighted degree centrality, hub & authority score, and S-core decomposition are suitable for capturing differences in terms of city hierarchies and spatial structure in a megaregion through the flow-based analysis (Table 4).

Policy implications
GBA is a fast-developing region with both national and global focuses.To provide support for regional cooperation and coordinated economic development and establish a vibrant and competitive urban agglomit is important to understand the regional structure and relations among the cities.Therefore, the findings of this study can inform spatial planning and policy making by describing the city hierarchies and spatial structure of the region from three perspectives: the most influential and interconnected cities, the connected cities and corridors, and the functional urban areas in the region.
This study examined the structural and functional organization of cities through the lens of complex network.The results showed that the network polycentricity of the GBA was highly consistent with the functional structure in 2019.Therefore, regional integration could be further enhanced in the fields of business, transportation, and other aspects (Chong and Pan 2020).One way to further strengthen the integration of GBA is the "hyperlink" strategy, which means to build a strong economic network among the cities of Guangdong Province by taking advantage of globalization to combine the different flow types, such as telecommunications, transportation, big data, and enterprises (Yang, Song, and Lin 2015).

Conclusions
Given that the polycentricity in a megaregion is characterized by hierarchical structures at different spatial, this study enriched the knowledge of the formation of a megaregion by addressing two research questions: (1) which network indices are suitable for measuring city hierarchies and megaregional spatial structure?(2) How to interpret the results obtained from the indices?
Three indices, weighted degree centrality, hub & authority score, and S-core decomposition, were selected in this study after reviewing the most related indices in the field of complex network.Using Guangdong Province as a case, we found that Guangzhou and Shenzhen were the leading cities in the area according to the weighted degree centrality, which provides a useful measure of the strength of cities in attracting daily mobility flows.The hub & authority score helped us get a better sense of the geography of intercity connectivity and which cities played an important role in forming intercity relationships.By integrating the link strengths of intercity flows into selecting groupings of nodes, S-core decomposition is a suitable index for identifying functional urban regions in the Guangdong Province.
The second question dealt with how different indices may help clarify the divergent spatial structure and functional relationships among cities in the megaregion in different ways.We observed that: (1) the polycentric structure (two well-developed poles that Guangzhou and Shenzhen).( 2) the core circle composed of Guangzhou-Shenzhen-Dongguan.(3) the functionally and physically integrated region (Guangzhou-Foshan-Shenzhen-Dongguan-Huizhou) in Guangdong Province.These findings can provide deep insights into the intricate interplay of cities and enrich the knowledge of the hierarchy of cities by recognizing the network nature of cities and promoting regional coordination and planning strategies.
This study firstly examined the indices in the Guangdong Province, while future studies could further investigate spatial structure in other megaregions worldwide to validate the applicability of the metrics in the weighted city network.Secondly, the relationships between producer services, inter-firm links, and other intercity economic ties are also important in determining city hierarchies in a megaregion, along with the flows of people between cities.Therefore, a more precise resolution of regional analysis can be reached via the comparative findings.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
The work is supported by the National Natural Science

Notes on contributors
Yanyan Gu is currently doing postdoctoral research in Shenzhen University.His research focuses on complex network analysis, urban land use, and human mobility.
Run Shi is currently a PhD candidate at the of Hong Kong.Her research interests include big data and urban planning.
Yan Zhuang is a research fellow at Wuhan University.He researches currently focuses on analyzing and modeling of geo-spatial data.
Qingquan Li is a professor at Shenzhen University, China.His research areas include spatiotemporal data dynamic modeling and precise engineering survey.
Yang Yue is a professor of urban informatics, head of department of Urban Informatics, Shenzhen University.Her research interests include urban modelling with big data, urban transportation, and urban sociology.

Figure 1 .
Figure 1.Hierarchical structure of the subgraphs based on S-core decomposition.

Figure 2 .
Figure 2. GDP and population of 21 cities in the Guangdong Province.

Figure 3 .
Figure 3. Overview of 21 cities and their intercity flows.

Figure 4 .Figure
Figure 4. Daily inflows and outflows among the 21 cities.

Figure 6 .
Figure 6.City ranks by weighted degree centrality.

Figure 8 .
Figure 8. Network structures of 21 cities based on S-core decomposition.

Table 1 .
Indices for identifying vital nodes used in the complex network theory.Degree centrality is more effective at identifying those vertices whose removal most significantly impact the structure of the network.For empirical networks, closeness proves to be the most effective means of targeting vertices for removal.

Table 2 .
The percentage of total daily flows in 2019.

Table 3 .
Comparisons of three indices in determining city hierarchies and structures.

Table 4 .
Foundation of China-Joint Programming Initiative Urban Europe [grant number 71961137003] and the National Natural Science Foundation of China [grant numbers 42171449, 42101464].Comparisons of three indices with previous studies.Indices Notation in complex network Observations in this study Empirical studies Weighted degree centrality Centrality: Control and influence Polycentric structure Yeh, Yang, and Wang (2015); Ma, Li, and Huang (2021); Zhang et al. (2021) Hub & authority score Connectivity: Control over links and information Spatial connection network pattern Feng, Growe, and Shen (2020); Zhou, Liu, and Li (2021); Zhang et al. (2021) S-core decomposition Structure: Core subgraphs in the networks Functional urban areas Chen and Gar-On Yeh (2022) Zhang et al. (2018)