Abstract
British scholar Peter Taylor constructed the World City Network by analyzing the office networks of multinational companies, enabling a network perspective on world cities. However, this method has long been hindered by data deficiencies and update delays. In this study, we utilized publicly available, realtime updated data on global routes to construct the World City Network, thereby addressing the issues of data insufficiency and delayed updates in the existing model. For the first time, advanced Graph Convolutional Networks were employed to analyze the World City Network, and we introduced GCNRank. Finally, we compared GCNRank with other centrality measures and found that GCNRank provides a more detailed representation of city rankings and effectively avoids local optima.
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Introduction
Cities are integral components of transportation and trade networks, and their interconnectedness gives rise to complex city networks, culminating in the formation of a globally integrated World City Network (WCN)^{1,2}. The WCN exerts a profound influence on local, national and international economies, and indirectly contributes to the spread of diseases such as influenza and, more recently, Coronavirus Disease 2019 (COVID19)^{3}. Therefore, comprehending the structural intricacies of the WCN and identifying its pivotal propagation nodes are imperative tasks. However, finding a solution to this complex problem is challenging.
British scholar Peter Taylor pioneered the construction of a WCN by scrutinizing the office networks of multinational companies^{4,5}, offering a networkbased perspective on world cities. Nevertheless,^{6} critically examines and questions the validity of officelocationbased methodologies in delineating the WCN, highlighting the inherent data inadequacies and update latency. Even the proposed Interorganizational Project Approach (IOPA) confronts similar hurdles. Meantime,^{7,8,9} delve into the realm of Air Transport Networks (ATNs), leveraging publicly accessible and realtime updated datasets to scrutinize global or regional air routes through the lens of complex networks. However, many researchers have predominantly concentrated on analyzing airport networks interconnected by flights, neglecting the broader scope of the WCN, and have relied heavily on conventional Complex Network analysis techniques such as degree and betweenness centrality.
Deep learning models have been demonstrated their efficacy across various applications^{10}. Specifically, Graph Convolutional Networks (GCNs), as deep neural networks tailored for graphstructured data, have exhibited exceptional performance in domains such as social networks, bioinformatics, and recommendation systems^{11}. The multihop message aggregation and propagation mechanism of GCNs effectively captures relationships between nodes over multiple hops within complex networks. For example,^{12} successfully employed GCNs to classify entire graphs of complex networks. Graph Neural Networks (GNNs) represent a series of potent and rapidly advancing intelligent tools, though their performance can vary significantly across different application scenarios. In this study, We initiate the use of GCNs to analyze the WCN, which will aid future researchers in developing GNNs specifically tailored for the WCN.
Our contributions
In this study, publicly available and realtime updated data on global routes were utilized to construct the WCN, thereby addressing the issues of insufficient data and delayed updates in previous WCN models. For the first time, we employed advanced GCNs to analyze the WCN and proposed the GCNRank algorithm. We conducted a comparative analysis of the global roles of cities in terms of connectivity, centrality, influence and propagation. Our findings indicate that GCNRank provides a more detailed representation of city rankings and effectively avoids local optima.
This paper aims to contribute to WCN research by providing: Effective Methodology: Graphs are ubiquitous data structures that model objects and their relationships within networks^{13}. GCNs excel at processing nonEuclidean spatial data and their complex features. Appropriate Theory: We posit that world cities selforganize and evolve into a complex network according to certain fundamental rules. Interpretable Mechanism: A generalized random graph model with a truncated powerlaw distribution can generate complex networks that conform to specific parameters.
Results
Several researchers view the WCN as an ecosystem^{14,15}, wherein individual cities continuously interact and exert mutual influence through their selforganizing behaviors^{16,17}, ultimately evolving into a complex network. GNNs^{18} have garnered significant attention due to their superior performance in processing nonEuclidean spatial data and extracting complex features. GCNRank converges to a truncated powerlaw distribution, characterized by a few nodes with exceptionally high ranks, analogous to the persistent prominence of cities like London and New York as key hubs.
The node characteristics of the WCN are iteratively smoothed by GCNs while still adhering to a truncated powerlaw distribution^{19,20}. The cutoff exponent reaches up to \(\gamma = 1.6655\). The powerlaw exponent remains within the range of 1 and 3, with \(\alpha = 2.6627\). The foundational principle of complex networks is selforganizing behavior: each node optimizes its connection decisions to maximize individual benefit and effectiveness; nodes consistently compete for new connections and exhibit a preference for linking to highly connected nodes. This selforganizing behavior among cities precipitates a Matthew effect, culminating in the formation of an orderly network^{21,22}.
Global role of cities
The statistics of these four metrics for the top 30 cities are illustrated in Figure 7. The analysis reveals that in the Degree, PageRank, and GCNRank metrics, London consistently ranks first, followed by Atlanta. Paris, Chicago, Shanghai, Beijing, New York, Moscow, Los Angeles, and Istanbul also exhibit relatively high rankings.
Degree refers to the sum of the weights of all edges associated with a node, quantifying the direct connections to other cities. It is calculated using \(G.degree(weight='weight')\) in Python. According to this metric, London ranks first (see Figure 7) and Atlanta ranks second. London serves as a major international air transport hub, boasting the busiest city airspace in the world. Heathrow Airport is the busiest airport in Europe and the second busiest globally for international passenger traffic. Atlanta, on the other hand, is the largest air transportation hub in the United States and ranks first worldwide in both passenger and cargo traffic volume. HartsfieldJackson Atlanta International Airport is the busiest airport globally and is home to the headquarters of Delta Air Lines.
Betweenness centrality represents the weighted frequency with which the shortest paths traverse a node, indicating the city’s bridging role within the network. It is calculated using \(nx.betweenness\_centrality(G, normalized=True, weight='weight')\) in Python. According to this metric, Amsterdam ranks first (see Figure 7), and Anchorage ranks second. Amsterdam holds the most central position in this WCN, yet this centrality does not translate into significant influence. On the other hand, Anchorage serves as a vital community transportation hub, nearly equidistant from New York City, Tokyo, and Murmansk (directly across the Arctic), with less than 10 hours of air traffic to nearly \(90\%\) of the northern regions globally. Despite its sparse population, remote geographic location, and limited external connections, Anchorage boasts a high airport density, functioning as a major gateway that connects other cities in Alaska to the external world.
PageRank is a network ranking algorithm that iteratively converges to stable rankings of node authority through a firstorder Markov chain, evaluating a city’s influence. It is computed using \(nx.pagerank(G, alpha=0.85, max\_iter=5000, tol=1e9, weight='weight')\) in Python. According to this metric, Moscow ranks second after London (see Figure 7), while Denver holds the 13th position. Moscow, as the largest city in Europe, serves as a vital transportation hub on the Eurasian continent. Sheremetyevo International Airport, the busiest airport in Russia, is ranked as the second busiest airport in Europe. On the other hand, Denver, an inland city, stands as the largest within an 800kilometer radius, almost equidistant from major cities such as Chicago, St. Louis, Los Angeles, and San Francisco. Denver International Airport is the sole major airport built in the United States in the past 29 years. Through collaborative efforts and mutually beneficial partnerships with surrounding counties, Denver has achieved economic prosperity and exerted a significant impact on surrounding counties.
GCNRank employs the information aggregation and propagation mechanism of GCNs to iteratively smooth the distribution of nodes in the network, achieving convergence to stable rankings and assessing the city’s propagation power. This algorithm is generally consistent with degree (see Figure 7). Through information aggregation and propagation, GCNRank effectively captures the multihop node relationships in the network, providing a more nuanced representation of urban rankings. For instance, despite Los Angeles and Frankfurt having equal degree values, the GCNRank of Los Angeles is marginally higher than that of Frankfurt. Both Betweenness centrality and PageRank perform poorly for singlelink nodes, with low values, sometimes even zero; however, they perform relatively well for nodes with many singlelink neighboring nodes, exhibiting higher values. This makes them susceptible to local optima in weakly connected networks. The GCNRank algorithm mitigates this issue by adding selfloops to each node, enabling the aggregation and propagation of selfinformation, effectively dealing with singlelink nodes and disconnected networks. Moreover, compared to other centrality measures, GCNrank has good scalability in node attributes, enabling consideration of multidimensional node attributes, thereby enhancing its expressive capability.
Discussion
We posit that the smoothing effect observed in GCNs arises from the network structure’s ability to smooth node features, an inherent characteristic of GCNs^{23}. As the number of convolutional network layers increases, the hidden layer representations of nodes do not converge to the same value unless the network structure itself is uniform, as seen in a fully connected network. In our study, we assumed equal original rankings for all cities. However, after more than 2600 iterations, the GCNRanks eventually converge to a truncated powerlaw distribution. This suggests significant differences in GCNRanks, a phenomenon determined by the network structure.
“The world is hustling, all for nothing but benefit; the world is bustling, all for nothing but interest.” The pursuit of interests fosters cooperation and competition among cities, leading to the formation of a WCN. It is increasingly apparent that the centrality of cities profoundly influences their economic and social development^{24}. The ageold adage that “no city is an island” holds truer than ever before. The WCN represents a distinctive social and economical network: expansive in scale, encompassing numerous entities, and characterized by intricate relationships. Our study approaches the analysis of the WCN by simplifying global air routes, aiming to enhance related understanding both quantitatively and qualitatively.
Globalization is not an end product but an ongoing bundle of processes^{25,26}. While we cannot accurately forecast future scenarios, it is evident that comprehending such transformations necessitates a robust empirical understanding of the contemporary WCN. This study represents the inaugural examination of intercity relations on a global scale in this fashion. In forthcoming research endeavors, we aspire to transcend this crosssectional analysis and delve into the dynamic evolution of the WCN^{27,28,29}.Only through this approach can we grasp the evolving nature of the network and its diverse impacts on cities. The centrality of cities within the WCN is heavily contingent upon the collective endeavors of urban policymakers^{30,31}, who must prioritize both the overarching framework of the WCN and the dynamics within local networks^{32,33}.
The current WCN model comprises a relatively simplistic weighted undirected graph, lacking considerations such as flight frequency and passenger numbers, leading to somewhat coarse outcomes. In subsequent research, we aim to incorporate more granular factors, including flight frequency, passenger numbers, and flight directions. This endeavor will facilitate a more exhaustive analysis by constructing a refined weighted directed graph.
Interpretation
1. Cities serve as aggregators of local markets by creating favorable conditions for corporate success, including the establishment of new offices, airports, and investments in electronic communication infrastructure, among other factors^{34,35}.
2. Highstatus cities possess more connections than lowstatus cities. The majority of other cities are well connected to London and New York, reflecting their global significance^{2,36}.
Data and methods
Data acquisition on global airports and air routes
The connectivity of 3217 global cities as of June 2014 was analyzed. Regional clusters of Asian, North American, and European cities were clearly observed (see Figures 1 and 2). The dataset of airports and routes was sourced from the OpenFlights website^{37}, which includes 67,663 routes between 8,107 airports. Interestingly, 53 pairs of different cities were found to share the same name, such as London in Canada and London in the United Kingdom, Albany in Australia and Albany in the United States. After data cleaning, 17,852 undirected connections were identified among 3217 cities globally.
Obtaining the backbone network of the WCN via global air routes
The connectivity within the WCN has generally been increasing. To some extent, a city’s power is a function of its position within the WCN. Certain cities have a greater capability to affect the circulation of resources than others. For instance, New York and London have consistently dominated the world city hierarchy.

London: London ranks as the most connected city, providing an optimal environment for multinational companies to achieve global success.

New York: New York serves as the gateway to North America, characterized by high levels of inward concentration and outward radiation. It operates as both the control center and service hub of global capital.

Hong Kong: Hong Kong’s connectivity closely follows that of London and New York. As China’s primary ‘gateway city’, it functions as the bridge between China and the rest of Pacific Asia.

Singapore: Singapore, the gateway to Southeast Asia, is a significant global entrepot and an aviation center connecting Asia, Europe, Africa and Oceania.

Shanghai: Shanghai, renowned as a global financial center, has historically fostered stronger ties with London, New York, and European cities in comparison to Beijing.

Beijing: Beijing, the capital of China, exhibits stronger connections with political world cities such as Washington and Brussels in contrast to Shanghai.

Dubai: Serving as the capital of the Emirate of Dubai, Dubai stands as the economic and financial epicenter of West Asia, functioning as the primary transportation hub for both passengers and goods in the region.

Paris: Paris has long been acknowledged as one of the world’s premier hubs for finance, commerce, culture, and fashion. Its most robust connection lies with London, often drawing parallels between the two cities as depicted in “A Tale of Two Cities” within Europe.

Tokyo: In the year 2020, Tokyo secured the fourth position in the Global Financial Centers Index, trailing only behind New York City, London, and Shanghai. It maintains the strongest connections with the four other major East Asian cities.
These nine Alpha cities are evenly distributed across three regions: the United States, Europe, and Pacific Asia. London and Paris are situated in Europe, while New York is located in the United States. Hong Kong is grouped with Singapore, Shanghai, Beijing, and Tokyo in Pacific Asia. This distribution pattern can be understood in terms of varying degrees of political fragmentation. In Pacific Asia, multinational corporations often establish offices in multiple cities to ensure comprehensive coverage. For instance, they typically set up offices in Hong Kong for China, Singapore for Southeast Asia, and Tokyo for Japan. Conversely, in the United States, a single city usually suffices to cover the entire region. In Europe, despite increasing political unity, numerous national markets still exist, preventing any single city from achieving the level of dominance observed in New York. While London holds significant prominence, it cannot attain absolute dominance due to the diversity of markets within the region.
Obtaining the WCN and GCNRank
Air traffic serves as a critical infrastructure network that underpins the global economy and society^{38}. Graph structures offer an effective means of representing a variety of complex relationships within social systems. GNNs are adept at capturing the intricate features inherent in graphstructured data, while GCNs excel at effectively aggregating and propagating node information.
We established a WCN with cities as nodes, connections as edges, and routes as weights, under the assumption of equal contribution from all routes between cities^{10}. The weighted undirected graph \(\mathcal {G} = \{ \mathcal {V}, \mathcal {E}, \textbf{A} \}\) comprises a set of nodes \(\mathcal {V}\) with \( \mathcal {V}  = n\), a set of edges \(\mathcal {E}\) with \( \mathcal {E}  = m\), and an adjacency matrix \(\textbf{A}\) with \( \textbf{A}  = n \times n\). The entry \(\textbf{A} (i,j) = k\) indicates the presence of k routes between node i and node j, where \(\textbf{A} (i,j) = \textbf{A} (j,i)\). We employed the Python toolkit NetworkX to visualize the WCN (see Fig. 3 and Fig. 4), with different colors representing distinct geographical areas and node size being proportional to its weighted degree.
The degree matrix of the adjacency matrix \(\textbf{A}\) is denoted as a diagonal matrix \(\textbf{D}\), where \(\textbf{D}(i,i) = \sum _{j=1} ^{n} \textbf{A} (i,j)\). To facilitate the propagation of its own information, a selfloop is added to each node, yielding \(\widetilde{\textbf{A}} = \textbf{A} + \textbf{I} \times (w_{\text {max}}+1)\), where \(\textbf{I}\) is an identity matrix and \(w_{\text {max}}\) is the maximum weight in the WCN. The diagonal matrix \(\widetilde{\textbf{D}}\) represents the degree matrix of \(\widetilde{\textbf{A}}\). Subsequently, the corresponding symmetric normalized matrix is computed as \(\widehat{\textbf{A}} = \widetilde{\textbf{D}}^{ \frac{1}{2}} \widetilde{\textbf{A}} \widetilde{\textbf{D}}^{ \frac{1}{2}}\).
We employed a vector to represent the node set and multiply it with the symmetric normalized matrix to aggregate and propagate adjacent node information (see Figure 5). Initially, we assumed that all nodes share an equal rank of \(\frac{1}{n}\). Thus, for the original rank vector \(\textbf{h}^0\), \(\textbf{h}^0(i) = \frac{1}{n}\). In the lth graph convolution layer, denoted as \(\textbf{h}^{(l)} = \widehat{\textbf{A}} \textbf{h}^{(l1)}\), \(\textbf{h}^{(l1)}\) represents the input rank vector, and \(\textbf{h}^{(l)}\) denotes the output rank vector. At the beginning of each layer, the rank \(\textbf{h}^{(l1)}(i)\) of node \(\textbf{v}_i\) is updated with the weighted average rank of its firstorder neighboring nodes and itself (see Figure 5(c)). At the end of each layer, the rank \(\textbf{h}^{(l)}(i)\) of node \(\textbf{v}i\) is also updated with the weighted average rank of its firstorder neighboring nodes and itself (see Figure 5(b)). The layered aggregation rule of the GCN enables each node to aggregate the information of further neighboring nodes as the number of aggregation layers increases. For nodes in the core part, the node information is aggregated and propagated much faster than for other nodes^{39}. Finally, we obtain the GCNRank vector that conveys the information of all nodes, represented as \(\textbf{h}^{\infty } = \lim _{n \rightarrow \infty } \widehat{\textbf{A}}^n \textbf{h}^0\).
Fitting truncated powerlaw and heavytailed distributions
The node degree distribution of complex networks typically conforms closely to a powerlaw distribution^{40}. As the network’s node count increases, so does the degree value of the hub^{41}, underscoring the scalefree characteristic of such networks.
In this context, a truncated powerlaw distribution refers to a powerlaw distribution with an exponential cutoff. Essentially, it is a powerlaw distribution multiplied by an exponential function. The probability density function is expressed as \(p(k) = C k^{\alpha } e^{\gamma k}\), where k denotes the node degree or rank, and C, \(\alpha \), and \(\gamma \) are constants. The truncated powerlaw degree distribution is delineated by two independent parameters: the power exponent \(\alpha \) and the cutoff exponent \(\gamma \). The exponential cutoff offers the technical advantage of ensuring the distribution is normalizable for all \(\alpha \). The constant C is defined as \(C = \frac{1}{Li_{\alpha }(e^{\gamma })}\), where \(Li_{n}(x)\) denotes the nth polylogarithm of x, expressed in the software Mathematica as PolyLog[n, x].
The truncated powerlaw distribution is, in fact, a general expression encompassing the uniform distribution, the exponential distribution, and the powerlaw distribution.
Akaike Information Criterion (AIC) is employed alongside maximum likelihood estimation (MLE). MLE seeks an estimator \(\hat{\theta }\) that maximizes the likelihood function \(L(\hat{\theta }data)\) of a given distribution. The AIC is computed as \(AIC = 2 \log \left( L(\hat{\theta }data)\right) + 2K\), where K denotes the number of estimable parameters in the approximating model. Following the calculation of the AIC value for each fitted distribution, normalization is carried out as follows: initially, the difference between different AIC values is obtained as \(\Delta _i = AIC_i  AIC_{\text {min}}\). Then, the Akaike weights \(W_i\) are determined as:

Various distribution models were explored for fitting: using powerlaw.Fit.exponential() to fit the exponential distribution, \(powerlaw.Fit.power\_law()\) to fit the powerlaw distribution, powerlaw.Fit.lognormal() to fit the lognormal distribution, and \(powerlaw.Fit.truncated\_power\_law()\) to fit the truncated powerlaw distribution (refer to Table 2 in the Appendix and Figure 6)^{42}.

The loglikelihood values for the exponential distribution were calculated using powerlaw.Fit.exponential. loglikelihoods() and its AIC was derived. Similarly, the loglikelihood values for the powerlaw distribution were calculated using \(powerlaw.Fit.power\_law.loglikelihoods()\) and its AIC was derived. For the lognormal distribution, powerlaw.Fit. lognormal.loglikelihoods() was used to calculate the loglikelihood values and derive its AIC. For the truncated powerlaw distribution, \(powerlaw.Fit.truncated\_power\_law.loglikelihoods()\) was used to calculate the loglikelihood values and derive its AIC.

Subsequently, the AIC weights were calculated, and the distribution model with the highest weight was selected as the bestfitting model. In this study, the truncated powerlaw distribution was determined to be the bestfitting model (see Table 3). Each distribution is best fitted by a truncated powerlaw distribution, with Akaike weights approaching 1. Akaike weights are normalized criteria for distribution selection, ranging between 0 and 1, where higher values signify betterfitted distributions.
Generalized random graph model with truncated powerlaw distribution

Theoretical model construction: First, we constructed a generalized random graph model that adheres to a truncated powerlaw distribution. The probability density function is expressed as:
$$\begin{aligned} p(x) = C x^{\alpha } e^{\gamma x}. \end{aligned}$$(3)We defined the model’s parameters, including the minimum value \(x_{\min }\), power exponent \(\alpha \), and cutoff exponent \(\gamma \).

Parameter estimation: To estimate the parameters, we use \(powerlaw.Fit.truncated\_power\_law.xmin\) for \(x_{\min }\), \(powerlaw.Fit.truncated\_power\_law.alpha\) for the power exponent \(\alpha \), and \(powerlaw.Fit.truncated\_power\_law.\) parameter2 for the cutoff exponent \(\gamma \). The formula for calculating the power exponent \(\alpha \) is as follows:
$$\begin{aligned} \alpha = 1 + n \left( \sum _{i=1}^{n} \ln \frac{x_{i}}{x_{\min }} \right) ^{1}, \end{aligned}$$(4)where \(x_{i}\) represents the observed values with \(x_{i} > x_{\min }\), and n denotes the sample size.

Simulation data generation and comparison: Using the estimated parameters, we generated simulated data and compared their distributions with the actual data. Figure 6 presents the actual frequencies of node degree, betweenness centrality, PageRank, and GCNRank^{43,44}. It also illustrates the probability density curves for the simulated node degree, betweenness centrality, PageRank, and GCNRank. Notably, the green points represent the actual data, which closely aligns with the simulated red curves.
The node degree, betweenness centrality, PageRank, and GCNRank within this WCN adhere to truncated powerlaw distributions, wherein a small number of nodes exhibit very high values, forming hubs within the network. Furthermore, as the number of nodes increases, the sizes of these hubs also increase.
Ethical approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Data availibility
We utilized the publicly accessible dataset of airports and routes provided by OpenFlights (https://openflights.org/data.php). Additionally, the datasets generated and analyzed during our research are available in the GitHub repository: https://github.com/Tina333333/WorldCityNetwork.
Code availability
The source code used for fitting models and comparing results is available in the GitHub repository: https://github.com/Tina333333/WorldCityNetwork.
References
Taylor, P. J. & Derudder, B. World city network: A global urban analysis (Psychology Press, 2004).
Taylor, P. J. Specification of the world city network. Geogr. Anal. 33, 181–194 (2001).
Meagher, K., Achi, N. E., Bowsher, G., Ekzayez, A. & Patel, P. Exploring the role of city networks in supporting urban resilience to Covid19 in conflictaffected settings. Open Health 2, 1–20 (2021).
Derudder, B. et al. The GAWC perspective on globalscale urban networks 601–617 (Edward Elgar Publishing, 2021).
GaWC. The world according to GAWC 2020 (2020)
Neal, Z. P. Fallacies in world city network measurement. Geogr. Anal. 53, 377–382 (2021).
Guimera, R., Mossa, S., Turtschi, A. & Amaral, L. N. The worldwide air transportation network: Anomalous centrality, community structure, and cities’ global roles. Proc. Natl. Acad. Sci. 102, 7794–7799 (2005).
Wang, J., Mo, H., Wang, F. & Jin, F. Exploring the network structure and nodal centrality of China’s air transport network: A complex network approach. J. Transp. Geogr. 19, 712–721 (2011).
Dai, L., Derudder, B. & Liu, X. The evolving structure of the southeast Asian air transport network through the lens of complex networks, 1979–2012. J. Transp. Geogr. 68, 67–77 (2018).
Zhang, S., Tong, H., Xu, J. & Maciejewski, R. Graph convolutional networks: A comprehensive review. Comput. Soc. Netw. 6, 1–23 (2019).
Chen, W., Castillo, C. & Lakshmanan, L. V. Information and influence propagation in social networks (Springer, Berlin, 2022).
Xin, R., Zhang, J. & Shao, Y. Complex network classification with convolutional neural network. Tsinghua Sci. Technol. 25, 447–457 (2020).
Wu, Z. et al. A comprehensive survey on graph neural networks. IEEE Trans. Neural Netw. Learn. Syst. 32, 4–24 (2021).
Acuto, M. & Leffel, B. Understanding the global ecosystem of city networks. Urban Studies 58, 1758–1774 (2021).
Abdullah, H. & GarciaChueca, E. Cacophony or complementarity? the expanding ecosystem of city networks under scrutiny. City Diplomacy: Current Trends and Future Prospects 37–58 (2020).
Kalantari, S., Nazemi, E. & Masoumi, B. Emergence phenomena in selforganizing systems: A systematic literature review of concepts, researches, and future prospects. J. Organ. Comput. Electron. Commer. 30, 224–265 (2020).
Fourati, H., Maaloul, R., Chaari, L. & Jmaiel, M. Comprehensive survey on selforganizing cellular network approaches applied to 5g networks. Comput. Netw. 199, 108435 (2021).
Singh, P. K. Data with nonEuclidean geometry and its characterization. J. Artif. Intell. Technol. 2, 3–8 (2022).
Corral, Á. & González, Á. Power law size distributions in geoscience revisited. Earth Space Sci. 6, 673–697 (2019).
Nuermaimaiti, R., Bogachev, L. V. & Voss, J. A generalized power law model of citations. In 18th International Conference on Scientometrics and Informetrics, ISSI 2021 (International Society for Scientometrics and Informetrics, 2021).
Scott, M. & Pitt, J. Interdependent selforganizing mechanisms for cooperative survival. Artif. Life 29, 198–234 (2023).
Lykourentzou, I. et al. Selforganizing teams in online work settings. arXiv preprint arXiv:2102.07421 ( 2021).
Chen, D. et al. Measuring and relieving the oversmoothing problem for graph neural networks from the topological view. In Proceedings of the AAAI conference on artificial intelligence 34, 3438–3445 (2020).
Neal, Z. P., Derudder, B. & Taylor, P. Forecasting the world city network. Habitat Int. 106, 102146 (2020).
Hay, C. Globalization and its impact on states (2020).
Beck, U. What is globalization? (Wiley, 2018).
Cheng, J. et al. A dynamic evolution mechanism for IOV community in an urban scene. IEEE Internet Things J. 8, 7521–7530 (2020).
Segar, S. T. et al. The role of evolution in shaping ecological networks. Trends Ecol. Evol. 35, 454–466 (2020).
Yaman, A. & Iacca, G. Distributed embodied evolution over networks. Appl. Soft Comput. 101, 106993 (2021).
Bollens, S. Urban peacebuilding in divided societies: Belfast and Johannesburg (Routledge, 2021).
Allam, Z., Sharifi, A., Bibri, S. E., Jones, D. S. & Krogstie, J. The metaverse as a virtual form of smart cities: Opportunities and challenges for environmental, economic, and social sustainability in urban futures. Smart Cities 5, 771–801 (2022).
Joss, S., Sengers, F., Schraven, D., Caprotti, F. & Dayot, Y. The smart city as global discourse: Storylines and critical junctures across 27 cities. J. Urban Technol. 26, 3–34 (2019).
Wen, H., Zhang, Q., Zhu, S. & Huang, Y. Interand intracity networks: How networks are shaping china’s film industry. Reg. Stud. 55, 533–545 (2021).
Upadhya, C. Assembling Amaravati: Speculative accumulation in a new Indian city. Econ. Soc. 49, 141–169 (2020).
Marvin, S. & LuqueAyala, A. Urban operating systems: Diagramming the city. Int. J. Urban Reg. Res. 41, 84–103 (2017).
Taylor, P. J. & Derudder, B. NYLON 2020: The changing relations between London and New York in corporate Globalisation. Trans. Inst. Br. Geogr. 47, 257–270 (2022).
Openflights.org. Airport, airline and route data. https://openflights.org/data.html (2014).
Strohmeier, M., Olive, X., Lübbe, J., Schäfer, M. & Lenders, V. Crowdsourced air traffic data from the Opensky network 2019–2020. Earth Syst. Sci. Data 13, 357–366 (2021).
Wu, F. et al. Simplifying graph convolutional networks. In International conference on machine learning, 6861–6871 (PMLR, 2019).
Vasconcelos, G. L. et al. Power law behaviour in the saturation regime of fatality curves of the covid19 pandemic. Sci. Rep. 11, 4619 (2021).
Zhao, K., Musolesi, M., Hui, P., Rao, W. & Tarkoma, S. Explaining the powerlaw distribution of human mobility through transportationmodality decomposition. Sci. Rep. 5, 1–7 (2015).
Tian, L., Zhao, K., Yin, J., Vo, H. & Rao, W. The levy flight of cities: Analyzing socialeconomical trajectories with autoembedding. Discr. Dyn. Nat. Soc. 2022(1), 8180953 (2022).
Chien, E., Peng, J., Li, P. & Milenkovic, O. Adaptive universal generalized pagerank graph neural network. arXiv preprint arXiv:2006.07988 (2020).
Bojchevski, A. et al. Scaling graph neural networks with approximate pagerank. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining, 2464–2473 (2020).
GaWC. The world according to gawc 2020. https://www.lboro.ac.uk/microsites/geography/gawc/world2020t.html (2020).
Wikipedia. List of countries by the united nations geoscheme. https://en.wikipedia.org/wiki/List_of_countries_by_the_United_Nations_geoscheme (2023).
Acknowledgements
The paper is partially supported by National Key Research and Development Program of China (No.2022YFE0208000, 2023YFB4301904).
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Weixiong Rao and Kai Zhao provided guidance for the experiments and writing, whereas Linfang Tian collected the data, conducted the experiments, and analyzed the results. All authors contributed to the manuscript review.
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Tian, L., Rao, W., Zhao, K. et al. Analyzing world city network by graph convolutional networks. Sci Rep 14, 18933 (2024). https://doi.org/10.1038/s41598024694941
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DOI: https://doi.org/10.1038/s41598024694941
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