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 (COVID-19)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.

Figure 1
figure 1

Global airport distribution. Regional clusters of Asian, North American and European cities are clearly observed.

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British scholar Peter Taylor pioneered the construction of a WCN by scrutinizing the office networks of multinational companies4,5, offering a network-based perspective on world cities. Nevertheless,6 critically examines and questions the validity of office-location-based methodologies in delineating the WCN, highlighting the inherent data inadequacies and update latency. Even the proposed Inter-organizational Project Approach (IOPA) confronts similar hurdles. Meantime,7,8,9 delve into the realm of Air Transport Networks (ATNs), leveraging publicly accessible and real-time 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.

Table 1 Dataset statistics.
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Deep learning models have been demonstrated their efficacy across various applications10. Specifically, Graph Convolutional Networks (GCNs), as deep neural networks tailored for graph-structured data, have exhibited exceptional performance in domains such as social networks, bioinformatics, and recommendation systems11. The multi-hop 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 real-time 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 networks13. GCNs excel at processing non-Euclidean spatial data and their complex features. Appropriate Theory: We posit that world cities self-organize and evolve into a complex network according to certain fundamental rules. Interpretable Mechanism: A generalized random graph model with a truncated power-law distribution can generate complex networks that conform to specific parameters.

Figure 2
figure 2

Global air route distribution. Regional clusters of Asian, North American and European cities are clearly observed.

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Results

Several researchers view the WCN as an ecosystem14,15, wherein individual cities continuously interact and exert mutual influence through their self-organizing behaviors16,17, ultimately evolving into a complex network. GNNs18 have garnered significant attention due to their superior performance in processing non-Euclidean spatial data and extracting complex features. GCNRank converges to a truncated power-law 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.

Figure 3
figure 3

The backbone network of the WCN: It comprises the top 30 cities defined by GaWC. The structure of the WCN is shaped by multiple factors, such as geography, politics, and economy. Cities belonging to the same subregional groups typically demonstrate a certain degree of concentration.

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Table 2 The designed collection of distribution models.
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The node characteristics of the WCN are iteratively smoothed by GCNs while still adhering to a truncated power-law distribution19,20. The cutoff exponent reaches up to \(\gamma = 1.6655\). The power-law exponent remains within the range of 1 and 3, with \(\alpha = 2.6627\). The foundational principle of complex networks is self-organizing 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 self-organizing behavior among cities precipitates a Matthew effect, culminating in the formation of an orderly network21,22.

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. Hartsfield-Jackson 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.

Table 3 Akaike weights of fitted distributions. It is evident that each of them is best fitted by a truncated power-law distribution, with the Akaike weights being quite close to 1. Akaike weights serve as normalized distribution selection criteria, ranging from 0 to 1. A higher value indicates a better-fitted distribution.
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PageRank is a network ranking algorithm that iteratively converges to stable rankings of node authority through a first-order Markov chain, evaluating a city’s influence. It is computed using \(nx.pagerank(G, alpha=0.85, max\_iter=5000, tol=1e-9, weight='weight')\) 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 800-kilometer 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.

Figure 4
figure 4

World City Network. The structure of the WCN is shaped by multiple factors, such as geography, politics, and economy. Cities belonging to the same subregional groups typically demonstrate a certain degree of concentration.

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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 multi-hop 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 single-link nodes, with low values, sometimes even zero; however, they perform relatively well for nodes with many single-link neighboring nodes, exhibiting higher values. This makes them susceptible to local optima in weakly connected networks. The GCNRank algorithm mitigates this issue by adding self-loops to each node, enabling the aggregation and propagation of self-information, effectively dealing with single-link nodes and disconnected networks. Moreover, compared to other centrality measures, GCNrank has good scalability in node attributes, enabling consideration of multi-dimensional 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 GCNs23. 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 power-law distribution. This suggests significant differences in GCNRanks, a phenomenon determined by the network structure.

Figure 5
figure 5

Information aggregation and propagation process. (a) Undirected graph. (b) Information aggregation for the node in red from its 1st layer neighboring nodes. (c) Information aggregation for nodes in red from their neighboring nodes.

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Table 4 Parameters of truncated power-law distributions.
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“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 development24. The age-old 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 processes25,26. 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 cross-sectional analysis and delve into the dynamic evolution of the WCN27,28,29.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 policymakers30,31, who must prioritize both the overarching framework of the WCN and the dynamics within local networks32,33.

Figure 6
figure 6

The distribution fit and scatter plots of degree, betweenness centrality, PageRank and GCNRank. Please note that all figures are presented as double-logarithmic plots. The green dots represent the actual values, while the solid red curves represent the truncated power-law distribution.

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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 factors34,35.

2. High-status cities possess more connections than low-status cities. The majority of other cities are well connected to London and New York, reflecting their global significance2,36.

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 website37, 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.

Figure 7
figure 7

Degree, betweenness centrality, PageRank and GCNRank. 1 indicates the world city rankings according to GaWC 202045. 2 indicates the result obtained by multiplying the original value by 1000.

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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 society38. 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 graph-structured 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 cities10. 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 self-loop 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 l-th graph convolution layer, denoted as \(\textbf{h}^{(l)} = \widehat{\textbf{A}} \textbf{h}^{(l-1)}\), \(\textbf{h}^{(l-1)}\) represents the input rank vector, and \(\textbf{h}^{(l)}\) denotes the output rank vector. At the beginning of each layer, the rank \(\textbf{h}^{(l-1)}(i)\) of node \(\textbf{v}_i\) is updated with the weighted average rank of its first-order 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 first-order 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 nodes39. 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 power-law and heavy-tailed distributions

The node degree distribution of complex networks typically conforms closely to a power-law distribution40. As the network’s node count increases, so does the degree value of the hub41, underscoring the scale-free characteristic of such networks.

Table 5 List of world cities used in Figures.
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In this context, a truncated power-law distribution refers to a power-law distribution with an exponential cutoff. Essentially, it is a power-law 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 power-law 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 n-th polylogarithm of x, expressed in the software Mathematica as PolyLog[n, x].

The truncated power-law distribution is, in fact, a general expression encompassing the uniform distribution, the exponential distribution, and the power-law distribution.

$$\begin{aligned} P(k)= & {} C k ^{- \alpha } e^{- \gamma k} = \left\{ \begin{array}{ll} C, &{} \alpha =0, \gamma =0 \\ C e ^{- \gamma k}, &{} \alpha =0 \\ C k ^{- \alpha }, &{} \gamma =0 \end{array} \right. \end{aligned}$$
(1)

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:

$$\begin{aligned} W_i= & {} \frac{exp(-\Delta _i / 2)}{\sum _{r = 1}^{R} exp(-\Delta _i / 2)}. \end{aligned}$$
(2)

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

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

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

Generalized random graph model with truncated power-law distribution

  • Theoretical model construction: First, we constructed a generalized random graph model that adheres to a truncated power-law 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 GCNRank43,44. 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 power-law 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.