The impact of inter-city mobility on urban welfare

While much effort has been devoted to understand the role of intra-urban characteristics on sustainability and growth, much remains to be understood about the effect of inter-urban interactions and the role cities have in determining each other's urban welfare. Here we consider a global mobility network of population flows between cities as a proxy for the communication between these regions, and analyze how these flows impact socioeconomic indicators that measure economic success. We use several measures of centrality to rank cities according to their importance in the mobility network, finding PageRank to be the most effective measure for reflecting these prosperity indicators. Our analysis reveals that the characterization of the welfare of cities based on mobility information hinges on their corresponding development stage. Namely, while network-based predictions of welfare correlate well with economic indicators in mature cities, for developing urban areas additional information about the prosperity of their mobility neighborhood is needed. For these developing cities, those that are connected to sets of mature cities show markedly better socio-economic indicators than those connected to less mature cities. We develop a simple generative model for the allocation of population flows out of a city that balances the costs and benefits of interaction with other cities that are successful, finding that it provides a strong fit to the flows observed in the global mobility network and highlights the differences in flow patterns between developed and developing urban regions. Our results hint towards the importance of leveraging inter-urban connections in service of urban development and welfare.


I. INTRODUCTION
Given the recent trend of rapid urbanization wherein the majority of the global population now resides in urban centers [1,2], cities are at the center of innovation and technological advancement [3,4].The current relevance of cities has fueled the birth of the so-called science of cities aimed at uncovering the physical and structural features driving their growth and function [5][6][7].Among the different topics addressed by this discipline, one recurrent question has been that of quantifying how human mobility shapes urban dynamics and provides reliable information on different socioeconomic indicators.Along this line, recent studies have used intra-city mobility flows to understand spatial city organization and its connection to urban characteristics such as facilities, jobs and services that are crucial for city livability and sustainability [8,9].Other works have investigated accessibility in urban systems considering city topology and infrastructure that mediate interactions and activities of inhabitants [10][11][12].More generally, mobility flows have been used to study the structure of urban areas and dynamics taking place within them [13][14][15][16][17].
The welfare of a city is just one example of the emergence of success within a system composed of a large set of similar elements among which there are relations of competition and cooperation.The study of the mechanisms driving the emergence of cases of success has been recently addressed in a variety of fields.For example, luck and randomness have been found to be crucial features behind the prosperity of scientific [18,19] or creative careers [20].
Apart from the impact of intrinsically stochastic events, the roots of individual success in different disciplines has been addressed through the lens of network science.Specifically, in [21], the authors capture institutional prestige by studying a co-exhibition network where nodes represent art galleries and the links are determined by the movement of artists among institutions.Similarly, in a recent work [22], the network of professional relationships of start-ups is built with the time-varying flows of employees among world-wide companies.This network-based approach has shown to have predictive power of the future success of companies.Finally, in [23], the authors construct a Workforce Mobility Network among metropolitan areas in the US and predict diverse socioeconomic outputs of urban areas such as number of New Patents, the number of R&D establishments, and Total Wages.
"What makes a city successful?" is an intriguing and complicated question as there are multiple factors shaping the success of urban systems.Here we aim at addressing this question by leveraging the close relationship between human mobility and the functioning of cities.To this end, we propose a network science approach to shed light on the heterogeneous distribution of success, as measured by various socioeconomic indicators, observed among cities worldwide.We use anonymous and aggregated inter-city flows to construct a global mobility network between cities, with the aim of capturing this success using the observed flow patterns.We start by ranking cities with respect to various network centrality measures-in particular connectivity, PageRank, and eigenvector centrality-which capture not only the strength of connections but also their quality, and compare these centralities with socioeconomic indicators in 268 global metropolitan areas.We demonstrate that the welfare of an urban area is largely influenced by the welfare of its neighborhood, that is, those cities with which it has the highest volume of inter-urban population flows.This impact becomes even more visible when grouping cities into various categories according to geographic region or development level, to distinguish flow profiles of cities in different groups.We observe that the international connections for a city in a less developed category (or geographic region) contribute significantly to its prosperity.Finally, we build a simple model for the outgoing flows from a city, finding that the success of a city is a major element in attracting flows from others, and that less developed cities place greater priority on directing flows to highly successful cities.

Mobility Data
The Google Aggregated Mobility Research Dataset contains anonymized mobility flows aggregated over users who have turned on the Location History setting, which is off by default.This is similar to the data used to show how busy certain types of places are in Google Maps, which helps to identify when a local business tends to be the most crowded.
The dataset aggregates flows of people from region to region weekly.
To produce this dataset, machine learning methods are applied to logs data to automat-ically segment it into semantic trips [24].To provide strong privacy guarantees, all trips are anonymized and aggregated using a differentially private mechanism [25] to aggregate flows over time [26].The analysis in this paper is done on the resulting heavily aggregated and differentially private data.No individual user data was ever manually inspected, only heavily aggregated flows of large populations were handled.
All anonymized trips are processed in aggregate to extract their origin and destination location and time.For example, if n unique users traveled from location a to location b within week w, the corresponding cell (a, b, w) in the mobility tensor would have a value of n ± η where η is noise drawn from a Laplace distribution with mean 0 and scale 1/0.66.
We then remove all metrics for which the noisy number of users is lower than 100, following the process described in [27].This automated Laplace mechanism yields a ( , δ)-differential privacy guarantee of = 0.66 and δ =2.1 × 10 −29 per metric.The parameter controls the noise intensity in terms of its variance, while δ represents the deviation from pure -privacy.
The closer each value is to zero, the stronger the privacy guarantees.The resulting mobility flows are encoded in an origin-destination matrix T whose elements T ij represent the flows from location i to j aggregated over the full year.

Socioeconomic indicators
To quantify the economic success of cities, we collected data of city-level socioeconomic variables from the company Jones Lang LaSalle IP, Inc. (JLL) for the year 2018 [28].The report published by JLL gives the GDP, Total Real Estate Investment, and a Cross-Border Real Estate Investment and Commercial Attraction Index (BHI) for 300 cities worldwide.In particular, the BHI is a composite measure that accounts for key real estate indicators (investment volumes and commercial real estate stock), as well as socio-economic and business indicators such as economic output, population, corporate presence, and air connectivity.
Besides these continuous variables, there are categorical variables in the dataset that divide cities into different subsets based on their level of development and geographical region.
The indicated levels of development (in increasing order) are Early Growth, Developing, Transitional, and Mature, which are assigned following information concerning the city's real estate liquidity, the depth of its corporate occupier base, and the quality and range of For our analysis we consider 268 metropolitan areas, excluding China for which we do not have mobility data.(See Figs.S1, S2 and Table S1 for more details about the urban areas used in this study.)

B. Relating welfare and PageRank centrality
While we have metadata for 268 cities, for the purposes of our analysis, we analyze the mobility network of N = 1, 774 nodes representing global metropolitan areas with population P > 100, 000 and E = 39, 868 directed edges.To incorporate, in a tunable way, the contribution of long-distance travel to economic activity, we define the edge weights of the mobility network as: where β ≥ 0. In this expression T ij is the total flow from city i to j and the d ij is the Euclidean distance between them, which allows us to account for the effect of distance as a cost in population movement [29,30].Incorporating this allows for including the fact that long-range connections indicate greater levels of implied connectivity between the cities, given the associated cost of travel.The optimal value of the exponent β is determined later.
The effect of distance on the flow-weights is illustrated in Fig. 1  Next we rank cities according to their importance based on their position in the mobility network.An ideal metric for that is the PageRank centrality, a network-based diffusion algorithm, used by Google and other web search engines to estimate the importance or quality of web pages.The algorithm takes into account both the number of connections as well as the importance of the neighbors a node connects to [33,34].For a weighted directed network, the PageRank of a node i is computed as [35]: where s out j is the total strength of outgoing links from node j and α is a "reset" parameter in the range [0, 1] (we use α = 0.85).In addition to webpages, the metric has been successfully employed to rank scientists based on their citation patterns [36,37], disease-causing genes based on protein-protein interactions [38], roads or streets in terms of traffic [39], ecological species based on their position in the food web [40], highlight cancer genes in proteomic data [41] among other applications.
We start our analysis by ranking cities in the mobility network by their value of PageRank and plot it against one of the composite socioeconomic indicators, the BHI.The strength of association is measured by the Spearman's correlation coefficient ρ s that measures the strength and direction of association between two ranked variables.In Fig. 1c we plot the case for β = 0 finding a monotonic relationship with ρ s = 0.53.Including distance in the edge-weights reveals an increase in the association between the two variables for any non-zero β.In Fig. S3 panels a and b we plot ρ s as well as the Pearson's correlation coefficient ρ p between log(BHI) and log(P R) as a function of β, finding a peak value for both correlation coefficients at β = 2.19 (ρ s ) and β = 1.88 (ρ p ).In indicate that (i) PageRank is correlated with all socioeconomic measures (the strongest being BHI) and (ii) this association is enhanced when both the flow and distance is incorporated in the edge-weights.We also show the results for two other centrality measures (weighted degree and eigenvector centrality).Taken together, it appears that both the position of a city in the global mobility network, as well as the strength and number of connections is a reasonable predictor of its welfare.
Next, we dis-aggregate the cities based on their level of development and geographical region to check whether the results are consistent across these categories.In Fig. S5 we plot the PageRank against BHI for four levels of development: Mature, Transitional, Developing and Early Growth.We find a progressive decrease in the strength of association from Mature cities (ρ s = 0.86) through transitional and developing cities (ρ s = 0.73, 0.88, respectively) and finally a weak correlation in early growth cities (ρ s = 0.38).Indeed, the differences become rather stark when looking at the other socioeconomic indicators as seen in Fig.The richness parameter ρ(r) (Eq.( 3)) a for the connectivity r = k out and b weighted out-degree r = s out .The monotonically increasing trend of ρ(r) with r indicates subsets of high-connectivity cities that connect to each other more than would be expected as merely a consequence of the distribution of links and edge-weights.c The number of neighbors of each node, with a higher out-degree than the node itself, as a function of the node's out-degree.The scatter plot has a peak separating cities into two clusters of core (in red) and peripheral (in blue) nodes.d The distribution of BHI in the core and peripheral cities, showing that cities in the core typically have much higher values of BHI than those in the periphery.

C. Effect of position in the mobility network
To check whether these observed differences based on maturity and location are consequences of higher-order topological features of the mobility network, we next employ the k-core decomposition, that identifies strongly connected core-nodes and sparsely connected peripheral nodes, revealing meso-scale network structure [42,43].Applying this analysis we The existence of large core clusters concentrated in Western Europe and North America (all Mature or Transitional cities) suggest that they form a densely interconnected backbone of the mobility network and thus control most of human capital and resource flow.This can be quantified by studying the weighted rich-club organization [44] which is defined as where r is a richness parameter.Two flavors of the parameter that we consider are the out-degree k out and the out-strength s out .Given this, φ(k) is a fraction that measures the number of edges between nodes of degree ≥ k out compared to the edges they would share if all these nodes were connected to each other.The quantity φ null (k out ) is the corresponding measure under out degree-preserving reshuffling of links.For s out , φ null (s out ) is computed through both edge and weight reshuffling.Values of ρ(r) > 1 indicate a set of nodes that are connected to more cities, with higher exchange of populations, than one would expect merely as a consequence of the distribution of links and edge-weights.In Fig. 2a, we plot ρ(k out ) and in b ρ(s out ), finding that nodes with high connectivity and strength of connections are between two to three times more connected to each other than would be expected by random chance.The results indicate that Mature and Transitional cities form a rich-club dominating mobility flows in the network.
Finally, while the k-core decomposition allows for assigning nodes to layers, there is no clear way to define a boundary between the core and periphery.To do so, we use a method introduced in [45]: First, we sort nodes with respect to k out in increasing order.Then, for each node, we count the number of its neighbors that have out-degree higher than itself.
In Fig. 2c, we plot the sorted degrees of all nodes and their corresponding number of connections to higher-degree nodes.There is a clear peak in the scatter plot that allows us to distinguish 2 clusters of cities being in the core (in red) and in the periphery (in blue) [46].The figure indicates that the top 30% of nodes in terms of their out-degree are in the core of the mobility network, consistent with the two dense cores identified by the k−core method in Fig. S9.Having identified the core and periphery we plot the distribution of BHI in the two regions of the network in Fig. 2d finding that cities in the core are in general more prosperous than peripheral cities.

D. Interplay between welfare and mobility neighborhood
The results thus far point towards strongly inter-connected sets of core cities that dominate mobility flow (and therefore exchange of human resources) that are successful in terms of socioeconomic indicators.This success can be reasonably well predicted by their centrality, or just their topological features in relation to the mobility network.For cities in the periphery, and those that are not so tightly connected, their level of development cannot be simply captured by topological aspects.In particular PageRank is a poor measure to capture such cities' welfare, given that it is noisy for nodes with low connectivity and edgeweights [34].Furthermore, it does not capture an important feature, that is the maturity level of the cities from which travelers come to a given city.For instance, it stands to reason that for two developing cities with similar connectivity profiles, if one of them has visitors from mature cities, and the other only from developing cities, then the former may benefit more from the flow of intellectual and human capital from more developed regions.
To account for this effect, we define the quantity where S j denotes the success/welfare of a given city j (measured through the socioeconomic indicators) and W I contains only international flows from j to i.We restrict ourselves to international flows, given that in our dataset, cities within a country have by and large the same maturity levels.The quantity S i can then be interpreted as the weighted average of the success of the international visitors to city i.For instance if we take BHI as the measure of success, then this number will be high if most visitors are from Mature cities, and low if where P R i is its PageRank and γ is a tunable parameter that measures the extent to which the level of development of a city's neighbors plays any role in predicting its own development.
For γ 0, the node's network topological properties are the best indicator of its welfare, whereas for γ ≥ 0 incorporating information about the development levels of its neighbors provides a better estimate.

E. The influence of success on mobility flows
The enhanced ability to predict the true socioeconomic indicator of a city in the less developed categories by incorporating the maturity level of its neighborhood, suggests that perhaps success plays a role in how flows are allocated.To check for this, we construct a simple model for the allocation of the outgoing flows of a city among its destinations that accounts for both the success of the target destinations, as well as the distance between these targets and the source city.Suppose a city i has a certain amount of total outflow c i , which it can allocate towards different cities j within some set of target cities ∂ i -its mobility network neighborhood.We assume that i allocates its outgoing flow c i to neighbors j ∈ ∂ i based on the success S j of city j, as well as the distance d ij , so that Tij (µ, ν) ∝ S µ j d ν ij , where µ and ν are free parameters, and Tij (µ, ν) is the predicted flow from i to j from the model (whose dependence on µ and ν has been made explicit).More precisely, we have If the success S j makes travel to city j more attractive to city i, then one would expect µ ≥ 0. On the other hand, if the cities are far apart in distance d ij , then it would make it more costly to travel from i to j, and in that case ν ≤ 0. The ratio | µ ν | then determines the relative importance of these factors in determining Tij .Given that Tij (µ, ν) is invariant to the scale of S j and d ij , the relative importance of these measures can be compared directly using this ratio regardless of units.
The outflow c i can be inferred from the raw (unweighted) origin-destination matrix using the expression c i = j∈∂ i T ij , where the neighborhood ∂ i is restricted to the subset of cities j that i connects to that also have BHI metadata S j .We then fit Eq. 6 to the true observed flows T ij to find the exponents μ and ν that optimize the Pearson correlation between the true and predicted flows T ij and Tij for connected city pairs i, j.This optimization is nontrivial but can be done approximately with a variety of methods, and here we choose a basin hopping algorithm [47].Higher values of |μ/ν| indicate that cities place high relative importance on success when allocating their outflows, and low values of |μ/ν| indicate that city outflows are much more strongly influenced by distance than by success.To determine whether the city subgroup (development stage or geographical region) affects the influence of success on outgoing flows, we identify separate exponents {μ, ν} for each subgroup by optimizing the Pearson correlation between Tij and T ij for all flows leaving cities i in the subgroup.
We also determine whether or not the success covariate S j (BHI) significantly improves our predictive model of outflows over a baseline model, where only the distance between cities is considered.In this way, one can check if cities tend to place substantial importance on success when choosing how to allocate their flows.To do this, we identify the exponent ν 0 that optimizes the correlation between the true and predicted flows while ignoring the success parameter (µ = 0).Since the model with µ has one more free parameter than the model with µ = 0, the fit between the true and predicted flows will always be better according to the Pearson correlation ρ p .Thus for a fair comparison, the more complex model needs to be penalized using some model selection criteria.Here we opt for both the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) [48], which are derived from the log-likelihood of the model fit, as well as the number of data points in the sample.Table S2 includes all model fit results for each category of cities. Applying both penalties indicate that the model performs significantly better with the inclusion of µ for all subgroups.We find that μ > 0 and ν < 0 for all city subgroups, indicating that cities allocate more flow to cities with higher BHI, and allocate less flow to cities that are farther away, though the extent to which this happens depends on the level of development, with the effect most pronounced in Early Growth and Developing cities, and least so in Mature and Transitional cities.
In Fig. 3(a), the relative magnitude |μ/ν| of the coefficients μ and ν is plotted along with the coefficient of determination r 2 for each city subcategory.As discussed, the ratio |μ/ν| quantifies the relative importance of success S j versus distance d ij in determining how a city i allocates its flow c i .According to the model fits, in general developing cities place more importance on success than more developed cities, as indicated by the shaded red box.More specifically, cities in the Developing, Early Growth, and Transitional subgroup place higher relative priority on the success of the cities they allocate flows to as compared to cities in the Mature subgroup.We also see a geographic dependence with North America and (Western and Eastern) Europe placing relatively lower importance on BHI and having weaker model fits in general.This suggests that other factors are more important in determining the corresponding flows (perhaps consistent with these regions having many developed cities).
As insets, we illustrate the variables used in the flow model (Fig. 3(b)), as well as show an example of the improved weight prediction after the inclusion of BHI for the Australasia subgroup (Fig. 3(c)).In this third panel, we plot the flow as predicted by Eq. 6-with μ and ν as the model parameters-along the x-axis, and the true observed flow along the y-axis.The weights condense significantly around the line of equality when including S j as a co-variate, as compared to using only d ij .
These results indicate that cities in earlier stages of development or in more developing regions place greater importance on their neighbors' success when considering their mobility outflows.These results compliment our previous analysis by suggesting that perhaps such cities strategically allocate their outflows in order to maximize the benefits they receive by connecting with more successful cities.On the other hand, for developed cities outflows are more strongly influenced by travel distance.

III. DISCUSSION
We have presented here, a comprehensive analysis on the role of inter-city global mobility flows apparent in driving the welfare and success of urban areas.At the macroscopic level, we report the existence of a tightly connected core of cities that form a rich-club dominating mobility flow in the network.Cities in this core tend to have higher levels of development, and are primarily located in Western Europe and North America.For these cities, their development-level is well predicted by their network topological properties, in particular centrality measures such as PagerRank.On the other hand developing and early cities are located in the peripheral layers and are scattered across multiple regions (Asia, Eastern Europe, Latin America, Middle East and Sub-Saharan Africa).Their level of development is partially connected to their network centrality in some cases, and very weakly so in others, in particular in Africa.
For those regions, whose success is weakly connected to their centrality, we find that adding information on their mobility neighborhood (that is the average development level of the cities they are connected to) leads to a marked increase in correlations with their socioeconomic indicators.This effect is less pronounced for more developed cities, whose socioeconomic indicators are connected primarily with their network centrality.Once again Sub-Saharan Africa is an outlier, its development being weakly connected to network properties and the socioeconomics of their mobility neighborhood.To check whether the connection of cities to other cities is indeed influenced by their development-level we propose a simple model that disentangles the relevant aspects influencing the success/welfare of urban areas.
The model assumes that the outflow of cities is distributed among destinations in proportion to their relative benefit (maturity) and cost (distance).The predicted outflows are compared to the empirical outflows to determine the contribution of each individual component of the model.We find that all cities incorporate the development level of the cities they connect to, and are less inclined to connect to cities at great distance.However, the former is much more pronounced in developing regions as compared to developed regions.
Taken together, our analysis indicates that the welfare of cities and the inter-urban mobility network are strongly correlated, but that this correlation depends on their level of maturity.The global urban ecosystem appears to be a combination of a well-established set of core urban areas whose interconnections describe their performance according to network science principles, and a subset of developing cities whose welfare is influenced by the extent of connections to the core.While we do not have data on individual mobility-flows the results are akin to that seen in [21], where artists climb the ladder, as it were, by strategically positioning themselves in the network of galleries exhibiting their work.

IV. LIMITATIONS
These results should be interpreted in light of several important limitations.First, the Google mobility data is limited to smartphone users who have opted in to Google Location History feature, which is off by default.These data may not be representative of the population as whole, and may vary by location.Importantly, these limited data are only viewed through the lens of differential privacy algorithms, specifically designed to protect user anonymity and obscure fine detail.Moreover, comparisons across rather than within locations are only descriptive since these regions can differ in substantial ways.

V. DATA AVAILABILITY
The Google Aggregated Mobility Research Data-set used for this study is available with permission from Google LLC.Links to all other data sources have been provided in the manuscript.

Supporting Information
The impact of inter-city mobility on urban welfare S. Mimar et al.Table 2: Results for flow model applied to both maturity and global sub-region city subgroups.The number of cities in each subgroup as well as the number of flows outgoing from these cities that were considered for the correlation analysis are shown.The inferred parameters from Eq. 8 are also shown, along with the corresponding coefficient of determination r 2 .We additionally display the inferred parameter ν0 for the restricted model where µ = 0, as well as the corresponding coefficient of determination r 2 0 .The column 'selected' denotes whether or not both the AIC and BIC difference between the two-parameter and one-parameter models are less than zero, indicating that the twoparameter model is preferred by these criteria.Finally, we show the ratio |μ/ν|, which indicates the relative importance of success versus distance in determining the flows for the corresponding subgroup.

FIG. 1 .
FIG. 1. Connecting network position and welfare.Weights W ij , of the global mobility network for a the raw mobility flows (β = 0 in Eq. (1)) and c when incorporating the effect of distance in the weights (β = 2.19), shown for North American cities.Node sizes are proportional to their PageRank in both maps.By including distance in the flows, we give stronger weight to long-range trips.b BHI vs PageRank for β = 0, with a Spearman correlation coefficient ρ s = 0.53.d BHI vs PageRank for β = 2.19, yielding ρ s = 0.77.
figure shows, incorporating the effect of distance makes long-range flows more prominent; colors indicate magnitude of flow, increasing from yellow to red, and node sizes correspond to PageRank.(Maps generated using Shapely [31] and GeoPandas [32] packages in Python.)

Fig. 1
Fig. S4 we show the connection of PageRank with other socioeconomic indicators (total real estate investment, GDP, and cross-border real estate investment) for a variety of choices of the edge-weights: the raw flow T ij , the euclidean distance d ij as well as Eq.(1).The results

S7.
In mature cities the PageRank is strongly correlated with all indicators, in transitional cities the correlation with real-estate investment weakens significantly, and moving to the other two categories the correlation is weak with all indicators.Differences also exist when grouping cities based on their geographical location as shown in Fig.S6.In prosperous regions (North America, Australasia and Western Europe) the association between PageRank and BHI is quite strong (ρ s = 0.86, 0.93, 0.76 respectively), in areas with mixed levels of prosperity such as Latin America, Asia and the Middle East, the correlations are slightly weaker (ρ s = 0.72, 0.86, 0.73) with the weakest correlations being in Sub-Saharan Africa (ρ s = 0.38).

FIG. 2 .
FIG.2.Rich-club organization and core-periphery structure of the mobility network.The richness parameter ρ(r) (Eq.(3)) a for the connectivity r = k out and b weighted out-degree Fig.S9.We see a monotonically decreasing trend of the strength of correlation as one moves from the core to the peripheral layers indicating that PageRank become a poor predictor of welfare for cities located more in the outer-layers of the mobility network.

FIG. 3 .
FIG. 3. Interplay between benefit (success) and cost (distance)a |μ/ν| and coefficient of determination r 2 for all city subgroups considered, using the model in Eq. 6 to fit the outflows of all cities in the subgroup.Black circles indicate development subgroups, and red triangles indicate geographical subgroups, while the size of the marker is proportional to the number of cities in the subgroup.Cities with r 2 > 0.8 and |μ/ν| > 0.1 are highlighted as having high relative flow dependence on BHI due to high model correlations and contributions from S µ j .All fits demonstrated significant improvement through the inclusion of the S µ j term, as evidenced by the corresponding AIC and BIC values (Table.?? in SI).b Model schematic, showing the variables involved in Eq. 6 around a central node i for which outflows Tij are being predicted.c Example predicted and true outflows from cities in the Australasia subgroup, showing significant improvement from including the BHI in the model fit.Only flows with T ij > 100 are displayed, for clearer visualization, and the line of equality is shown for reference.
a l R e a l E s t a t e I n v e s t m e n t G D P C r o s s -B o r d e r R e a l E s t a t e I n v e s t m e n t PR T ij

Figure
Figure S4: Correlation summary of mobility network centrality measures for clobal cities in the dataset, namely Pagerank (PR), Weighted-Degree (WD) and eigenvector centrality (EIG) versus socioeconomic indicators: BHI, Total real Estate investment, GDP and Corss-Border Real Estate Investment.Metrics are computed with various edge-weights to show the performance of flow (T ij ), distance d ij and flow × distance (T ij d ij ).The correlation metric used here is the Spearman's coefficient ρ s .
a l R e a l E s t a t e In v e s t m e n t G D P C r o s s -B o r d e r R e a l E s t a t e In v e s t m e n t PR Tij PR d 2.19 ij PR Tij × d 2.
a l R e a l E s t a t e In v e s t m e n t G D P C r o s s -B o r d e r R e a l E s t a t e In v e s t m e n t PR Tij PR d 2.19ij

Figure S7 :Figure S8 :
Figure S7: Correlation summary of mobility network centrality measures, namely Pagerank (PR), Weighted-Degree (WD) and eigenvector centrality (EIG) versus socioeconomic indicators: BHI, Total real Estate investment, GDP and Corss-Border Real Estate Investment.Metrics are computed with various edge-weights to show the performance of flow (T ij ), distance d ij and flow × distance (T ij d ij ).The correlation metric used here is the Spearman's coefficient ρ s .

Figure S9 :Figure S12 :
Figure S9: Correlation between BHI and PageRank (Spearman correlation coefficient is on the y-axis) in different clusters in the same core (having the same coreness value identified by k−core decomposition) where coreness is shown in the x-axis (from periphery to core left to right).k−core method identifies 2 dense clusters in the core of the mobility network with sizes 40 and 48, corresponding to North American and European cities as shown in Fig. S8 on the map.PageRank becomes less predictive for clusters of cities located in the outer layers of the mobility network.The numbers in each point represent the number of cities in each cluster.

TABLE I .
Role of mobility neighborhood for cities grouped by maturity Values of γ opt for cities based on their maturity, optimizing the Spearman's correlation coefficient between the actual success S i (BHI) and the estimator Ŝi Eq. (5).Size denotes the number of cities in each category.ρs (γ) and ρ p (γ) are the Spearman's and Pearson's correlation coefficients for a given γ.Note that Pearson's correlation coefficients are computed by taking the logarithm of both S and Ŝ.Development-Stage Size γ opt ρ s (γ opt ) ρ p (γ opt ) ρ s (0) ρ p (0) they are from Early Growth cities.Based on this we define an estimator for a city i's success

Table I
The role of the average success of neighbors S is the most prominent for Early growth cities, and practically non-existent for Mature cities (see Fig.S12for in-flow breakdown of Early Growth and Developing cities originating from national and international cities).Table II contains the same information for cities categorized by Geographical Region.Mature cities in Western Europe and North America display the lowest values for γ, whereas regions such as Asia, Latin America and the Caribbean-that contain a mix of cities in terms of their development contains the optimal value of the exponent γ (see Figs.S10 and S11), maximizing the Spearman correlation ρ s (γ), between the estimator Ŝi , and a city's actual level of success S i as quantified by the BHI which captures the city's maturity level.A clear pattern emerges, whereby we find a decreasing γ with increasing levels of development.levels-show intermediate values for γ.The highest values are seen for the Middle East region and Central Europe that are home to fast rising cities.Interestingly Sub-Saharan Africa shows the same trend as Asia, though it is an outlier, as it seems neither the Pagerank nor
19a l R e a l E s t a t e In v e s t m e n t G D P C r o s s -B o r d e r R e a l E s t a t e In v e s t m e n t B H I T o t a l R e a l E s t a t e In v e s t m e n t G D P C r o s s -B o r d e r R e a l E s t a t e In v e s t m e n t PR Tij PR d 2.19 ij PR Tij × d 2.19