## Introduction

Wealth and income inequalities are growing1, slowing development, economic growth, and technological progress2,3,4 while fostering radicalization and the advance of political populism5,6. These disparities have historical roots; unequal access to education, technology, and public services are self-reinforcing mechanisms by which economic inequality compounds across generations7,8. Less is known about how structural factors that influence inequality interact with one another. Two prime examples of such factors are social networks and geography.

Research on social networks emphasizes that social relations provide individuals with essential access to economic opportunities9. Social networks are claimed to maintain and even amplify inequalities when economic status plays a role in how social relations are established10,11. For example, a major micro-level mechanism for social-tie formation is homophily, the tendency for similar individuals to become friends12. Triadic closure, the phenomenon that friends of friends are more likely become friends13, compounds the effect of homophily in tie formation14. Since wealth is the one of the most significant boundaries to social relations in most societies15, these micro-level mechanisms can result in social segregation at the macro scale: groups with different socioeconomic status are separated from each other in social networks16. This kind of macro-scale network topology can lead to divergence of economic potentials between groups if access to resources or information runs through the network11.

Social networks are themselves embedded in geography, which itself has fundamental connections with inequality. The location of an individual’s home, for example, predicts a large share of their future economic outcomes17. A consequent divergence of outcomes across neighborhoods is observed even within relatively small geographical units such as cities and towns18,19. The local bias of social ties, by which individuals are more likely to connect with those who are close to them20,21, and the observation that social connections are less frequent across physical or administrative boundaries22,23,24, together suggest that the primary way geography influences economic outcomes is through its effects on social network structure. Indeed, spatially bounded social relations, limit both individual25 and collective prosperity26,27 because access to diverse resources provided by physically distant social connections is a key element for progress28.

In this paper we study how social networks and geography interact, and together what relationship they have with inequality outcomes. Despite the rich literature relating both features with inequality directly, there has to date been little empirical work examining this interaction, likely due to the difficulty in connecting data on spatially embedded social interaction and economic outcomes. To address this gap, we analyze an online social network of roughly 2 million individuals located in about 500 towns in Hungary. We relate the degree of fragmentation in town-level social networks to income inequality, finding a vicious cycle: higher social network fragmentation compounds income inequality over time. (From here on throughout the paper we also use the term fragmentation to refer to social network fragmentation.) Next, we describe how physical urban geographic features, which are rather static and unchanging in the short run, relate to fragmentation. Using a two stage model, we show that the relationship between a town’s geographic features and income inequality is mediated by social networks. We interpret our results as suggesting that when a town’s geography facilitates sorting and segregation, our models predict greater inequality.

Our empirical approach rests on the observation that cities and towns are prime arenas of social interaction29. Previous research in city science has devoted major efforts to explain social network structure with urban characteristics, distribution of individuals, and human behavior in cities30,31. For example, there is significant evidence that co-location is necessary for social ties to form and prosper32. Both home locations and urban activities are shown to cluster by income level, ethnic groups, and age and gender33,34,35,36. However, due to intensive mobility within cities37, the emergence of spatial communities in cities have not been observed on individual level networks38, only after aggregating connections, for example to ZIP codes in the United States26.

It has long been theorized in economics and social sciences that the relationship between inequalities and social networks in cities are mediated by physical space39. For example, following Alonso40, job-related information is shared in the city center, access to which is unequally distributed across the urban population. Social networks of the wealthy reinforce their geographic clustering around this interaction center41. Alternatively, self-isolation of the wealthy and powerful in peripheral neighborhoods or suburbs is another way in which social segregation manifests and economic information and opportunity is restricted42. In the urban sociology research, spatial barriers and boundaries are known to facilitate social segregation43. For instance, administrative boundaries such as school districts44 and physical borders such as railroads or rivers45,46 serve as landmarks that facilitate discrimination, the differential provision of public goods, and sorting. The emergent islands of segregation are difficult to bridge by social ties because poor and rich neighborhoods are typically located far from each other47.

To better understand how the structure of built environment relates to income inequalities through social relations, we use open source geographic data and develop three measures of urban segregation of towns: (1) the average residential distance from the town center, (2) the extent of spatial concentration of amenities in towns, and (3) the degree to which physical barriers divide residential areas. Each of these indicators are significantly related to social network fragmentation. Using a machine learning approach, we find that these geographic indicators are better predictors of social network fragmentation than other social indicators of segregation.

We then deploy these indices of urban geography as instrumental variables for social network fragmentation in a regression model predicting economic inequality. Falsification tests of the instrumental variable approach support our interpretation that the geographic topology of towns has a significant relationship with economic inequality via its relationship with social network fragmentation.

## Results

### Social networks and the dynamics of inequality

We first investigate the levels and changes of income inequality from 2011 to 2016 in all 474 Hungarian towns with at least 2500 inhabitants. We exclude capital city Budapest from the analysis because it is a unique settlement in several ways. It contains twenty administrative subunits which serve as weak social and political barriers—we do not observe this granularity in our social network data. It is also an order of magnitude outlier in population, density, and physical size and would introduce significant leverage in our regression models. The Hungarian Statistical Office provides binned data on personal income tax filings in each town in our sample including the total amount of income and the number of taxpayers in each bin (see Materials section). As an example, in Fig. 1a we compare the cumulative distribution of gross income across these bins in 2011 for a low (Ajka, in gold) and high inequality town (Gödöllő in dark blue), both having around 30,000 inhabitants.

We measure income inequality using the Gini index based on these bin distributions (see Supplementary Note 1) and then relate it to our measure of social network fragmentation at the town level. The cumulative distribution of income in Ajka (gold) is above the one of Gödöllő (dark blue), indicating lower inequality in the Ajka. We denote the Gini index of town i in year t by Gi,t, i.e. GAjka,2011 < GGödöllő,2011.

To capture social network structure within towns, we use data retrieved from a Hungarian online social network (OSN) iWiW, a once popular social media platform used by nearly 40% of the country’s population. Similar OSN data retrieved from other platforms (e.g. the Dutch OSN Hyves and Facebook) have been used to model income in geographical areas in the Netherlands and in the US26,27. In iWiW, we have access to the location of users at the town level and can analyze more than 300 million friendship ties the users have established by the end of 2011. Previous research demonstrated that geographical factors explain registration rates on the website48, that administrative and geographical boundaries constitute spatial borders on the iWiW network49,50 and found relation between network structure indicators and social outcomes such as the prevalence of corruption in towns51. A more detailed description of iWiW presented in Supplementary Notes 2 and 3, including its approximation for social and economic representativity and potential biases.

When studying social network fragmentation within a town, we consider only those links between iWiW users that are both from that town. We apply the community detection method known as the Louvain algorithm52. This method partitions the individuals of the network in town i into groups by optimizing a measure called modularity Qi that compares the density of edges within groups to the density across groups53. Mathematically, $${Q}_{i}=\mathop{\sum }\nolimits_{k = 1}^{{K}_{i}}\left[\right.\frac{{L}_{k}^{w}}{{L}_{i}}-\left(\right.\frac{{L}_{k}}{{L}_{i}}{\left)\right.}^{2}\left]\right.$$, where Ki is the number of communities for the partition and Li is the total number of edges in town i, Lk is the number of edges adjacent to members of community k, and $${L}_{k}^{w}$$ is the number of edges within community k.

Because Qi is highly dependent on the size and density of the network, following54, we scale it by the theoretical $${Q}_{i}^{max}$$ that would be achieved if all edges were within the communities. The ratio

$${F}_{i}={Q}_{i}/{Q}_{i}^{max}$$
(1)

for the town networks provides a good quantitative characterization of their fragmentation51. Here we use the values of fragmentation Fi observed for ties created by the end of 2011.

The structure of the social networks in the sample towns Ajka and Gödöllő is illustrated by a filtered, random sample of well-connected nodes from their social networks in Fig. 1. The precise filtering is described in Supplementary Note 2. The network in the relatively unequal town Gödöllő in Fig. 1b is rather fragmented and falls into loosely connected subnetworks compared to the network of Ajka, a town with low inequality, visualized in Fig. 1c. Figure 1d illustrates the positive correlation (Pearson’s ρ = 0.29 for all towns) between Gi,2011 and Fi meaning that income inequalities are generally higher in those towns where the social network is fragmented.

Turning to the dynamics of inequality, Fig. 1e illustrates the strong correlation (ρ = 0.9) between inequality in 2011 and 2016. We observe a slight increase in the overall level of inequality in most towns from an average Gini index of 0.474 in 2011 to an average of 0.484 in 2016 (significant according to a Mann–Whitney U-test, p < 0.001). There are examples of towns with both growing and falling inequality.

To analyze how network fragmentation is related to these dynamics, we apply the following ordinary least-squares (OLS) regression:

$${G}_{i,2016}=\alpha \times {G}_{i,2011}+\beta \times {F}_{i}+\gamma \times ({G}_{i,2011}\times {F}_{i})+{Z}_{i,2011}+\epsilon$$
(2)

where the coefficient γ of the interaction term measures the interaction effect of inequality and social network fragmentation. Zi,2011 refers to a matrix of control variables (population density and fraction of iWiW users in total population). Here β is the regression coefficient for Fi and the total contribution of fragmentation to income inequalities can be estimated from the partial derivative of Gi,2016 with respect to Fi using the formula

$$\frac{\partial {G}_{i,2016}}{\partial {F}_{i}}=\beta +\gamma \times {G}_{i,2011}.$$
(3)

Figure 1f presents the relationship between social network fragmentation and the change of town income inequality between 2011 and 2016 and plots β by levels of Gi,2011. We find that the interaction between inequality in 2011 and fragmentation has a positive and statistically significant relationship with inequality in 2016. However, the marginal effect of Fi informs us that social network fragmentation is positively related to future levels of income inequality only if the initial levels of inequalities are high. Fi has no significant relation with Gi,2016 at the lowest levels of Gi,2011. One standard deviation change in Fi is associated with 0.1 and 0.4 standard deviation change in Gi,2016 at the mean and maximum values of Gi,2011. This result provides empirical support to the theory that social networks can increase inequalities when individuals sort based on their initial endowments11. The estimates of Eq. (2) can be found in Supplementary Note 4.

Having established a relationship between social network fragmentation and income inequality, we now turn our attention to potential geographic drivers of such fragmentation, namely the topology of urban space. Divisions or inequalities in geography have long been considered fundamentally related to economic inequality46,55. Our goal is to better understand this relationship by observing how geographical factors relate to inequality through their relationship to social networks.

### The role of urban topology

To test the hypothesis that urban topology is related to income inequality via its relationship to social network fragmentation, we apply a two-stage least square (2SLS) regression model on income inequality. Though we cannot claim that the estimates we derive represent causal effects, our approach minimizes the risk of omitted variable bias56. We also exclude a variety of alternative explanations through a series of falsification and robustness tests46.

In the first stage of the 2SLS model, we estimate social network fragmentation using the formula:

$${F}_{i}=\delta +\gamma I{V}_{i}+\delta {N}_{i}+{e}_{i}$$
(4)

where IVi, short for instrumental variable, denotes our urban topology indicators to be introduced below, Ni is the fraction of the population of a town i on iWiW, and ei is an error term, assumed to be normally distributed.

The urban structure indicators are created using data from open-source geographic databases. This allows the replication of our measurements in other countries. The geographic features were observed in 2017, lagging our estimates of income inequality and fragmentation. As the geographic features change slowly over the course of many years, the risks of reverse causality are limited. Indeed, many manifestations of segregation and sorting by class were documented in Hungarian towns in the 1970s and 80s57. The following indicators are proposed to quantify three dimensions of spatial segregation, the details of which are described in the section on methods.

#### Average distance from the center (ADC)

Co-location is important for social tie creation and the probability of links decreases as distance grows49. Therefore, large distances between neighborhoods of towns can lead to fragmented social networks, because distant individuals are less likely to meet26. However, certain locations in towns facilitate the integration of distant individuals. For example, downtown is assumed to be and indeed functions as the major hub for social interaction in most towns and cities40. Therefore, to quantify the role of distance, we measure the average distance of randomly sampled neighborhoods from the center of gravity in towns. Because the form of urban polygons are rarely circular, and may even contain disjoint fragments of neighborhoods that are far from each other (e.g. Esztergom in Fig. 2a) the center of gravity might not coincide with downtown. In these cases the ADC index rather captures the large distances across neighborhoods than actual distance from the center. Nevertheless, we expect that social network fragmentation is higher in towns where ADC is large and lower where the index value is low.

Although we cannot test a causal effect of ADC on social network fragmentation, we do argue that reverse causality is not likely. City growth is a complex phenomenon depending on land use, regulations, economic attractiveness, and transport58. Hence, it is not likely that the presence of segregated social groups drives town growth and hence increases distances, especially not in the short or medium term.

#### Spatial concentration of amenities (SCA)

Individuals go out and interact in places that are not necessarily located downtown59. The spatial concentration of such amenities is related to the location of rich and poor in cities. Unlike in the US, amenities concentrated in the center of European urban areas have been found to attract the rich to and push the poor from central locations60. Consequently, when amenities are spatially concentrated, residents in peripheral areas of the town might be excluded from majority of social interaction61. However, the question how this concentration is related to social network fragmentation is still open, since amenities distributed evenly across neighborhoods can also facilitate interaction among peripheral neighbors and increase local cohesion to the detriment of overall connectedness in the town62.

To understand how the SCA is related to network fragmentation, we apply point of interest data (POI) that covers restaurants, bars, pharmacies, cinemas etc. To do justice between the two alternative expectations, SCA quantifies the concentration of amenities across its spatial groups defined by a density-based clustering algorithm. This measure takes high value if amenities are concentrated in few spatial clusters and low value if they are evenly scattered across spatial clusters (see description in the Methods section).

We find a significant positive correlation with SCA and network fragmentation (ρ = 0.253), which suggests that network segregation is higher in towns where amenities are spatially concentrated. Therefore, we falsify the alternative hypothesis and expect a relation between SCA and inequalities through network fragmentation. However, we cannot rule out reversed causality, since the location of amenities are based on demand that depend directly on local purchasing power.

#### Segregation by physical barriers (SPB)

Both built structures such as major roadways and railroad tracks and natural barriers like rivers are known to facilitate segregation in cities55. The effect of such barriers is thought to be an exogenous factor facilitating segregation. Because they can be considered exogenous, they have been used as instrumental variables to measure the effect of racial segregation on disparities in income46. This measure encodes the colloquial phenomenon that some neighborhoods are on the “wrong side of the tracks”. The effects of physical segregation on socio-economic outcomes are remarkably persistent. For example, neighborhoods in the eastern parts of postindustrial British cities have lower incomes today because they were less desirable places to live in the 19th century when the wind (blowing west to east) concentrated pollution there63. Negative externalities such as air pollution are known to cluster in poorer neighborhoods64. In our specific context, we expect that social networks are more fragmented in towns that are spatially segregated both because physical barriers decrease the probability of face-to-face interaction49 and because they facilitate sorting of new arrivals by providing clear boundaries to neighborhoods65.

Though we cannot demonstrate a causal effect of SPB on network fragmentation, reverse causality is unlikely because the barriers are either natural and unchanging or were planned and constructed many years ago. For example, the backbone of Hungarian transportation infrastructure was designed and built in the 19th and early 20th centuries66, after which very few new railroad tracks have been built. Major new road construction within settlements is rare, and rivers have not been redirected in recent decades.

For the purpose of comparing towns of various sizes, we scale the ADC, SCA, and SPB measures by the total residential area of towns46. Figure 2 presents these three measures using three pairs of towns. To illustrate the ADC measure we contrast Kazincbarcika, an industrial town in a valley with a concentrated residential area, with Esztergom, a town with two distinct populated areas. Kazincbarcika has a low ADC score compared to Esztergom. The SCA measure is lower in Ajka, an industrial town that sprawls along a major road, than in Szentendre, a historical town in which amenities (cafes, restaurants, museums, etc.) concentrate in the downtown area. Finally, the SPB measure is low in Veszprém: the nearby major roadway wraps around the town and its train station is positioned on the outskirts. In contrast, Kaposvár has a high SPB score. The river Kapos and the town’s rail link cut the settlement from east to west, while two major roadways intersect near the center of the town.

Results of the estimation specified in Eq. (4) are presented in Table 1 and confirm a significant positive correlation between social network fragmentation Fi and all ADC, SCA, and SPB dimensions of urban segregation. All three indicators capture different facets of potential geographic sources of social segregation. In Supplementary Note 5, we use principal component analysis to construct a composite indicator of urban topology. This composite indicator captures a multidimensional notion of geographic segregation (it is high when all three elementary indicators are high). Description, distribution, and correlation of control variables are described in Supplementary Note 6. Supplementary Note 7 illustrates that all three urban structure indicators and the composite index outperform alternative segregation proxies (measuring ethnic, religious, educational, political heterogeneity) in predicting fragmentation by applying a machine learning approach.

The second stage of the 2SLS estimation follows the formula

$${G}_{i,2016}=\alpha +{\beta }_{1}{\hat{F}}_{i}+{\beta }_{2}{X}_{i}+{\varphi }_{k}+{e}_{i}$$
(5)

where $${\hat{F}}_{i}$$ is the predicted value of fragmentation estimated from Equation 4, φk refers to county-level fixed effects and ei is the error term. Xi is a matrix of control variables including the level and change of foreign-direct investment, unemployment rate, population density, and distance to the closest border.

Results presented in Table 2 confirm that social network fragmentation, instrumented by urban structure indicators, is positively related with income inequality. This result represents strong evidence of our proposed relationship between social network fragmentation and inequality. It also suggests that urban structure is an important indicator of social network outcomes that coincide with inequality. The control variables suggest that densely populated towns have lower levels of inequality than sparsely populated towns, which is in line with previous findings67, and inequality in towns close to the border (which tend to be peripheral towns in the case of Hungary) is above average. Supplementary Note 8 reports complete regression tables of the second stages of the 2SLS models.

Model statistics in Table 1 suggest that urban topology indicators are strong instruments of social network fragmentation, confirmed by an F-test of the first stage regression. A Wu–Hausman test confirms that they are not significantly correlated with the second stage dependent variable: income inequality. With the exception of the SCA regression, the instrumental variable models provide better fit than OLS regressions using the original urban structure measures instead of fragmentation as confirmed by a Sargan test. Robustness checks reported in Supplementary Note 9 confirm that results are mostly stable against restricting the observations to larger towns. Supplementary Note 10 contains further robustness tests, to check whether the effects of urban topology on inequality are mediated by the fragmentation variable, we regress fragmentation and inequality on all explanatory variables from Eqs. (4) and (5). In these comprehensive models we find that SPB is the most robust IV and fragmentation remains a statistically significant predictor of inequality.

It should be emphasized that we cannot prove causal relationships with our modeling approach via robustness and falsification tests. However, comparing the robustness of the instrumental variables allows us to exclude a variety of confounding factors and helps us understand the spatial dimension of social network segregation and its role in income inequality. We also carried out a series of regressions to rule out alternative hypotheses following the falsification strategy of Ananat46. In these, we test the correlation of urban topology indicators with other factors that predict town segregation or other dimensions of inequality, for example the level of economic efficiency of a town, measured by business tax receipts (reported in Supplementary Note 11). We find that the SPB measure does not correlate with any other proxies of inequality we consider, replicating the previous finding of Ananat that division of physical space by railroads is a sound instrumental variable for the analysis of the effects of segregation on inequality outcomes. ADC and to a greater extent SCA do correlate with some proxies for inequality, but we note these proxies do not effectively substitute for the urban topology indicators in our primary models. We acknowledge that we cannot exclude all alternative paths of cause and effect between social network fragmentation and economic inequality.

## Discussion

In this paper, we demonstrated that the fragmentation of social network structure is positively associated with income inequality in cities and towns. Moreover, we have found that the relationship is dynamic—the interaction of fragmentation and existing inequalities predicts a significant growth in inequality in the future. The physical arrangement of a city’s residential areas, the loci of its social interactions, is also connected to social network fragmentation. We observe a tendency: if the urban fabric contains significant distances, physical barriers, or spatially concentrated amenities, social networks tend to be more fragmented. The relationship between geographic division and inequality manifests in this fragmentation.

Our analysis suggests how and why urban planning can be an effective tool to moderate inequalities in the long-run. While it has long been known that segregation is often an implicit goal of urban planning, for example, Detroit’s Eight Mile Wall, the barrier that long separated suburban New Haven from Hamden in Connecticut68, or the widespread phenomenon of gated communities69, our work suggests that even innocent design choices can lead to bad outcomes. Conversely, certain policies may facilitate mixing and block the compounding of inequality by fragmentation.

For instance, insuring access to places of interaction is known to improve emotional well-being in neighborhoods70. Yet evidence for the effectiveness of mixed-income public housing projects in reverting segregation is more limited71. Smart policy is especially important when cities are growing: urban sprawl in Beijing has generated significant segregation between economic strata and locals and newcomers72. Other work suggests that experimentation and innovation in urban planning is needed to foster the accumulation of good forms of social capital73.

We describe three urban topology indicators that capture different dimensions of social segregation in cities and can be ordered in terms of changeability. Average Distance from the Centre (ADC), Spatial Concentration of Amenities (SCA), and Segregation of Physical Barriers (SPB) each offer their own insight for policy. Physical barriers is the most robust instrument for social network fragmentation; however, this is the least changeable factor. Its policy implications offer guidance for where to lay railroad tracks and primary roads within cities. In contrast, the distribution of amenities, which can change relatively quickly is a less robust instrument for social network fragmentation. Yet it is more relevant to day-to-day urban planning because public planners make frequent decisions about zoning and building permits for amenities. These might influence the spatial dimension of social interaction and consequently the dynamics of inequalities.

We cannot prove the following story of cause and effect: that poorly designed cities fragment the social network and hence amplify economic inequality. There may be confounding variables that explain our results. Indeed, the long-term evolution of neighborhoods is a complex phenomenon including mechanisms and feedback loops that we can not evaluate in this paper. Nevertheless our observations give us the confidence to propose that the rise of inequalities in towns may be fruitfully blunted by wise urban planning. We hypothesize that improving access across neighborhoods, facilitating mixing within them, and supporting a more equal distribution of services can mend broken social networks and improve economic outcomes across the board.

## Methods

### Materials

Our data access to the iWiW online social network is controlled by a non-disclosure agreement. The data, besides other information, includes self-reported location of 2.8M users and their social connections reported on the OSN website. Access can be requested in email addressed to: lengyel.balazs@krtk.hu.

Town-level aggregate information including income distributions, population distribution according to school, age, ethnic and religious groups, population density, unemployment, distance from border and foreign-direct investment was collected from https://www.teir.hu.

Corine Land Cover (CLC) data of built up residential areas including continuous and discontinuous urban fabric according to CLC 2012 were collected from the https://land.copernicus.eu/pan-european/corine-land-cover website. Geographic data on the location of residential areas, rivers, railroads, and major roads was collected from Open Street Map https://data2.openstreetmap.hu/hatarok/. Data on POI listed as “amenities” was downloaded using the https://download.geofabrik.de/website.

### Methods

To calculate ADC, we randomly located points on the polygons of residential zones in the CLC database. The number of points in each town refers to its total population and, because we aim to create the measure reflecting on urban topology, the number of points in a polygon is a function of the polygon’s area. Based on this randomized spatial distribution, we estimated the center of gravity for the town and calculated the average distance of points from it, following:

$$AD{C}_{i}=\frac{\mathop{\sum }\nolimits_{p}^{P}{D}_{p,c}}{P}/{S}_{i}$$
(6)

where Dp,c denotes the distance of points p to the estimated center of gravity c out of P points and Si refers to the size of the town’s area. The value of ADC is small for compact settlements and is large in towns with remote population fragments.

To calculate SPB, we adapt the measure of the physical division of the residential areas of cities known as the Railroad Division Index46. We source data on residential-zoned areas in each settlement in the OSM dataset and cut the polygons by the rivers, major roads, and railroads in the settlement. Then, we calculate the resulting dispersion of its residential area across disconnected components:

$$SP{B}_{i}=1-\sum _{a}{({S}_{a}/{S}_{i})}^{2}$$
(7)

where Si refers to the size of the town’s area and Sa denotes size of area a after barrier dispersion. The value of SPB is small for settlements that are not divided by barriers and large for those where barriers disconnect large fractions of residential areas.

To measure how much spatially concentrated the amenities are in the town, we identify spatial clusters of POI by applying a DBSCAN algorithm with 500 m radius. This algorithm groups those amenities together that are in the close neighborhood of each other. The spatial concentration of the groups is then quantified with the probabilistic entropy of the size distribution of spatial clusters multiplied by minus 1:

$$SC{A}_{i}=\frac{{\sum }_{c}({p}_{c}\times \mathrm{log}\,{p}_{c})}{n(c)}/{S}_{i}$$
(8)

where c refers to spatial clusters and pc is the number of POIs in c over the total number of clusters in the town n(c). The value of SCA is high for settlements where amenities are concentrated in few spatial clusters and small for those where amenities are evenly.

These calculations have been made in Python (version number 3.7.2).

### Reporting summary

Further information on research design is available in the Nature Research Reporting Summary linked to this article.