We use data from OpenStreetMap (OSM, osm.org), a large participatory project to collect and share spatial information such as buildings, transportation networks and POI data. Just like Wikipedia, the development of OSM is collaborative such that users can enter and edit information. OSM is based on a network of individual nodes where each node is classified into certain categories regarding infrastructure or landscape such as street, building, or river. Furthermore, POIs are divided into several classes (public, health, leisure, catering, accommodation, shopping, money, tourism, etc.), which are themselves subdivided into tags. For example, OSM distinguishes bars from restaurants and fast-food outlets. OSM is well-documented and further information on tags and underlying data structures can be found in the OSM wiki pages (wiki.openstreetmap.org).

To carry out our various analyzes, we use data from the Filosofi⁷ system (French localized fiscal and social income). This INSEE database divides the territory into squares of 200m×200m, thus overcoming administrative boundaries. It provides, among others, variables such as the age pyramid of the inhabitants, their income and the year of construction of the buildings.

In order to analyze the city composition around POIs, we define aggregated categories of services, based on OpenStreetMap tags:

Details of the OpenStreetMap tags in each category are listed in Supplementary Table 1. For further details on OSM tags, we refer the reader to the OSM wiki pages for shops⁸ and amenities.⁹

Inspired by Knap et al. (2023) and the policy plans of the Mayor of Paris, we focus on the composition of cities driven by services accessibility in a walking distance of 15 min, which roughly corresponds to a 1-km radius at a 5 km/h speed as defined in the OECD accessibility framework of cities.¹⁰

Accessibility scores are an application of the two-step floating catchment area method (2SFCA; Luo and Wang, 2003). It has been first used to measure health services accessibility, but it can be applied to any sort of extensive variables like the number of amenities for instance. The idea behind 2SFCA is to measure for each service provider the surrounding demand. Then, as a second step, for each person (or place) asking for this service, we calculate the surrounding supply by considering that each service provider divides itself up on the previously calculated demand.

For each category (established on Section 2.1.3) or for each sub-type of amenities (OSM tags), we count the total number of items in each square in a grid (square of 200m×200m), which represents the supply S_j as defined in this section. This grid—named INSPIRE—comes from the Filosofi data set previously presented. We obtain, for instance, maps like in Figure 2A for restaurants or Figure 2C for schools in Paris. Then, for each category, we compute the 2SFCA score as shown in Figures 2B, D with, here, the number of restaurants and schools per square as the supply (and like for all computations, the number of inhabitants per square as the demand), respectively.

The figures depicting the number of restaurants and schools in Paris (Figures 2A, C) and the corresponding accessibility scores on the grid (Figures 2B, D) clearly show the importance of taking into account not only the total number but also accessibility measures of amenities. While the number of restaurants shows a strong trend toward the city center of Paris (arrondissement 1–4), the schools seem to be relatively well distributed among the city (Figure 2A). However, for schools (Figure 2D) the accessibility score clearly shows a concentration in the Passy district located in the South-West of Paris (16th), in the 5th/6th arrondissements (below the Seine) and in the 2nd/9th arrondissements (above the Seine, near the Opéra district). On top of the accessibility score for each category of amenities, we also calculate the accessibility score for housing and social housing thanks to the data provided by the Filosofi dataset.

In more detail, we note by j a square representing a supplier, while we denote by i a square consuming services (each square is both supplier and consumer). We also denote by k the index for the squares in the 1-km zone around j. We call D_k the demand (measured in inhabitants) in the square k, S_j(p) the supply in the square j, measured in number of amenities of category p, W_kj the permeability coefficient or distance weight function of demand from k to go to square j as defined in Equation 1, and ℙ_kj the probability that inhabitants in square k visit square j (e.g., ℙ_kj∝W_kj).

Here, we choose the following distance weight function, with d_kl the distance between k and l, being inversely proportional to the squared distance between two geographical units, also depicted in Figure 3:

The distance weights are calculated using the PySAL library (Rey and Anselin, 2009) using the DistanceBand method, a continuous weight corresponding to the distance from the POI. We only consider a radius of 1km around the POI in order to look at amenities reachable within 15 min of walking as described above. Hence, we only calculate the distance weights within a distance of 1, 000m.

We can define the aggregated demand 𝔻_j received by the square j.

From that aggregated demand taking into account the zone around square j, we can compute the supply per inhabitant and amenity category p by taking their ratio R_j(p).

And lastly, the accessibility indicator for the square i counted in number of services accessible in a one-kilometer radius per inhabitants, also denoted by 2SFCA_i score :

As depicted in Figure 4, the aggregated accessibility score CS_i as shown in Equation 6 is calculated by the following step-wise process given the demand and supply for each square as measured in the number of inhabitants and the number of its amenities per category, respectively [Figure 4 (0)].

Knap et al. (2023) derived the amenity weights from the trip distributions calculated based on travel patterns in the their data. Our data sets did not allow for such calculations. To make up for this shortcoming, we have put forward the following hypothesis: the rarer a type of amenity, the more important it is. Schools or hospitals are less numerous than restaurants, because their capacity and importance is greater. We therefore calculate, for each type of amenity (i.e., for each category), the ratio between the number of items for the category N_p on the total number of amenities N and give an amenity weight opposite to this frequency.

The idea is that a scarcer amenity has greater importance. This amenity weight shows the inverse probability of finding an amenity of a certain category within all amenities in the area. Nevertheless, this hypothesis tends to overestimate certain amenities, such as museums, which are less numerous than some other categories but not necessarily more important to individuals.

This gives the aggregated 2SFCA_i for cell i, denoted by CS_i:

where

is the min-max normalization of the accessibility score 2SFCA_i(p) of cell i for amenity of type p ∈ P.

Regression analysis produces econometric results for Paris and the Petite Couronne that are comparable among different cities or regions. To do so, we set up a methodology as close as possible to the one introduced by Knap et al. (2023) for Utrecht.

The spatial weight matrix¹¹ makes it possible to account for the geographical relationships and influences that exist between the different units in the database. There are two types of weights: contiguity weight and distance-based. Given the characteristic of Paris and our database (each unit is a square) we use only the first type. Three popular types of contiguity weights are called bishop, rook or queen weight as they take into account the adjacent squares of a grid depending on the figure's possible moves on a chess board (Loonis, 2018). As our dependent variables are demographic and social and in a highly urbanized context, the queen contiguity weight is justified based on the handbook (Loonis, 2018) stating that “the neighborhood in the sense of contiguity is often used to study demographic and social data, in which it may be more important to be on either side of an administrative boundary than to be located at a certain distance from one another.”

One way of checking whether a particular weight is relevant to a particular phenomenon is to calculate the Local Moran Index (LISA; Li et al., 2007). This indicator of spatial auto-correlation makes it possible to check whether a phenomenon is distributed randomly or, on the contrary, according to the spatial interactions between each unit. If it is close to 1 (resp. −1), there is a perfect spatial auto-correlation (resp. dispersion). We use here the local Moran Index in order to see where differences or outliers are located instead of receiving a single value from the global index which represents a summary of the relationships on the map.¹²

After calculating the local Moran index for our model on the accessibility score with queen weight for Paris, we find an index equal to 0.95 and a p-value equal to 0.01. This means that there is a strong spatial auto-correlation and that the hypothesis of a random distribution of the aggregate accessibility indicator (H₀ : random distribution) can be rejected at the 95% threshold which confirms our choice for the queen weight. As for the tests performed on Paris, we obtain for the Petite Couronne a Moran index that is very close to 1 (0.97) and a very low p-value (0.001) which leads us to choose a queen weight for these regressions.

For our regression analysis, we use a model called Spatial AutoRegressive with additional AutoRegressive error structure (SARAR; Kelejian and Prucha, 1988; Anselin and Florax, 1995). We chose the model as the best fit for our purposes among several possibilities, for more information we refer to the Supplementary material. To implement our model we use the spreg library which computes this model using the generalized method of moments (GMM).¹³

To select our model we perform a Lagrange-Multiplier (LM) test which successively tests for the presence of spatial lag (robust and non-robust) (H₀ : p = 0), the presence of spatial error (robust and non-robust) (H₀ : λ = 0) and the joint presence of spatial error and spatial lag (H₀ : p = λ = 0) (Anselin, 1988). As stated by Anselin et al. (1996), the LM test is a “simple diagnostic test for spatial dependence.” Here, λ corresponds to the spatial correlation effect of errors (spatial autocorrelation) and ρ is the endogenous interaction effect (spatial autoregressive). Moreover, the p-values obtained during the Lagrange-Multiplier tests are very close to 0, which confirms our choice of the SARAR model as the appropriate econometric model, see also Table 1.

We have chosen to take into account heteroskedasticity to avoid errors in the significance of the coefficients. As in statistics, heteroskedasticity occurs when the variance of the residuals depends on the value of the variable of interest, i.e., the variance of the residuals decreases with the variable of interest. For example, as we analyze accessibility to amenities, we have to take into account that a person with a higher monthly income has a larger choice and thus, larger variability, of amenities than a person with a low income.

The results are presented in Table 3. Please see Section 2 in the Supplementary material for more details.

We run the regression of our aggregated 2SFCA on different socio-economic variables based on Filosofi data. The variables we work with are summarized in Table 2 below. We do not have some of the variables used in Knap et al. (2023), notably percentage of people receiving unemployment benefits, percentage of migrants, and distance to the nearest transport.

Complementary to the regression analysis, we applied hierarchical clustering being an unsupervised method to create a typology of neighborhoods based on the accessibility measurements. The idea is to cluster on the different categories' accessibility as defined earlier, added to the accessibility of housing and social housing, to then define the main functions of different neighborhoods. To conclude, we correlate those functions with the socio-demographic characteristics of the population that inhabits the found clusters.

We carry out a clustering on accessibility measures on the aggregated categories we defined first (and not on the whole big dimensional space, for computation time reasons). We use MiniBatchKMeans clustering (Sculley, 2010). To choose the number of clusters, we carry out an elbow method with the distortion score as shown on Supplementary Figure 1: it suggests taking five clusters. See Section 3 in the Supplementary material for more details. MiniBatchKMeans is preferable because it is faster and also because it does not aggregate continuous squares such as AgglomerativeClustering (Nielsen, 2016). This particular fact allows to identify similar neighborhoods in different cities. The clustering has been done on around 2,000 squares for Paris and around 14,000 squares for Petite Couronne.

The results of the regression analysis for Paris, as listed in Table 3 (left), indicate that the higher the percentage of the population below 17 years old or above 65 years old, the lower the CS_i score. This outcome may seem a little surprising but is consistent with the results found in Utrecht (Knap et al., 2023). One explanation could be that families and oldest households prefer to live in less “dynamic” areas like the 15th arrondissement. These neighborhoods are quieter during the day and at night, i.e., there are fewer bars.

However, the percentage of households living below the social minimum, the percentage of buildings built before 1945 and the average income are not significant at the 5% level regarding the results of the regression analysis. This indicates that, in Paris, accessibility to essential amenities cannot be explained by the poverty rate of a neighborhood and hence, economic criteria are not useful to predict the level of accessibility to essential amenities in Paris. With regard to the buildings built before 1945, this can be explained by the presence of almost all buildings dating from the nineteenth century in Paris. This also explains the high p-value of the percentage of buildings built after 1990. Because of the homogeneity of density in Paris, there is no relationship between our metric and the density of the neighborhood, in contrast to what the INSEE study of 2017 seemed to indicate throughout France as a whole (Trevien, 2017).

We have seen that in Paris, young people under 17 have relatively less access to services than adults under 65 ceteris paribus. But while socio-economic-criteria do not shape inequality in accessibility to services, for the Petite Couronne we observe that the coefficient of %_soc.minimum is significant and negative: the more households living below the minimum social threshold, the lower the accessibility score, as listed in Table 1 (right). If Paris is a relatively homogeneous city in terms of socio-economic level, then the poorest households will tend to move to the outskirts of Paris where the cost of living is lower. Performing the analysis on the Petite Couronne and not only on Paris therefore allows us to take into account socio-economic inequalities that are not noticeable within Paris.

In addition, the coefficients on the percentage of buildings built before 1945 and after 1990 are significant: geographic units containing more buildings built before 1945 and after 1990 have a better accessibility score to services than units containing buildings built between 1945 and 1990. This could be a consequence of the urban policies implemented during the Trente Glorieuses period,¹⁵ aiming to respond to the post-war housing crisis: in 1953 with the Courant plan¹⁶ these policies, based on the principle of zoning,¹⁷ resulted in the construction of large residential areas on the outskirts of Paris far from shops and services. Moreover, the positive relationship between the percentage of buildings built after 1990 and the CS_i metric could be due to the presence of this type of building in freshly renovated neighborhoods and more office spaces. Eventually, we observe that geographic units containing more buildings built after 1990 have a relatively poorer accessibility score than units containing pre-war buildings: this could reflect the disparities between Paris and its suburbs observed in Figure 8, as the majority of Parisian buildings were built under Haussmann.¹⁸ Paris is a fairly old city whose architecture is fairly well-protected by law: most of the Haussmannian districts are protected, and there has been little destruction since the 1980's. For more details, see Clerval (2016). It can also be noted that the coefficients of %_residences and density are significant: the more houses rather than apartment blocks are located in an area, the lower the accessibility score. In addition, the more densely populated an area, the higher its accessibility score.

Regarding the clusters obtained for Paris as depicted in Figure 11, we can clearly differentiate various areas. The green cluster (P-commercial) corresponds to the commercial function previously identified. Indeed, with a closer look, we see that it matches les Halles (1st arr.), well-known for its shopping center, and the Opéra neighborhood (9th arr.), well-known for its shops. The orange cluster (P-cultural) seems more centered on cultural amenities compared to the green one, that's why it includes Le Marais (3rd arr.) and Saint Germain-des-Prés (6th arr.), two neighborhoods with plenty of art galleries, and the Quartier Latin (5th and 6th arr.), which is not only an educational center but also a cultural one, as shown by (Apur, 2023). Those impressions are quantitatively confirmed as shown on Supplementary Figure 2. Here, we can see that the clusters P-commercial and P-cultural are standing out for the overall amenities' accessibility, though cluster P-commercial is more inclined toward fashion, food shops, restaurants and supply shops whereas cluster P-cultural has higher accessibility to culture and arts and to education.

The external cluster is being divided into three clusters : the red (P-border) one which corresponds to the border, the blue (P-intermediate1) one and the violet (P-intermediate2) one. The cluster P-border stands out. Its geographical location is on the border of Paris, and it is the poorest in terms of accessibility apart from social housing. Multiple explanations can be given to that cluster. At first, it could be an artifact due to the arbitrary limit of the data to Paris itself. However, historically the first real development of social housing in Paris - the so-called Habitat Bon Marché, literally cheap habitat—occurred precisely in those areas at the edge of Paris, as shown in a study by the Atelier Parisien d'Urbanisme (APUR; (Arènes et al., 2019)). This historical fact (Stèbè, 2022) could explain the importance of social housing on the border of Paris (excluding the edges of the Bois de Boulogne, 16th arr). Also, Paris and its suburbs have been separated by the périphérique, an important ring road around the city. So the lack of amenities can also be the physical effect of that ring road. However, this cluster can also be found on some edges of the Seine and on some edges of the cimetière du Père Lachaise (20th arr) in the North. To conclude, the P-border cluster is probably the mixed consequence of being on the edges (of Paris, of the Seine, of parks) and the history of Parisian planning where the first social housing have been in place on those edges. This history triggers socio-economic consequences, comparatively isolating poorer communities from the rest of amenities.

The comparison between clusters P-intermediate1 and P-intermediate2 is easier. Apart from social housing and housing, P-intermediate1 has better accessibility in every aspect, even though it is still behind P-commercial and P-cultural. Regarding the socio-demographic description of each cluster, as shown in the different violin plots of Supplementary Figure 3, the P-border cluster is clearly the poorest cluster by looking at the percentage of poor households and the mean standard of living, followed by P-intermediate2. This relative poverty is probably partly due to the fact that the red cluster includes more vulnerable families, like single-parent and very large households. We can also see that this cluster is home to mostly renters. Social housing, renting, presence of socially difficult situations and poverty therefore co-occur.

The cluster P-intermediate2 seems to host a similar amount of poor people, but the mean income is lower. This cluster is probably home to a small middle class. We can also look at the fact that it is hosting the denser squares, which is corroborated by the geography of the cluster : the 13th, the 15th, the 19th, the 20th are the most populated arrondissements. By contrast, P-commercial and P-cultural are the least dense and the richest clusters. In particular, P-cultural one is hosting the richest square in all Paris. Barplots describing the distribution of the aggregated accessibility score and of the socio-economic variables are shown in Supplementary Figures 2, 3.

We perform the same clustering as for the city of Paris for the whole region of the Petite Couronne, at first including Paris as depicted in Figure 12 and then without Paris in order to see if the data on Paris introduces a strong bias, see Figure 13.

Regarding the complete Petite Couronne with Paris (Figure 12), we quickly find back the same center (in green, PCP-center) as in the cluster analysis of Paris (P-commercial and P-cultural), showing that it does stand out, no other center seems to match it. Some other city centers can be identified in the violet (PCP-intermediate2) cluster, which is the cluster of Paris “middle ground” neighborhoods, P-intermediate2. Globally, the same city centers are standing out as in the map obtained with the aggregated accessibility score and highlighted in Figures 9A, C. They are for instance Reuil-Malmaison, Saint-Maur, Montrouge, Boulogne-Billancourt, see also Figure 1A. The department of Seine-Saint-Denis in the north of the Petite Couronne seems to be the only territory without any well-equipped city center. In this part of the region, there is the largest part of less equipped clusters (orange, PCP-sparse) and red (PCP-extended-border). The PCP-sparse cluster is the least equipped of all clusters (even in terms of housing and social housing). This cluster is probably representative of a suburb (banlieue pavillonaire) with low density of housing and amenities accessible by walking. On the contrary, the blue (PCP-social-housing) and red (PCP-extended-border) clusters are denser, with more housing and social housing, and are more compatible with the 15-min city because more amenities are accessible by walking. Barplots describing the distribution of the aggregated accessibility score and of the socio-economic variables are shown in Supplementary Figures 4, 5.

Clustering on the Petite Couronne excluding Paris essentially finds the clusters predicted by the aggregated accessibility score as depicted in Figure 13. The orange cluster (PC-city-centers) stands out from the others: access to amenities is particularly high, but there's no particular socio-economic profile. However, it seems to match the city centers of the suburban cities.

The green cluster (PC-social-housing) stands out particularly strongly from the others: it has a younger, poorer population. Access to social housing is particularly high, and access to amenities is relatively lower than in the other clusters. Both clusters called social-housing correspond to each other, the green cluster (PC-social-housing) in Figure 13 matches the blue cluster (PCP-social-housing) in the previous clustering analysis, depicted in Figure 12. Barplots describing the distribution of the aggregated accessibility score and of the socio-economic variables are shown in the Supplementary Figures 6, 7.