Graph Representation of the 15-Minute City: A Comparison between Rome, London, and Paris

Abstract

We studied the structure of the 15-minute city by measuring the distances from the services on spatial graphs. While the concept of the 15-minute city is rapidly spreading, its operative definition can be of help for planning and understanding the possibilities of the general idea. For three European cities (Rome, Paris, and London), we developed a method to calculate pedestrian travel time to reach services for all the possible paths on urban graphs, finding that the 15-minute city generally has multiple connected components and that the services have not necessarily been part of it. This algorithm was used to to make a quantitative comparison between the cities, showing that Paris and London have a stronger 15-minute characterization than Rome. By generalizing the method, it was possible to define a 15-minute urban index, which quantitatively characterizes this city feature. The results seem to be promising because, at the cost of the massive use of computational time, a digital model for the city can be generated: a planning tool to simulate urban solutions and a rigorous criterion for evaluating how much a city can be considered a 15-minute city.

1. Introduction

The 15-minute city, “where locals are able to access all of their basic essentials at distances that would not take them more than 15 min by foot or by bicycle”, is a possible criterion for realizing sustainable urban regeneration [1,2]. Although Carlos Moreno’s model was preceded by other experiences (complete neighborhoods, 20-minute neighborhoods, and local life circle parameters) [3], the 15-minute city (hereafter 15-MC) has achieved popularity through its adoption as a policy in Paris, France [4]. Afterwards, both academics and professionals tried to provide a more accurate and quantitative definition of the general concept [3,5]. The theory of “chrono-urbanism” [6] has been studied as a way to improve the quality of life [7], and the 20-minute city model [8] has been analyzed as a possible strategy to shape smart cities [9].

While there is consensus on the relevance of the proximity to services to improve the urban environment, the services that should be taken into account are still being debated [10]. Recent analyses have suggested, for example, giving preference to grocery stores [11], while the C-40 network published the core principles of a 15-minute city: “residents of every neighbourhood have easy access to goods and services, particularly groceries, fresh food and healthcare” [12].

The recent COVID-19 pandemic motivated several studies on the proximity city and the need for its reorganization [13]. Still, the energy crisis due to post-pandemic events have reinforced the need for solutions to access services while consuming the minimum amount of energy [14]. Because the 15-MC is a possible approach, many authors have identified recommendations for its realization [15,16].

The analysis of the literature on this subject (see, for example, a review [2]) shows that a quantitative characterization of 15-MC is needed to implement the recommendations. For this purpose, many authors have used the urban graphs approach to calculate walking time and define proximity [2,7]. Walking time can be calculated from a census block [17] or from the geographic center of a block, while some authors suggested using travel sheds around utilities [8]. Moreover, the spatial patterns on graphs as a function of different variables can also be considered [7].

The necessity of directly calculating the 15-MC on a graph without using blocks, centroids, or spatial approximations is therefore evident. In the same way, it seems necessary to have a method to consider the inhomogeneity of services. To solve these problems, one can measure the shortest path on the graphs starting from the service and subsequently calculate the intersections of different graphs, as shown in the next sections.

In order to identify a generic t-minute zone, a buffer can be drawn on the map of a city: a circle or a polygon centered on the service with a radius smaller than the distance that a pedestrian can travel in t minutes, but GIS-based measurement tools can provide more detailed and usable results using the distances on the graphs [18]. Therefore, in the last few years, representation by means of urban graphs has been used to refine the concept of the 15-minute city [19], even if often the large amount of computer time needed to find all the shortest paths on the graphs has favored the choice of Euclidean measures rather than the distances on the graphs [19]. The Euclidean approach, however, does not take into account the detailed topology of urban road networks, which often shows a complex structure, where the shortest path to a service may take longer than t inside the circle. Venice, Italy, is perhaps one of the extreme examples of this complexity [20], but even modern cities that have spontaneously grown require tortuous paths to reach neighboring points in the Euclidean sense [21,22]. By identifying all the possible routes to a given service, the result is a capillary network for which it is possible to accurately measure the travel times and act locally to improve accessibility.

In order to transform the concept of the 15-minute city into an effective planning tool, a graph representation of the city is used [23,24]. This representation is traced back to Euler’s solution to the Königsberg bridge problem [25], which in recent times still provides with useful results and improves the understanding of the structure and functions of cities [20,26,27]. In this framework, it is quite simple to give a quantitative connotation of the 15-MC sustainability criterion; but, as noted in [19], the approach involves a large computational effort for the the search for the shortest path on the graph [28]. However, as the following paragraphs show, the possibility of parallelizing the algorithms and the availability of fast processors make the computation possible even for very large cities, i.e., for graphs with a large number of nodes.

This paper is organized as follows: The next section is devoted to defining the general t-minute city (hereafter t-MC) on an urban graph with respect to a given set of services. In Section 3, the theoretical results are used to compare the structures of Rome, Paris, and London. The following secion includes the results, in particular the introduction of a 15-MC index. Finally, the model and its possible developments are discussed together with its possible developments.

2. A Graph Model of the 15-Minute City

Graph theory has often been used to model urban environments, for example, to analyze traffic congestion using Poissonian [27] graphs or for the study of the topology of the London road network by means of Erdös-Rényi random graphs [29]. A planar graph $G(V,E)$ is a set of vertices V and a set of edges $E\subseteq V\times V$. The planarity is granted by Euler’s polyhedron formula $\left|V\right|-\left|E\right|+A=2$, where A is the number of the faces [30]. The nodes of G are the intersections of the roads, and the lengths of the edges are proportional to the travel time with a coefficient, which is the speed of the pedestrians.

When placing a service on the node $f\in V$, as shown in Figure 1a, it is possible to find the shortest paths to reach it from any vertex of the graph (Figure 1b) and therefore the corresponding travel time.

In Figure 1b, the (possibly overlapping) shortest paths starting from nodes 10, 15, and 30 to f are shown. By applying the shortest path search algorithm to all the nodes of the graph, we find all the vertices of G that are less then 15 min away from f, as shown in Figure 1c.

If two services $f_1$ and $f_2$ of the same type are available, a pedestrian can reach one or the other indifferently, as shown in Figure 1d. To generalize the procedure, $f^i=(f_1^i,f_2^i,\dots ,f_{N_i}^i)$ is the list of $N_i$ services of type i, and $C_k^i=(C_1^i,C_2^i,\dots ,C_{N_i}^i)$ are the nodes that reach $f^i$ in less than 15 min. The set of vertices that form a 15-MC with respect to services of type i is given by the union of all these vertices

$$C^i={\cup }_{k=1}^{N_i}{C}_k^i\subseteq V$$

(1)

If two services of the same type $f_1$ and $f_2$ are far enough, the intersection of $C_1\cap C_2$ could be empty because there are no points that reach both services in less than 15 min, and the corresponding graph is not connected. If the services $f_1$ and $f_2$ are close enough, the intersection is not empty, and the set of 15-minute points is larger than that accessing a single service. This simple observation also shows that if the 15-MC is calculated on the urban graph, one can immediately recognize the places where it is necessary to intervene to reconnect, for example, the two sections of the city. Finally, one can consider the case of two different services f and g which must both be reached in less than 15 min. In this case, Equation (1) can be used to compute the two sets of vertices that reach f and g in less than 15 min, and then the intersection is found by selecting only the points that access both, as shown in Figure 2.

It can be observed that, in general, the services do not necessarily belong to the intersection set. In general, when there are K types of services, the set of the 15-minute vertices is

$$C={\cap }_{i=1}^K{C}^i={\cap }_{i=1}^K\left({\cup }_{j=1}^{N_i}{C}_k^j\right)\subseteq V$$

(2)

Given the two graphs $G(V,E)$ and $G^{\prime }(V^{\prime },E^{\prime })$, $G^{\prime }$ is a subgraph of G if $V^{\prime }\subset V$ and $E^{\prime }\subseteq E$, and we write $G^{\prime }\subseteq G$. Therefore, if $G_C$ is the graph induced on G by the vertices C, then the 15-MC graph is the subgraph $G_C$ of the urban graph G [31].

Despite the simplicity of this formula, it can reveal hidden urban characteristics, such as disjointed components of subgraph $G_C$. The set C depends on the travel time t, the service matrix $\mathbf{s}$, and the travel speed v: the 15-MC of cyclists is different from the 15-MC of pedestrians. The graph of the 15-MC can be defined as

$$G_C=G_C(C,E_C;t,\mathbf{s},v)\subseteq G$$

(3)

From this equation, it is evident that there is no single 15-minute city, but each set of services defines a possible city. Moreover, we can see that by varying the parameter t, the area can be expanded to obtain a hierarchical map of the t-MC, as shown below. In summary, the definition (3), together with the algorithm to calculate it, can be used in real cases to verify if an urban area is a 15-MC and to find the optimal location of new services as well as corrective actions.

3. The 15-MC Graph and Urban Analysis

Equation (3) was used to check the characteristics of the 15-MC by comparing three European cities: Rome (Italy), Paris (France), and London (United Kingdom). The choice of the cities was based on their structural and historical homogeneity: a river as a physical division of the territory, the demolition of historical areas, and the expansion of weakly connected suburbs. The three cities also show similar evolution from theor ancient to modern urban layout.

Modern London expanded with the rapid growth of suburbs in the 1800s as a result of large population growth. Medieval Paris, partially restored in the 18th century and redesigned by Haussmann in the 19th century, expanded in the 1900s with the construction of new suburbs to cope with population growth. In the 19th and 20th centuries, some parts of the historic center of Rome were demolished to create large avenues. These cities seem suitable to exemplify and clarify the proposed method of analysis and enable comparison, which can be extended to any other city. The three cities were also chosen for their relationships with 15-MC. The transformation plan of Paris into 15-MC is one of the objectives of the current administration [4]; London is starting to move in the same direction [32]. Yet, the case of Rome is more complex due to the disorderly growth of the suburbs [33]. All the maps we used were taken from Open Street Maps [34]. The WGS-84 coordinates of the bounding boxes were $(41.7882,41.9952,12.3644,12.6281)$ for Rome, $(48.7947,48.9085,2.2467,2.4321)$ for Paris, and $(51.2473,51.6792,-0.4944,0.2774)$ for London.

As discussed in the previous section, we needed to know the vertices of the graph, their geographic coordinates, and the edges (pairs of vertices) connecting them. These data were easily obtained from Open Street Maps [34] using the OSMnx python [35] library. It is important to emphasize that the accuracy of the results is related to the accuracy and reliability of the OSM data [36]. However, while the scope of this study is mainly methodological, the OSM data perform well for demonstration purposes. In order to use the method for planning purposes, planners need to be equipped with accurate and reliable datasets that are generally available to city governments.

On the graphs, it is possible to identify all the sequences of adjacent edges starting from the closest node to the given service. This approximation is good enough because the paths between two nodes are usually short. However, in the case of very long streets (a feature that often characterizes urban sprawl [37]), to improve the model’s accuracy, it would be sufficient to insert additional nodes on the edges to segment them into shorter sections. The services selected for the analysis were pharmacies, post offices, and supermarkets: three services that correspond to the three primary needs of food, health, and administration, but any set of services can be chosen as parameters of Equation (3). The selected services are nonetheless significant due to their importance and diversity to clarify the methodological characteristics of the model. When assuming ≃1.22 m/s as the average speed of a [38] pedestrian, Equation (3) becomes

$$G_C\left(C,E_C;15\min ,(‘‘\mathrm{pharmacies}’’,‘‘\mathrm{post}\mathrm{offices}’’,‘‘\mathrm{supermarkets}’’),1.22\mathrm{m}/\mathrm{s}\right)$$

(4)

Rome, Paris, and London: A Comparison

To compare the 15-MC properties of the three cities, a 488 km² area was chosen inside a 22 km side-square. First, using the OSMx data, it was possible to identify the walkable streets with the edges of the graph and the intersections with the nodes, emphasizing that a road can also intersect itself by adding an extra node. The resulting graph is undirected because pedestrians can walk the streets in both directions.

The following step was finding the coordinates of the services on the map and placing them on the closest road by inserting an additional node, with a negligible positioning error. Finally, for each node of the same type of service, the set of points reachable in less than 15 min could be found. For the analysis, the slope of the roads or the ease of traveling, as suggested by some authors [39], was not considered as a first approximation, choosing instead the average value of 1.22 m/s for pedestrian speed [38], which is slightly lower than that used in [19]. After having calculated all the nodes that could be reached in less than 15 min, it is immediate to find the union of all these nodes sets according to Equation (1). It is worth pointing out that in this way, the 15-MC was calculated starting from services and not from houses.

Graph $G_C$ for the city of Rome is shown in Figure 3. It can be observed that while the city center is definitively a 15-MC, moving radially towards the periphery, this characteristic is progressively lost.

By comparing the properties of graph G with those of the 15-min $G_C$, information on the structure of the 15-MC was obtained. The G graph has $122,961$ edges, while the $G_C$ graph has $76,142$ edges. The $\left|C\right|/\left|E\right|$ ratio between the number of edges $\left|C\right|$ and the number of edges $\left|E\right|$ is 0.62; therefore, 62% of the pedestrian street network graph can be considered a 15-MC.

The few red zones in the central area are parks or archaeological areas, but moving towards the periphery, some inhabited urban areas do not belong to the 15-MC. By looking at the details of one of these areas, it is possible to identify the roads where reconnection interventions with the rest of the city are necessary. By way of example, Figure 4 shows a magnification of the area of Rome Magliana (WGS-84:$(41.8340,41.8682,12.4328,12.4822)$), where the inhabitants of the district cannot reach the services in less than 15 minutes.

The map of Figure 4 shows where to “mend” the urban fabric, for example by placing a missing service or by improving the connection. Still, the interventions do not necessarily have to be placed on the red areas, but rather the Equation (3) can be used as a diagnostic tool, positioning the interventions in a suitable place on the graph and repeating the simulation to verify the new 15-MC. The 15 min area is often considered one of the tools to reduce emissions and global warming [2], but to evaluate the effects it is necessary to carry out numerical simulations.

Repeating the same analysis for the city of Paris, the results are shown in Figure 5, where $\left|E\right|=192,633$, $\left|C\right|=171,369$, and the ratio $\left|E\right|/\left|C\right|=0.89$. Again, this number is affected by the presence of green spaces, so this ratio is only indicative. However, in the next section, a definition of this index that takes into account the possible biases is given.

Finally, the city of London has a 15-MC structure similar to that of Paris but different from that of Rome, as can be seen in Figure 6, with $\left|C\right|/\left|E\right|=0.82$. The overall comparison of the three cities shows that Rome is a 15-MC around its center, but this feature fades when moving towards the periphery. Paris, on the contrary, has a compact 15-MC structure with very few red zones corresponding mostly to green areas, while the center of London is more similar to that of Rome. Moving towards the suburbs, the 15-MC property does not break down, giving rise to a set of weakly connected 15-MC zones.

Table 1. Comparison between the number of edges $\left|C\right|$ and $\left|E\right|$ and their ratio for the cities of Rome, Paris, and London.

City	$\left|\mathit{C}\right|$	$\left|\mathit{E}\right|$	$\left|\mathit{C}\right|/\left|\mathit{E}\right|$
Rome	76,142	122,961	0.62
Paris	171,369	192,633	0.89
London	156,442	189,398	0.82

is a summary of these properties for the three cities.

4. Results

The analysis presented in the previous section provides maps of the 15-MC. Still, a more detailed analysis is essential to obtain a deeper understanding of the differences between cities and to define possible quantitative targets for urban organization. When selecting circles on the map and measuring the ratio $\left|C\right|/\left|E\right|$ within these areas, it is possible to obtain indications of the radial evolution of 15-MC. Thus, the definition of an index $\gamma (r,x_0;t,\mathbf{s},v)$, as a function of the radius r with respect to a given origin $x_0$, can be used to characterize the 15-MC and compare different cities or areas of the same city. Formally, the index $\gamma$ is given by

$$\gamma (r,x_0;t,\mathbf{s},v)\equiv |C(r,x_0;t,\mathbf{s},v)|/|E|$$

(5)

where $x_0$ is the center, and r is the radius. The center $x_0$ can be placed on a possible city center, consistent with the analyses of 15-MC, which confirm that the center tends to be more 15-MC than suburban areas [39]. The value of $x_0$ is the average value of the spatial coordinates of the services. For the city of Rome, this point is located at Esquilino Hill, very close to the place of foundation, confirming the urban structure’s historical radial development [40]. In Paris, the mean value of the services is near the Jardin du Luxembourg, while in London, it is at St. John’s Smith square, Westminster.

The plot of the values of $\gamma$ for different values of r with the parameters of Equation (4) is shown in Figure 7.

The index is still dependent on the green areas, the archaeological areas, or the brown fields [41], but as r increases, it very clearly shows the trend in the 15-MC characteristics. A possible generalization of the index takes into account the properties or weights $w\left(e\right),e\in E$ of each edge, such as the length of the path, the population density [39], or the slope of the streets

$$\gamma =\frac{{\sum }_{e\in E}w\left(e\right)}{{\sum }_{c\in C}w\left(c\right)}$$

(6)

and, therefore, Equation (7) becomes a special case where all $w(·)$ have the same value. If $w(·)$ is the population density, the edges of the parks and archaeological area have $w=0$, and the index $\gamma$ is not biased.

Finally, using Equation (3), it possible to draw a graph where the properties of t-MC with different values of t are highlighted. This use of the equation clarifies the potential of the proposed model because it allows hierarching t-MC urban properties. For this purpose, the algorithm in Equation (3) was used to compute the t-minutes graph with different values of t for the peripheral area centered on the Tor Bella Monaca district in Rome (WGS-84: $(41.8237,41.9066,12.5735,12.6988)$), which includes an internal and external part of the GRA, newly urbanized areas, and the campus of the University of Tor Vergata. The Tor Bella Monaca district was chosen as a paradigm of one of the many Roman suburbs created in the 1930s. The area is characterized by its remoteness from the center and the lack of public services and facilities. By progressively applying the method, each street was assigned a t-MC label indicating the level of connectivity with the rest of the city, as shown in Figure 8 for $t=\{15,16,\dots ,30\}$.

This map does not explicitly suggest to the planner where to modify the most disconnected edges but rather shows how to design an urban intervention and then to repeat the simulation to examine how the graph would transform.

5. Discussion

The proposed method generalizes the studies focused on the quantitative characterization of the 15-MC. The method requires a large computing capacity to find the graphs and a run time that increases with the number of services and the size of the area. On the other hand, the result is extremely detailed and can be used as a digital model of the city to test possible solutions for mobility, for the placement of services, and for planning [42], as well as to predict and measure their effectiveness.

Our analysis shows the convenience of computing the properties of the 15-MC starting from services and not from houses. This simple approach drastically reduces the computational time, giving detailed maps of the urban area. The availability of these maps is of great help for planning services for 15-MC [43], for example, by superimposing these maps on those of the emissions and measuring their statistical correlation. The centrality of the services is useful to rethink the urban layout of workplaces, especially due to the expansion of coworking places [44,45], while the reliability of the model can be easily verified experimentally by measuring the displacements of actual pedestrians.

The second relevant aspect of the model is that the 15-MC is calculated on a set of heterogeneous services, thus showing the possible conflict between two types of utilities on the graph. In this sense, the graph topology given by Equation (3) is also the map of possible space use conflicts.

The paths on the graph are the shortest paths, which are not necessarily chosen by pedestrians, and this is a possible limitation of the method. However, it must also be emphasized that the model is a sort of ground state for the 15-MC, while the development of the method should include other variables (for example, the slope or the dangerousness of a road).

Finally, the proposed tool is intended to provide indications on the places in which to intervene and to simulate the effects of an intervention. Further developments lead to the possibility of comparing different shapes of the 15-MC for different services. In fact, the 15-MC with respect to one set of services may conflict with the 15-MC with respect to another class of services, and at the moment there is no model that indicates how these different cities can interact and coexist. The 15-MC index can also be related to the population density, which may impact on the evaluation of the necessity to include an additional service or not.

Furthermore, due to the growing trend towards soft mobility and the use of electric vehicles, it is also interesting to investigate the different 15-MC for different means of transport and not only for pedestrian movement.

The possibility of labeling each street with its t-MC property also allows the exploitation of labels for the construction of a syntax model of t-MC space [46], following the line graph contraction method described in [26].

Finally, the results of this work contribute to urban planning assisted by numerical simulations. In fact, when placing a set of ideal services on the graph, it is possible to exploit the algorithm as an evaluation tool by simulating the city’s transformations.