Revisiting the coupling between accessibility and population growth

A central point in urbanism is the relation between accessibility and population growth rate. In other words, the question is to understand and to quantify the effect of a growing connectivity—or improving accessibility—on the population growth of cities inside an urban area. A large body of literature discuss the interaction (and coevolution) between transportation infrastructure and growth, and suggest that an increase in population (and/or jobs, land use, development etc) is expected for cities that are affected by this improved connectivity (see for example [1–6]). In general, the local population density is strongly affected by economic factors [7], and it is expected that the population growth rate will be larger in high accessibility areas. The largest part of evidences to support this fact comes from econometric approaches, starting with the seminal paper on accessibility [8]. Accessibility can be measured by different variables [9–13] and quantifies in general how easy it is to go around. Using population density as dependent variables, and accessibility measures amongst the explanatory variables, a regression analysis allows to determine the relative impact, or explanatory power of different transport modes [14] or of the temporal component of a given mode [15–18]. However, these approaches are exclusively empirical and clear theoretical foundations on accessibility and its link with population growth are missing despite their critical importance for identifying the principles that govern urban expansion [19] and the evolution of infrastructures [20]. In several models of urban dynamics (e.g. in land use transport interaction or LUTI models), accessibility is used as a key factor that determines growth [21, 22], but here also there are no clear theoretical justifications.

An important merit of the accessibility measure introduced in the paper [8] is its predictive power with respect to land use development: the development ratio of an area (defined as the ratio of the actual development of the area with respect to its probable development) depends as a power law of the accessibility with an exponent of order 3 [8] for the urban area of Washington and for various types of accessibilities. These striking results constitute the basis for accessibility studies, and triggered a wealth of studies and works in econometric regression analysis that are based on gravity law measures. In particular, it has been claimed [23] that gravity-type accessibility measures have the largest explanatory power, performing better than eight different types of accessibility measures. In several cases, the regression analysis is performed in terms of population density, and not in terms of used land [14, 17, 23]. In [17], using several measures of potential accessibility, it is shown how in the Netherlands the development of the railway network had a positive effect on population growth, especially at the beginning of the 20th century when industrialization took off. A similar study is carried out in [14], where the effect of the road and the railway networks on growth are studied for Finland in the period 1970–2007. The results of the study point out that in Finland the population distribution is mainly concentrated in areas with high road-based gravity accessibility. The largest statistical significant effect for the railway network is obtained when the time to the nearest station is used in combination with the potential accessibility by road network alone. In particular, the conclusion of the study is that railways had a positive effect on growth mostly at a local level, and mainly in the 1970s and in the period 2000–2007. Other studies with simpler accessibility measures but with more elaborate econometric analysis can also be found. For example, in the study [15], the authors study the impact of interstate highways on the growth of cities in the US between 1983 and 2003 and, using a two-stages regression analysis, concluded that a 1% increase in the stock of highway of a city causes about 0.15% increase of its employment, over the 20 years period considered. Along these lines, Mayer and Trevien [18] focused on the Paris metropolitan area and evaluated the impact of the Réseau Express Régional (RER) on growth in the period 1975–1990. Using various regression analysis, they found a strong impact of the RER on the number of jobs (between 7% and 11%), but did not find a significant impact on population density. The impact of the RER in the Paris metropolitan area was also studied in [16], where instead a positive impact on population density is found: with each additional kilometer a municipality is located closer to a RER station, employment increases by 2% and population increases by 1%. These effects are considerably stronger when a municipality is located less than 13 km from a RER station. In this case, the growth of employment and population is found to be of 12% and 8% per additional kilometer.

Most of these studies are based on a regression analysis of a quantity such as the population with accessibility measures. A first remark is that the impact of accessibility that is found in recent papers is usually very weak, and the effect difficult to exhibit from empirical observations, in sharp contrast with the original observations in [8] and pointing to the need for an explanation of this apparent discrepancy. Another important point is the lack of theoretical foundations and the lack of even a simple toy model that could point to the type of relation between these different variables. The recent availability of new sources of data allow us to envision the construction of such models and to test their predictions against empirical observations [24]. Here, we will present an analysis on the evolution of population of the 10 largest urban areas in France from 1968 to 2014 and we show that it is the accessibility variation that impacts the growth rate, an effect that decays relatively quickly in time and in space, as we show for the Paris urban area. Also in order to observe this effect we have to focus on the subset of cities that experienced a very large accessibility variation. This led us to propose a simple model able to explain empirical observations about the population growth.

2.1. The dataset

We will use data available from the National Institute of Statistics and Economic Studies (INSEE) [25]. The dataset contains the population of each municipality in France for the years 1968, 1975, 1982, 1990, 1999, and all years from 2006 to 2014. The number of municipalities is not the same every year, due to merging and separation of administrative units. We will consider here the specific set of the 2301 municipalities that belong to the urban area of Paris (i.e. the region Ile-de-France, see the figure SI1 (https://stacks.iop.org/JPCOMPLEX/01/025002/mmedia) for a map) that are present in the INSEE list for all the years in the database. For each municipality i, we consider its population P_i(t) for a given year t, and its population P_i(t + δt) for the following available year. We define the growth rate at time t, g_i(t) as

$$\begin{equation}{g}_{i}\left(t\right)=\frac{{P}_{i}\left(t+\delta t\right)-{P}_{i}\left(t\right)}{{P}_{i}\left(t\right)\delta t}.\end{equation} \tag{ 1 }$$

This definition of growth rate is inspired by Gibrat's rule of proportionate growth [32]. This is the standard definition for growth rates, and we observe that they depend very weakly on the population (see SI figure S2). It takes two years to define a growth rate and from the 14 years of available population data, we can compute 13 growth rates.

2.2. Homogenization in large urban areas

We first focus on the time evolution of growth rate in large urban areas such as the Paris urban area (the region Ile-de-France, see SI for a map and more details) and we will define the aggregate growth rate of a region, or for a generic collection of municipalities, as

$$\begin{equation}{g}_{\alpha }\left(t\right)=\frac{{P}_{\alpha }\left(t+\delta t\right)-{P}_{\alpha }\left(t\right)}{{P}_{\alpha }\left(t\right)\delta t}.\end{equation} \tag{ 2 }$$

where P_α = ∑_i∈αP_i is the total population of the region α. In the case of the Paris urban area, there are eight 'departments' (Paris proper, Seine-et-Marne, Yvelines, Essonne, Hauts-de-Seine, Seine-Saint-Denis, Val-de-Marne, and Val-d'Oise), over each of which we aggregate cities. In the left panel of figure 1 we report the observed aggregate growth rates for all the departments in Ile-de-France in the considered time window. Despite the fact that the region has shown a marked growth in the time window considered here with an increase of its total population from about 9.5 millions in 1968 to 12.4 million in 2014, the city proper of Paris (called 'intra-muros' in French) displays the opposite trend: in 1968 Paris hosted about 2.59 million people and in 2014 about 2.22 million inhabitants. This is a phenomenon of sub-urbanization where Paris proper has growth rates lower than those of surrounding departments, and points to urban sprawl in this urban area—an expansion of human population away from central urban areas, that has become prominent in urban areas since the 90s (see for example [26–29]). Another significant feature of figure 1 is the fact that aggregate growth rates of different departments converge to the same rate value, typically between 0% and 2% yearly. This observation is a first facet of a process of homogenization, according to which peculiarities of different regions (or cities) have become less relevant as years go by. Homogenization in this Ile-de-France region can also be observed visually in figure 1 (bottom) where we compare maps of growth rates in the period 1968–1975, versus the map of the period 2006–2014.

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We also show in figure 1 (top, right) the relative standard deviation (RSD) of the growth rates distribution, defined as the standard deviation of the growth rates of all cities inside each departments divided by the average. This variance displays a decreasing behavior with time which implies that the aggregate growth rate for each departments is always closer to the value of the growth rate for a typical municipality inside the region.

These results show that for municipalities belonging to the same urban area an overall homogenization trend where growth rate fluctuations disappear in time and in space.

2.3. Quantifying the coupling accessibility-growth rate

We now focus on the coupling between accessibility and population growth rate. We considered different measures of accessibility but we will show here the results obtained for the accessibility used by Hansen [8] which is defined below. Our results are however valid for other measures such as the inverse time to reach the closest train station or the inverse time to reach the center of Paris for example (see the SI, section 4 for details and discussions about various measures). The Hansen accessibility measure integrates the coupling between the infrastructure and the land-use component in a single expression which is expressed as [8]

$$\begin{equation}{A}_{i}=\sum _{j}\frac{{S}_{j}}{{T}_{i,j}^{\tau }}\end{equation} \tag{ 3 }$$

where the sum is taken over all areas j that can be reached from the ith area. Such an expression takes into account the land-use component S_i which characterizes the activity of the area (for instance the population, or the number of jobs), and the transportation component T_i,j which is the travel time between areas i and j. In our paper the land-use component is always the population of the ith municipality, and the travel time T_i,j contains all information about the transportation network (and its evolution along the years). The exponent τ weights how much the travel times between the areas impact on accessibility and is taken here equal to one. We have used this Hansen potential accessibility because it is one of the most used in the modern literature, especially in quantitative geography studies [14, 17], and because has been found to be the one with the largest explanatory power [23]. Instead of performing a regression analysis as it is usually done in most studies (for example [14–18]), we will exhibit directly the effect of accessibility on growth rates. We first study the impact of accessibility on growth rates and we observe a very weak dependence (see SI, figure S3), showing that there is virtually no impact. Such negative result is actually in line with most studies where only a very weak effect was found, in contrast with the impressive results of Hansen [8]. As expected from the previous section, growth rate fluctuations are usually small and we do not expect very important effects. In particular, in the case of the Paris urban area, we can understand the main reason behind the failure of accessibility to account for growth rates (see SI, figure S4 for more details and maps of growth rate and accessibility): most accessibility measures considered are essentially related to centrality, i.e. how close the municipality is close to the center (Paris here) and to denser areas. Growth rates do not have however this structure at all: on the contrary, we observe in the recent history, an inverted structure where further municipalities have a larger growth rate, which signals a suburbanization of the area.

These negative results lead us to consider cities that experienced a variation of accessibility. This seems reasonable as an improvement of transportation mode in a given area can indeed trigger a wave of newcomers. Empirically, if we first consider all cities together, we do not observe any significative trend. We then focused on municipalities with the largest accessibility variation such as the top 1% of cities who display the largest accessibility increase in a given period of time. In figure 2, we show the growth rate for the top 1% and for all cities for different time periods. We observe that cities with larger variations of Hansen accessibility display indeed a significantly larger average growth rate compared to the average (see figures S5 and S6 in the SI for a discussion with other accessibility measures). This effect is in particular present for the periods 1975–1985 and 1985–1995, while for the period 1995–2005 the effect is less significant. This impact identified in figure 2 is stable and significant only if we consider the municipalities that witnessed the largest accessibility variations (the cities in the top 1%–2%). In the SI (figure S7), we show that the difference between the growth rates of all cities and the selected subset of cities with a large variations becomes negligible as soon as more than 2% of the cities are considered. The effect observed here seems therefore to be relevant only for a small fractions of cities. This is probably related to the fact that in the periods considered here, the transportation networks did not evolve dramatically: for most of these periods, only 5% to 10% of new sections are added to the network (see figure S8 in the SI). In addition, we observe on figure 2 a rapid decrease of the difference between the growth rates for the two groups: for the municipalities that experience a growth in accessibility in the period 1975–1985, we observe that after 1990 their growth rates are similar to the ones of all the other municipalities. The same happen for the municipalities that experience a growth in accessibility in the period 1985–1995, for which there are no significative differences after 2010 approximately. These results therefore point to two important conclusions:

• (a)  
  First, instead of accessibility, it is the accessibility variation that acts as a control parameter on the population growth rates.
• (b)  
  Second, the impact of accessibility variation is limited in time and growth rates rapidly converge back to the average value for all cities in the considered area.

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These two remarks are obviously important pieces of the puzzle that we will use for constructing a simplified model of this effect.

In order to gain a further understanding and quantitative insights about the coupling between accessibility and growth rate, we introduce here a simple model. We start from the general diffusion equation with noise which reads

$$\begin{equation}\frac{\mathrm{d}{P}_{i}}{\mathrm{d}t}={\eta }_{i}{P}_{i}+\sum _{j}{J}_{ji}{P}_{j}-{J}_{ij}{P}_{i}\end{equation} \tag{ 4 }$$

This equation which was introduced in the context of wealth dynamics [30] has a natural interpretation in the case of cities [31]. The first term corresponds to the Gibrat model [32] and describes the stochastic growth of population (birth–death processes and other exogenous processes) and the random variable η_i is assumed to be a Gaussian noise, with average m and variance 2σ². The other terms describe migration of individuals from one city to another: J_ij is the migration rate from city j to city i. It has been shown that this equation provides a regularization of the Gibrat model and a natural explanation of Zipf's law, at least in the mean-field version where J_ij = J for all i and j [30, 31]. In the Gibrat case (J_ij = 0) the growth rate does not depend on the population and fluctuates around the average value

$$\begin{equation}\langle g\rangle {:=}\frac{1}{\langle P\rangle }\frac{\mathrm{d}\langle P\rangle }{\mathrm{d}t}=m+{\sigma }^{2}\end{equation} \tag{ 5 }$$

where ⟨⋅⟩ denotes the average over the noise η.

We now introduce a minimal model for the impact on population growth of increasing accessibility which consists of two cities, 1 and 2. We have in mind a large city 1 connected to a small peripheral city 2 with P₂ ≪ P₁ (city 1 can in fact be considered as the whole world outside city 2, see figure 3). We also assume that there is migration from city 1 to city 2 that is described by the rate J. We neglect both the outflow from city 1 to city 2 in the evolution of city 1, and the counterflow from 2 to 1 (we note that keeping a term of the form −JP₁ would simply renormalize the average growth of the city 1). The evolution of populations 1 and 2 is then given in this framework by

$$\begin{align}\hfill \begin{aligned}\hfill \frac{\mathrm{d}{P}_{1}}{\mathrm{d}t}& ={\eta }_{1}{P}_{1}\hfill \\ \hfill \frac{\mathrm{d}{P}_{2}}{\mathrm{d}t}& ={\eta }_{2}{P}_{2}+J{P}_{1}\hfill \end{aligned}\end{align} \tag{ 6 }$$

where we assume that both η₁ and η₂ have the same average m and variance 2σ². In the case where the cities are disconnected (J = 0), they grow with the same natural rate

$$\begin{equation}{g}_{1}={g}_{2}=\frac{\mathrm{d}\langle {P}_{1\left(2\right)}\rangle }{\langle {P}_{1\left(2\right)}\rangle \mathrm{d}t}=m+{\sigma }^{2}.\end{equation} \tag{ 7 }$$

When there is a flow J > 0 from city 1 to city 2, the formal solutions of equation (6) are

$$\begin{align}\hfill \begin{aligned}\hfill {P}_{1}\left(t\right)& ={P}_{1}\left(0\right)\left({\mathrm{e}}^{{\int }_{0}^{t}{\eta }_{1}\left({t}^{\prime }\right)}\right)\hfill \\ \hfill {P}_{2}\left(t\right)& ={P}_{2}\left(0\right){\mathrm{e}}^{{\int }_{0}^{t}{\eta }_{2}\left({t}^{\prime }\right)}+{\int }_{0}^{t}J\left({t}^{\prime }\right){P}_{1}\left({t}^{\prime }\right){\mathrm{e}}^{{\int }_{{t}^{\prime }}^{t}{\eta }_{2}\left({t}^{{\prime\prime}}\right)}.\hfill \end{aligned}\end{align} \tag{ 8 }$$

where we consider the general case where J depends on time, and where we used the fact that the equation for P₁ is decoupled. For understanding the impact of accessibility variations on the growth rate, we consider the following simple scenario for the time-varying migration rate J. For instance, if the city 2 can be coupled to city 1 at time t = t₀ and if we assume that this coupling lasts a finite duration δt:

$$\begin{align}\hfill \begin{aligned}\hfill J\left(t\right)& =0\quad \text{for} t{< }{t}_{0}\hfill \\ \hfill J\left(t\right)& =J\quad \text{for} {t}_{0}{< }t{< }{t}_{0}+\delta t\hfill \\ \hfill J\left(t\right)& =0\quad \text{for} t{ >}{t}_{0}+\delta t\hfill \end{aligned}\end{align} \tag{ 9 }$$

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For this simple scenario, we can compute the growth rate g₂ of city 2 and find in the limit where the number of newcomers in city 2 is much less than the population P₂ (see SI, section 7 for more details)

$$\begin{align}\hfill \begin{aligned}\hfill {g}_{2}& =m+{\sigma }^{2}\quad \text{for} t{< }{t}_{0}\hfill \\ \hfill {g}_{2}& =\left(m+{\sigma }^{2}\right)+J \frac{\langle {P}_{1}\left(t\right)\rangle }{\langle {P}_{2}\left(t\right)\rangle }\quad \text{for} {t}_{0}{< }t{< }{t}_{0}+\delta t\hfill \\ \hfill {g}_{2}& =m+{\sigma }^{2}\quad \text{for} t{ >}{t}_{0}+\delta t.\hfill \end{aligned}\end{align} \tag{ 10 }$$

where ⟨P₁(t)⟩/⟨P₂(t)⟩ ≈ P₁(0)/P₂(0) when the number of newcomers in city 2 is much less than the population P₂ (see SI7 for more details). This simple model thus predicts an increase for growth rates that is linear in J, and inversely proportional to the population of city 2. The growth rate is thus larger during the migration period and is back to its uncoupled value afterwards, in agreement with our empirical observations and which justifies this finite duration δt, see figure 2.

In order to connect this model to real-world data, we have to identify the migration rate J. From our empirical results, it seems natural to identify J with the accessibility variations ΔA and not with the accessibility A of the area considered. We also assume that the impact of a given accessibility variation is larger for a larger city which implies that J is an increasing function of the population P, and we will assume a simple power law form P^α. This leads us to the main assumption of our model that consists in writing the coupling J as

$$\begin{equation}J\propto {P}^{\alpha }{\Delta}A\end{equation} \tag{ 11 }$$

where P is the average population of city 2 and ΔA the accessibility variation experienced by this city. With this expression, the growth rate of city 2 is given by

$$\begin{align}\hfill \begin{aligned}\hfill {g}_{2}& =m+{\sigma }^{2}\quad \text{for} t{< }{t}_{0}\hfill \\ \hfill {g}_{2}& =\left(m+{\sigma }^{2}\right)+\kappa {P}^{\alpha -1}{\Delta}A\quad \text{for} {t}_{0}{< }t{< }{t}_{0}+\delta t\hfill \\ \hfill {g}_{2}& =m+{\sigma }^{2}\quad \text{for} t{ >}{t}_{0}+\delta t.\hfill \end{aligned}\end{align} \tag{ 12 }$$

where we see that the impact of the growth rate variation scales as κP^α−1ΔA (we assumed here that ⟨P₁⟩ is varying very slowly over the time scale δt and we integrate it in the constant κ). Using this result, we can determine the value of α from empirical data. Indeed, for fixed α, the growth rate variation of a city is proportional to P^α−1ΔA and we look at the value of α which gives the largest difference between the top 1% and the rest. We thus assume here that the best value of α is the one that maximizes the impact of the accessibility variation on the system. We thus select cities with the largest value κP^α−1ΔA and measure the difference of their growth rate with respect to the average. We show the results in figure 4 (top) which demonstrate that it is in the region α ≃ 1 where we observe the most significant differences among cities with the largest migration inflow and the rest. This result also confirms a posteriori our empirical analysis of growth rates versus accessibility variations.

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This model—together with the empirical result α ≈ 1—thus provides a basic mechanism for the coupling between accessibility and growth rate variations, and predicts that the growth rate variation depends linearly with the accessibility variations

$$\begin{equation}{\Delta}g\simeq \kappa {\Delta}A.\end{equation} \tag{ 13 }$$

where Δg is the difference between the growth rates after and before the accessibility change. As we have seen however, a significant effect seems to be quantifiable only for the few 0.5 or 1% of cities who experienced the largest variation of accessibility. We therefore focus on this subset of cities which experienced a significative change in accessibility, and test if such a linear dependence can indeed be observable. We plot in figure 4 (bottom) the linear coefficient of the regression for the different periods. As expected, we observe that this linear coefficient is significantly different from zero for the periods of observed accessibility variations (here this effect is significative only for the periods 1975–1985 and 1985–1995, see SI, section 8 for more details), and we observe that κ is of the order 10⁻¹ with values at most equal to 0.5 in our data.

Accessibility measures are expected to be a powerful tool to build explanatory variables that have a large predictive power on growth. There is a large literature in quantitative geography and spatial econometrics that points in this direction, making accessibility as an extremely useful tool to assess the impact of transportation modes on growth. However, we have shown here that using recent data from the urban area of Paris, the relevant variable seems to be the time variation of the accessibility (as already suggested in [18]) rather than accessibility itself. Also, it seems that the effect of such a variation decays in time (and space) and that cities recover relatively quickly to a level of growth similar to the average of the corresponding region. These observations highlight the importance of spatial and temporal scales in these studies, and could reconcile the difference between the large impact of accessibility observed in the seminal paper [8] and recent works, although additional results on different geographical areas would be required to prove this in greater detail. Such a historical study would be needed in order to understand clearly the reasons why such large variations are not observed anymore, but goes beyond the scope of the present paper.

These different empirical observations found here led us to propose a simple model where the important ingredient is the interurban flow that has a limited lifetime and which is proportional to the accessibility variation. The conditions of validity of such a model are difficult to assess but the simplicity and ubiquity of the mechanism lead us to think that this first step towards the modeling of the accessibility-growth rate coupling provides at least a framework that can be built upon. It will allow to go beyond regression analysis, and eventually to help planners for identifying critical factors for the evolution of cities.

The data that support the findings of this study are openly available at the following url (INSEE, 2015): www.insee.fr/fr/statistiques/2522602, under the title: Historique des populations légales. Recensements de la population 1968–2014.

VV thanks the IPhT for its hospitality.