2.1 Generative process

We are interested in modelling the process that generates data compatible with the scaling law (1). The starting point of our model is the widespread interpretation that Eq (1) reflects a change in people’s efficiency (or consumption) depending on the amount of interactions available to them [12]. Accordingly, we consider a generative process in which tokens (e.g. a patent, a dollar of GDP, a piece of infrastructure) are assigned to (produced or consumed by) an individual person j with probability p_p(j).

Consider j = 1, …, M persons living in i = 1, …, N cities, on which the population of the city i is given by x_i and . A total of Y ≡ ∑_i y_i tokens are (randomly) assigned to the X persons. In the absence of any other information, this defines our first (null) model:

1. (P) Per-capita model: All tokens Y are distributed with equal probability to all persons j as in a constant per-capita attribution, p(j) = 1/X. In this case, the probability p_c that a token is attributed to city i is given by (2) where c(j) is the city in which j lives and δ(x) = 1 for x = 0 (otherwise δ(x) = 0).

This model corresponds to a linear (trivial) scaling law, β = 1 in Eq (1). A super-linear β > 1 (sub-linear β < 1) scaling is obtained if a token is more likely to be assigned to someone living in a more (less) populous city. In this spirit, in Ref. [19] we assumed that the probability that a token is assigned to person j depends on the population around j as (3)

Here we generalize this idea to account for spatial interactions between j and other individuals j′ that live in other cities (i.e., c(j)≠c(j′)). We introduce a quantity A_j, defined as the total attractiveness of individual j due to all its interactions, and use it as a weight on the probability of assignment of a token as (4) where Z(β) is the normalization constant (i.e., ). If β = 1, the probability p_p(j) is the same for all j as in the per-capita model and we recover Eq (2). For β > 1, p_p(j) grows with the interactions A_j in line with a super-linear scaling. For β < 1, p_p(j) decays with A_j in line with a sub-linear scaling.

The attractiveness of an individual A_j certainly depends on a multitude of factors that could be included in the model, depending on data availability and research interest. Here, we focus on pairwise interactions a_j,j′ between individuals j and j′ separated by a distance d = d_j,j′. We obtain A_j as the total interaction of j and all other individuals j′ by summing over all j′ (5)

The distance d_j,j′ ≥ 0 does not need to be a distance in a mathematical sense and, in practice, depends on the availability of data. Below we use the geographic (geodesic) distance between cities (another natural choice would be the commuting time). The pairwise (spatial) interactions a_j,j′ is discussed below and will lead to three different specific models.

2.2 Spatial interactions

In order to explore the formalism above we now consider simple dependencies of the pairwise interaction a(d) on the distance d ≡ d_j,j′ between two persons j, j′. In general, we are interested in functions a(d) that monotonically decay with d from a(0) = 1 to lim_{d → ∞} a(d) = 0. Choosing another value at a(0) leads to the same results because of the normalization of p_p(j) in Eq (4). Our framework can be applied to any function a(d) suitable to model spatial relationships, data will reveal us which one is more suitable.

The simplest choice of a(d) is

1. (C) City model: (6) in which interactions occur only within the same city (d = 0). From Eq (5) we get A_j = x_c(j), i.e., we recover the scaling law (1) and Eq (3) (the model of Ref. [19], Sec. 4.2).

Spatial interactions beyond city limits can be incorporated using more general functions a(d). Here we start this investigation with functions a(d;α) that depend on a single parameter α that is measured in the same units of d (e.g., km) and sets a scale for spatial interactions such that a(α;α) = 1/2 (i.e., at a distance d = α the interactions decay to a factor 0.5 of the interaction at the same city d = 0). Furthermore, we wish to recover the choice (6) in the limit of small α, i.e., a(d)→a_C(d) in Eq (6) for α → 0₊. Two choices of a(d;α) that satisfy these properties (and also a(0;α) = 1 and lim_{d → ∞} a(d, α) = 0 for any α) are:

1. (G) Gravitational model: (7) inspired by models of gravitational interactions (for large d the interactions decay as a ∼ 1/d², one can also replace the power 2 by an additional parameter) [3, 8, 15].
2. (E) Exponential model: (8)

For α → ∞, the distances do not matter, everyone is equally linked to everyone else, and the P-model is retrieved. Altogether, the four models discussed above are summarized in Table 1 and satisfy

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Table 1. The four models considered in this paper.

https://doi.org/10.1371/journal.pone.0243390.t001

2.3 Likelihood

We now discuss how the likelihood of our models can be computed from the data. We assume that (x_i, y_i) data is available at locations i = 1, …, N. We denote the locations i as cities but we stress that this does not need to correspond to any urban definition of cities as the spatial interaction between different regions can be accounted explicitly in our models by choosing an appropriate function a(d). We assume also that a measure of distance d_i,i′ between all pairs of cities is available (e.g., the geodesic distance between the centroid of the cities).

Besides their location (city), individuals are indistinguishable. Therefore, the probability p_c(i) that a token is assigned to city i is given by a sum of p_p(j) over persons j on city i (i.e. c(j) = i), which contains exactly x_c(j) ≡ x_i terms (9) where we used Eq (4) and consider that x_i ≫ 1 for all i. The last equation defines the attractiveness of an individual in city i as (10)

This can be thought also as the number of effective interactions available for an individual in city i so that A_i = x_i in the city model (6) and A_i ≥ x_i otherwise (e.g., for the gravitational and exponential models). It depends only on the population x_i of all cities and on the distances d_i,i′ between cities, e.g. through Eqs (7) or (8), and therefore A_i can be computed independently of the data y_i.

The expected number of tokens in city i is given by (11)

The probability of observing y_i tokens in each city of size x_i is a multinomial distribution (12)

This corresponds to the likelihood P(D|M, θ) of the data D ≡ {y₁, ⋯, y_N}—since the populations (x₁, ⋯, x_N) are fixed—for a given model class M and given parameters θ. It is convenient to write the log-likelihood as (13)

3.1 General framework

The models described above contain strong simplifying assumptions Our focus on the scaling relationship led to the assumptions that individuals are identical and that the token assignments are independent. and therefore our approach here is not to test whether the data is compatible with them (we know it is not While in linear fitting the number of observations equals to the number of cities, our model focus on individuals j and tokens of output y (X = ∑x_i, Y = ∑y_i) so that the number of observations is much larger and the expected fluctuations (for large cities) are much smaller than the fluctuations in the data. This accounts only to fluctuations of the (random) assignment of tokens and neglects fluctuations (present in the data) due to measurement imprecision and due to factors that are not part of our model.) but instead to compare the different models. This means that instead of the likelihood P(D|M, θ) that models generate the data D = {y₁, ⋯, y_N}, computed in the previous section, we should focus on what the data D tells us about the model class M ∈ {P, C, G, E} and their parameters θ = {α, β}. This is done based on the (posterior) probability (14) computed from the three terms in the right hand side:

• P(D) depends only on the data, act as a normalization, and does not affect the choice between models.
• P(M, θ) is the prior probability and is taken flat so that no a priori preference is given to any model. Specifically, we write P(M, θ) = P(θ|M)P(M) with P(M) = 1/4 and constant P(θ|M) in 0 ≤ β ≤ 2 and 0 ≤ α ≤ α_max, where α_max is an arbitrary maximum distance (we use α_max = 6, 371 km, Earth’s radius). This implies that P(θ|M) for our the models P, C, G, E are 1, 1/2, 1/2 α_max, 1/2 α_max, respectively.
• P(D|M, θ) is the likelihood and is evaluated numerically from Eq (13). This is facilitated by two observations: (i) the two first terms in the log-likelihood (13) are independent of the models so that the variation across M and θ depends only on the last term; (ii) in this last relevant term, the parameter α enters only in A_i through the dependence on a(d) so that for a fixed α the dependence of the matrix d_i,i′ is reduced to the vector A_i. It is thus computationally more efficient to fix α, compute A_i once, and then consider variations in β.

3.2 Estimation of parameters θ = {α, β}

The best parameters θ = {α, β} of a given model M are the ones that maximize the posterior P(θ|D, M). Since the priors are constant, this is equivalent to the maximization of the log likelihood (13) in the space of admissible parameters set by the priors.

3.3 Model selection

In the comparison of the different model classes M we account for the fact that models have different (number of) parameters θ by computing P(M|D), or equivalently, the description length (15) by integrating over all parameters θ of model M

The description length corresponds to the size (in number of bits, for based 2 logarithm) of the optimal encoding of data and model [22]. Since the priors P(θ|M) and P(M) are constant, the crucial computational step is the integration of the likelihood over the parameters θ. When the number of observations Y = ∑_i y_i is large (often the case for relevant urban scaling analysis), the likelihood is expected to be sharply peaked around the maximum-likelihood parameters θ. In this case, the description length is dominated by the maximum log-likelihood and further approximations can be used to compute (e.g., the Bayesian Information Criterion). However, one should be careful using these approximations to compare non-nested models (e.g., G and E) and around parameters θ in which the priors are discontinuous (as in the relevant case of α = 0).

4.1 Data

We apply the models and data-analysis methods described above to five datasets from two different countries. For Brazil, the data on three observables y—GDP, deaths due to external (non-natural) causes, and deaths due to AIDS—in the year 2010 is given for thousands of municipalities (administrative boundaries). For USA, the data on two observables y—GDP and miles of roads—in the year 2013 are given for hundreds of metropolitan areas. The USA cases can be considered as the paradigmatic examples of super- and sub-linear urban scaling laws [12]. In both countries, the average distance between two urban units is of thousands of km. The results of our analysis are reported in Table 2. The data and codes used in this paper are available in Ref. [23]. The data was collected from censuses and governamental agencies, was used in Refs. [12, 17], and is available with further documentation and all codes used in this paper in Ref. [23].

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Table 2. Results of the four models in five databases.

https://doi.org/10.1371/journal.pone.0243390.t002

4.2 The effect of α

We start investigating the central question of this paper: does spatial proximity between cities help to explain the observations y studied in urban scaling? And, if so, does it affect the scaling exponent β? The results in Fig 2 demonstrate that the answer to both questions is positive in most (but not all) cases. The top row of the figure shows that the value of the (maximum likelihood) exponent β for a fixed α changes significantly with α. The bottom row shows that often (Brazil GDP, USA Roads, but not in USA GDP) the best model is observed for α > 0. In these cases, there is an interval in α for which the model with geographic distance has a larger likelihood than the α = 0 case, compatible with the idea that the spatial scales we are accounting in this interval are meaningful (i.e., distances of 10–100 km that are relevant to interacting people).

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Fig 2. Spatial dependence affects the choice of the scaling parameter β.

The value of α (measured in kilometres) is varied and the most likely value of β is estimated for each α. The three left panels shows the value of β and the three left panels indicate the likelihood of the different models M indicated in the legend. The panels on the top correspond to GDP data from Brazil, the best model is M = G with α = 14.6 and β = 1.24. The panels on the centre correspond to GDP data from USA, the best model is the city model M = C (obtained for α = 0) with β = 1.12. The panels on the bottom correspond to data from road miles in the USA, the best model (largest likelihood) is M = G with α = 20.4 and β = 0.75.

https://doi.org/10.1371/journal.pone.0243390.g002

The addition of spatial interactions does not trivialize the urban scaling law, differently from the effect of city boundaries reported in Ref. [14]. In fact, the non-linear scaling exponent β is often enhanced by the spatial relation α, i.e., super- (sub-) linear scalings β > 1 (β < 1) in the usual approach (at α = 0) show an even larger (smaller) value of β for the maximum-likelihood value of α. For instance, for Brazil GDP the estimation of β in the non-spatial models are 1.05 (linear fitting) and 1.17 (city model) while in the spatial models it is 1.24 (gravitational model) and 1.21 (exponential model). The same effect is observed in the case of sublinear scaling in the data for USA Roads Lengths, see Table 2.

4.3 Comparing different models

In all our five datasets the models with non-linear scaling (C,G, and E, for which β ≠ 1) are preferred over the per-capita (P) model (negative in Table 2). In four of the five datasets, the models with spatial interactions (α ≠ 0 in the G and E models) are preferred over the one (C-model) that ignores it. The exception is the case of USA GDP, for which the estimated value of α is zero for the G model and very small (1.65 km) for the E model. The description length of the C model is smaller than the one in the G, E models by 2 and 3 bytes, respectively, indicating that the largest likelihood of the data obtained with α = 1.65 in the E model is not sufficient to justify its increased model complexity.

The comparison of the Gravitational and Exponential models reveal that both show a very similar behaviour as a function of α (Fig 2), similar inferred model parameters α and β, and similar description lengths (Table 2). This indicates that the conclusions are not very sensitive to the functional form of a(d), used to account for spatial interactions (as long as they satisfy the natural constraints we used to propose a(d)). The most important distinction we found is between models that ignore spatial interactions (linear fitting, C model, and α = 0) and those that account for it (α > 0 in the G and E models).

4.4 Increased interactivity

We now investigate how the spatial models introduced here change the number of effective interactions of individuals. In the introduction we discussed how the GDP of cities close to large urban areas were underestimated. Our analysis reveals that spatial interactions were not a strong factor in the USA GDP data overall. This was different for Brazil GDP, where the best model is the Gravitational model with α = 14.6. For these parameters, in Fig 3 we show the increased attractiveness—or number of interactions, A_i in Eq (10)—that individuals in different cities in Brazil experience. It fluctuates significantly from city to city because it is an intricate function of the location of all cities, but it is clear that smaller cities are more affected than larger cities.

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Fig 3. Accounting for spatial interactions increase the attractiveness of individuals in small cities.

The attractiveness A_i in Eq (10) divided by the population x_i is shown as a function of x_i for the different Brazilian municipalities i. The horizontal black line correspond to the case in which spatial interactions are ignored (α = 0 ⇒ A_i = x_i). The dots correspond to the result of the Gravitational model with the maximum likelihood parameters obtained for the case of GDP (see Table 2).

https://doi.org/10.1371/journal.pone.0243390.g003

For the case of “São Caetano do Sul”, the attractiveness of the inhabitants of this municipality is 43.6 times larger than assuming that interactions occur only within the municipality (i.e., A = 43.6 x for α = 14.6 in M = G). The GDP of this city is 11.0 BR$ (Billion reais), much larger than the per-capita expectation of 2.1 BR$. The city model improves this expectation to 2.7 BR$, still too low but better than the linear-fit estimation 1.6 BR$. The best spatial model (G model with α = 14.6 and β = 1.24) improves the prediction to 4.5 BR$. Therefore, we conclude that spatial interactions can explain a considerable amount of the GDP of this municipality, even more than the inclusion of the non-linear scaling (β > 0), but that other factors remain significant.