^{1}

^{2}

^{2}

^{3}

^{*}

^{1}

^{2}

^{3}

Edited by: Wei-Xing Zhou, East China University of Science and Technology, China

Reviewed by: Beom Jun Kim, Sungkyunkwan University, South Korea; Yougui Wang, Beijing Normal University, China

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The structure of the International Trade Network (ITN), whose nodes and links represent world countries and their trade relations, respectively, affects key economic processes worldwide, including globalization, economic integration, industrial production, and the propagation of shocks and instabilities. Characterizing the ITN via a simple yet accurate model is an open problem. The traditional Gravity Model (GM) successfully reproduces the volume of trade between connected countries, using macroeconomic properties, such as GDP, geographic distance, and possibly other factors. However, it predicts a network with complete or homogeneous topology, thus failing to reproduce the highly heterogeneous structure of the ITN. On the other hand, recent maximum entropy network models successfully reproduce the complex topology of the ITN, but provide no information about trade volumes. Here we integrate these two currently incompatible approaches via the introduction of an Enhanced Gravity Model (EGM) of trade. The EGM is the simplest model combining the GM with the network approach within a maximum-entropy framework. Via a unified and principled mechanism that is transparent enough to be generalized to any economic network, the EGM provides a new econometric framework wherein trade probabilities and trade volumes can be separately controlled by any combination of dyadic and country-specific macroeconomic variables. The model successfully reproduces both the global topology and the local link weights of the ITN, parsimoniously reconciling the conflicting approaches. It also indicates that the probability that any two countries trade a certain volume should follow a geometric or exponential distribution with an additional point mass at zero volume.

The International Trade Network (ITN) is the complex network of trade relationships existing between pairs of countries in the world. The nodes (or vertices) of the ITN represent nations and the edges (or links) represent their (weighted) trade connections. In a global economy extending across national borders, there is increasing interest in understanding the mechanisms involved in trade interactions and how the position of a country within the ITN may affect its economic growth and integration [

The above considerations imply that the empirical structure of the ITN plays a crucial role in increasingly many economic phenomena of global relevance. It is therefore becoming more and more important to characterize the ITN via simple but accurate models that identify both the basic ingredients and the mathematical expressions required to accurately reproduce the details of the empirical network structure. Reliable models of the ITN can better inform economic theory, foreign policy, and the assessment of trade risks and instabilities worldwide.

In this paper, we emphasize that current models of the ITN have strong limitations and that none of them is satisfactory, either from a theoretical or a phenomenological point of view. We point out equally strong (and largely complementary) problems affecting on one hand traditional macroeconomic models, which focus on the local weight of the links of the network, and on the other hand more recent network models, which focus on the existence of links, i.e., on the global topology of the ITN. We then introduce a new model of the ITN that preserves all the good ingredients of the models proposed so far, while at the same time improving upon the limitations of each of them. The model can be easily generalized to any (economic) network and provides an explicit specification of the full probability distribution that a given pair of countries is connected by a certain volume of trade, fixing an otherwise arbitrary choice in previous approaches. This distribution is found to be either geometric (for discrete volumes) or exponential (for continuous volumes), with an additional point mass at zero volume. This feature, which is different from all previous specifications of international trade models, is shown to replicate both the local trade volumes and the global topology of the empirical ITN remarkably well.

Before we fully specify our model, we preliminarily identify its building blocks by reviewing the strengths and weaknesses of the two main modeling frameworks adopted so far.

We start by discussing traditional macroeconomic models of international trade. These models have mainly focused on the volume (i.e., the value e.g., in dollars) of trade between countries, largely because the economic literature perceives trade volumes as being

Jan Tinbergen, the physics-educated^{1}

where GDP_{k} is the Gross Domestic Product of country _{ij} is the geographic distance between countries _{ij} and _{ji} can be different. An analogous _{ij} = _{ji} of bilateral trade. In the latter case, Equation (1) still holds but with the symmetric choice α = β. With this in mind, we will keep our discussion entirely general throughout the paper and, unless otherwise specified, allow all quantities to be interpreted either as directed or as undirected. Only in our final empirical analysis will we adopt an undirected description for simplicity.

More complicated variants of Equation (1) use additional factors (with associated free parameters) either favoring or resisting trade [

where the functional form of

It is generally accepted that the expected trade volumes postulated by the GM, already in its simplest form given by Equation (1), are in good agreement with the observed flows between trading countries. To illustrate this result, in _{ij}, the full probability distribution from which this expected value is calculated is not specified, and actually depends on how the model is implemented in practice. In the GM case, the distribution is chosen to be either Gaussian (corresponding to additive noise, in which case the expected weights can be fitted to the observed ones via a simple linear regression [

Parameter values for the traditional Gravity Model used in

1970 | 9.9·10^{8} |
0.91 | 0.81 |

1980 | 3.1·10^{9} |
0.83 | 0.89 |

1990 | 1.5·10^{10} |
0.97 | 0.93 |

2000 | 4.3·10^{10} |
1.05 | 0.93 |

Empirical non-zero trade flows vs. the corresponding expectation under the traditional Gravity Model. Log-log plot comparing the empirical volume (

A related but more fundamental limitation of the GM is that, at least in its simplest and most natural implementations, it cannot generate zero volumes, thereby predicting a fully connected network [

While there are variants and extensions of the GM that do generate zero weights and a realistic link density (e.g., the so-called Poisson pseudo-maximum likelihood models [

As we mentioned at the beginning, many processes of great economic relevance crucially depend on the large-scale topology of the ITN. In light of this result, the sharp contrast between the observed topological complexity of the ITN and the homogeneity of the network structure generated by the GM (including its extensions) call for major improvements in the modeling approach. In particular, in assessing the performance of a model of the ITN, emphasis should be put on how reliably the (global) empirical network structure, besides the (local) volume of trade, is replicated. In the network science literature, successful models of the ITN have been derived from the Maximum Entropy Principle [

In general, different choices of the constrained properties lead to different degrees of agreement between the model and the data. This can generate intriguing and counter-intuitive insight about the structure of the ITN. For instance, contrary to what naive economic reasoning would predict, it turns out that the knowledge of purely binary local properties (e.g., node degrees) can be more informative than the knowledge of the corresponding weighted properties (e.g., node strengths). Indeed, while the binary network reconstructed only from the knowledge of the degrees of all countries is found to be topologically very similar to the real ITN, the weighted network reconstructed only from the strengths of all countries is found to be much denser and very different from the real network [

The solution to this apparent paradox lies in the fact that, while the knowledge of the entire weighted network is necessarily more informative than that of its binary projection (in accordance with economic postulates), the knowledge of certain marginal properties of the weighted network can be unexpectedly less informative than the knowledge of the corresponding marginal properties of the binary network. In fact, it turns out that if the degrees of countries are (not) specified in addition to the strengths of countries, the resulting maximum-entropy model can(not) reproduce the empirical weighted network of international trade satisfactorily [

An important take-home message is that, in contrast with the mainstream literature, models of the ITN should aim at reproducing not only the strength of countries (as the GM automatically does by approximately reproducing all non-zero weights), but also their degree (i.e., the number of trade partners) [

Unlike the GM, maximum-entropy models of trade are

where δ is a free parameter that allows to reproduce the empirical link density. The model has been tested successfully in multiple ways [

The GM in Equation (1) and the maximum-entropy model in Equation (3) have complementary strengths and weaknesses, the former being a good model for non-zero volumes (while being a bad model for the topology) and the latter being a good model for the topology (while providing no information about trade volumes). An attempt to reconcile these two complementary and currently incompatible approaches has been recently proposed via the definition of an extension of the maximum-entropy model to the case of weighted networks [_{ij} and 〈_{ij}〉 as functions of the GDP [

Unfortunately, in the above approach the choice of country-specific constraints (degrees and strengths) only allows for regressors that have a corresponding country-specific nature. This makes the model in Almog et al. [^{2}_{ij}〉 and _{ij}, and highlight a limitation of current maximum-entropy models based only on country-specific constraints.

Combining all the above considerations, it is clear that an improved model of the ITN should aim at retaining the realistic trade volumes postulated by models based on Equation (2) (including the GM, the RM, and possibly many more), while combining them with a realistic network topology generated by (extensions of) maximum-entropy models. Such a model should also aim at providing the full probability distribution, and not only the expected values as in Equation (1), of trade flows and, unlike the GDP-only model in Equation (3) [

In this section, we introduce what we call the Enhanced Gravity Model (EGM) of trade. The EGM mathematically formalizes the two ingredients that, in light of the previous discussion, any “good” model of economic networks should feature: namely, realistic (trade) volumes and a realistic topology, both controllable by macroeconomic factors.

The first lesson we have learned is that Equation (2) is successful in reproducing link weights only after the existence of the links themselves has been preliminarly established. This implies that Equation (2), as a model of real-world trade flows, is actually unsatisfactory and should rather be reformulated as a conditional expectation of the weight _{ij}, given that _{ij} > 0. In other words, if _{ij} denotes the entry of the adjacency matrix _{ij} = Θ(_{ij}), i.e., _{ij} = 1 if _{ij} > 0 and _{ij} = 0 if _{ij} = 0), an improved model should be such that Equation (2) is replaced by

where 〈_{ij}|_{ij} = 1〉 is the conditional expected weight of the trade link from country

The second lesson we have learned is that, in analogy with Equation (4), Equation (3) should be generalized to allow for both dyadic (

where a crucial requirement is that _{ij} is monotonic in _{ij}. It is also worth noticing that the explanatory factors used in Equations (4) and (5) need not coincide. However, to avoid using different symbols for the arguments of the two functions, we adopt the convention that

We want our model to produce both Equation (4) as the desired (gravity-like) conditional expectation for link weights and Equation (5) as a realistic expected topology. To do so, we introduce the full probability _{ij}). We are free to choose whether _{ij} takes non-negative integer values [in which case _{ij}(_{ij} takes the particular value _{ij} > 0 indicates the presence of a trade link (i.e., _{ij} = 1). By contrast, the event _{ij} = 0 indicates the absence of a trade link (i.e., _{ij} = 0) and is also included as a possible outcome in _{ij}(_{ij}(_{ij} and _{kl} between two distinct country pairs, or equivalently the factorization of _{i,j}_{ij}(_{ij}) of dyadic probabilities. However, we will later find that the desired model has precisely this independence property. Importantly, unlike in the traditional GM, in our approach dyadic independence is a consequence and not a postulate.

We now look for the form of _{ij}(_{ij}(_{ij} that _{ij}(0) that they are not connected, i.e.,

where, for real-valued weights, _{ij}(0) denotes the point mass, i.e., the magnitude of the delta-like probability density function _{ij}(_{ij}(0):

We now relate _{ij}(

(note that the event _{ij} equals

and its expected value gives the conditional expectation of the link weight, given that the link exists:

Setting Equation (10) equal to Equation (4) leads to

Equation (11) carries an important message. It reveals that, while a superficial inspection of Equation (8) might suggest that the expected trade volume 〈_{ij}〉 is independent of the topology of the ITN, i.e., on _{ij}(0) or equivalently _{ij}(0) is coupled to the other values _{ij}(_{ij}〉 depends on both _{ij} depends on

Equations (7) and (11) fix two important properties we require for _{ij}(

We look for the form of

(where the sum extends over all weighted graphs with _{ii} = 0 for all _{ij}〉 and 〈_{ij}〉 (for all pairs _{ij} and β_{ij} as the (real-valued) Lagrange multipliers required to enforce the expected value of _{ij} = Θ(_{ij}) and _{ij} respectively [where Θ(

(representing a linear combination of the quantities whose expected value is being constrained) and the partition function ^{*}(_{ij}〉 and 〈_{ij}〉 is found to be

where, given

is the resulting (maximum-entropy) probability that the link from node ^{*}(

Importantly, while the constraints used in the maximum-entropy models of the ITN considered so far in the literature are observed topological properties (e.g., the degrees and/or the strengths of nodes), the constraints considered here are economically-driven expectations, namely Equations (5) and (11). This key step allows us to reconcile macroeconomic and network approaches within a generalized framework and represents an important difference with respect to previous models. In particular, we use Equations (6), (8) and (10) to express _{ij}, 〈_{ij}〉 and 〈_{ij}|_{ij} = 1〉 in terms of _{ij} and _{ij} [

The above expressions allow us to rewrite Equation (15) as

Now, equating Equation (16) to Equation (5) and Equation (17) to Equation (11) [or, equivalently, Equation (18) to Equation (4)] allows us to find the values of _{ij} and _{ij} solving the original problem:

Inserting Equations (20) and (21) into Equation (19), we finally get the explicit probability _{ij} given by Equation (5); if realized, this link acquires a weight

which is a geometric distribution representing the chance of _{ij}, followed by a failure with probability 1−_{ij}. The above result provides an insightful interpretation of the realized volumes in the model in terms of processes of link establishment and link reinforcement (see section 5).

We now take an econometric perspective and discuss how the model parameters can be chosen to optimally fit a specific empirical instance of the network. To this end, we use the Maximum Likelihood (ML) principle applied to network models [^{*} denotes the weight matrix (with entries ^{*}(^{*}). We therefore define the log-likelihood function as

(where we have dropped the dependence of

For probability distributions belonging to the exponential family, i.e., in the form given by Equation (14) like the one we are considering, the second derivatives of the log-likelihood coincide with (minus) the covariances between the constraints included in the Hamiltonian defined in Equation (13) (see for instance [

The above expressions, which are valid for _{ij}(_{ij}. As the monetary units in the data are changed arbitrarily (e.g., from dollars to thousands of dollars), so will the estimated mean and the resulting expected number of zeroes. By contrast, in our model the monetary units affect

The above results can be adapted in a straightforward, although more technical, fashion to the case when link weights are assumed to take non-negative real values. The entire derivation is reported in the

In the real-valued case, ^{*}(_{ij}〉 and 〈_{ij}〉 (for all pairs

where δ(_{ij} is still given by Equation (5).

The above expression shows that _{ij} at _{ij}|_{ij} = 1〉, connection probability _{ij} and unconditional expected trade volume 〈_{ij}〉 given by Equations (4), (5) and (11) respectively. Establishing a link from country _{ij} given by Equation (5); if realized, this link acquires a weight

which is now a purely exponential distribution with the desired (conditional) mean

The estimation of the parameters

We can finally test the predictions of our model against empirical international trade data. The datasets are described in the

We adopt an undirected network description (where the connection between two countries carries a weight equal to the total trade in either direction) to facilitate the definition of the topological properties characterizing the ITN. Previous work has shown that, given the highly symmetric structure of the ITN, the undirected representation retains all the basic properties of the network [

We choose _{ij}|_{ij} = 1〉 is the same as in the GM defined by Equation (1) (now interpreted as a conditional expectation). This means choosing

where we have set β ≡ α due to undirectedness. Similarly, we choose _{ij} is the same as in the model defined in Equation (3), i.e.,

With the above specification, the expected topology does not depend on any dyadic factor. This is the simplest choice that is found to reproduce the topology of the ITN very well [

Given the above model specification, for a given instance ^{*} of the empirical network we find the optimal parameter values ^{*}, α^{*}, γ^{*} and δ^{*} through the ML conditions given by Equations (23) and (24). Importantly, Equation (24) reads in this case ^{*} that ensures that the expected number of links

We first test the performance of the EGM in replicating the empirical trade volumes, i.e., the purely local (dyadic) structure of the ITN. In _{ij}|_{ij} = 1〉 under the EGM given by Equation (27). As mentioned above, for the EGM the parameters are obtained via the ML principle as prescribed by Equation (23) and their resulting values are reported in

Parameter values for the Enhanced Gravity Model calculated by considering integer link weights (equal to integer multiples of the monetary unit used in the dataset) and carrying out the corresponding ML estimation as prescribed by Equations (23) and (24).

δ | α, |
|||

1970 | 4.7·10^{5} |
1.0·10^{8} |
0.67 | 0.78 |

1980 | 1.1·10^{6} |
9.3·10^{8} |
0.77 | 0.75 |

1990 | 1.4·10^{6} |
5.4·10^{9} |
0.87 | 0.86 |

2000 | 3.3·10^{6} |
1.7·10^{10} |
0.91 | 0.90 |

Empirical non-zero trade flows vs. the corresponding expectations under the traditional Gravity Model and the Enhanced Gravity Model. Log-log plot comparing the empirical volume (

Importantly, comparing the values of the parameters α, β, γ reported in

In order to better understand the differences between the trade volumes predicted by the two models, in _{≥}(_{≥}(0) = 1 in order to include zero weights, corresponding to pairs of countries that are _{≥}(_{≥}(1−ϵ) = 1 to a value _{≥}(1 + ϵ) ≈ 0.53, where ϵ > 0 is arbitrarily small. Recalling that link weights take only non-negative integer values in our analysis, this discontinuity indicates that there are roughly 47% pairs of countries that are not connected (_{≥}(_{≥}(+∞) = 0, indicating that the only discontinuity we see at _{≥}(_{≥}(_{ij}, hence the expected number of positive weights, is identical in the discrete and continuous versions of the model). For positive weights, the real-valued EGM would continuously interpolate the discrete points of the integer-valued EGM because this is a generic property of geometric and exponential distributions with the same expected value. So, in either specification, the EGM nicely replicates both the empirical distribution of strictly positive link weights and the sharp peak “jumping out” from it, while the GM does not.

Empirical and model-generated cumulative distributions of trade flows. Log-linear plot comparing the empirical cumulative distribution of trade flows (normalized in order to include zero flows) in the ITN for the year 2000 (red) with the corresponding distributions obtained using the Gravity Model defined in Equation (1) (green, parameters estimated as reported in

We now want to check whether the trade links, besides being predicted in correct number by the EGM, are also placed between the correct pairs of countries by the same model. This means moving the focus of our analysis toward the purely binary, global topology of the ITN. As a first qualitative illustration setting the stage for this analysis, in _{ki〉GM} =

Country-based network configurations for year 2011 in the real ITN (red), the GM (green), and the EGM (blue). For three representative countries, we show the connections to all trade partners in the world. The total number of countries in the data (see _{ij}, so links change from realization to realization. The expected degree is however independent of the individual realizations and is close to the empirical one for all countries. We have selected a typical realization that produces a degree equal to the expected degree for each of the three countries.

We now consider higher-order topological properties as a more stringent and quantitative test. In the top left panel of _{i}) of country _{i}), i.e., the fraction of trade partners of country _{i}) of such partners. The empirical quantities are compared with the expected quantities under the GM and the EGM. The exact expressions for both empirical and expected quantities are provided in the _{i}) are on average highly connected, both to the rest of the world (large _{i}). By contrast, countries that trade with a high-degree country (large _{i}) are on average poorly connected, both to the rest of the world (small _{i}). For both properties, we find that the EGM is in excellent agreement with the empirical ITN, as opposed to the classical GM which systematically generates nearly constant and much higher values as a result of predicting a complete network.

Network properties in the real ITN (red), the GM (green), and the EGM (blue). Top left: average nearest neighbor degree _{i} for all nodes. Top right: clustering coefficient _{i} vs. degree _{i} for all nodes. Bottom left: average nearest neighbor strength _{i} for all nodes. Bottom right: weighted clustering coefficient _{i} for all nodes. All results are for the snapshot of the ITN in the year 2000. For all the other years in the analyzed sample, we systematically obtained very similar results. See

Having checked that the EGM does very well in separately replicating both the local link weights and the global topology of the ITN, we now perform a last and most severe test monitoring properties that combine topological and weighted information together (all definitions are again given in the _{i}) of the country _{i}) of country _{i}) trade a lot with the rest of the world (large _{i}) have instead low trade activity with the rest of the world (small

In this paper we have introduced the EGM as a novel, advanced model for the ITN and economic networks in general. Phenomenologically, the EGM allows us to reconcile two very different approaches that have remained incompatible so far: on one hand, the traditional GM that is well established in economics and successfully reproduces non-zero trade volumes in terms of GDP and distance but fails in predicting the correct topology [

The agreement between the EGM and trade data calls for an interpretation of the process generating the network in the model. In this respect, we notice that Equations (15) and (22) allow us to interpret the realized trade volumes in the EGM as the outcome of two equivalent processes (a serial and a parallel one) of link creation and link reinforcement. In the serial process, for a given pair of countries _{ij} and then increment its volume in unit steps, each with success probability _{ij}. After the first failure, we stop the process for the pair of countries under consideration and start it again for a different pair, and so on until all pairs are considered. In the equivalent parallel process, all pairs of countries simultaneously explore the mutual benefits of trade and engage in a first connection, each with its probability _{ij}. Then, all pairs of nodes for which the previous event has been successful reinforce their existing connection by a unit weight, each with its probability _{ij}. The process stops as soon as there are no more successful events. In either case, Equation (15) gives the resulting probability that the realized volume is

Importantly, Equation (19) shows that _{ij} which is in general different from the probability _{ij} of each of the _{ij} in Equation (13) is set to zero, i.e., if the constraint on the expected value of Θ(_{ij}) (the expected topology) is removed as in the standard GM. In such a case, _{ij} becomes equal to _{ij} (i.e., link creation and link reinforcement become equally likely) and therefore

Consistently with the fact that trade volumes are typically reported as integer multiples of some indivisible monetary unit (e.g., dollars), the above discussion and most of our analysis has been assuming non-negative integer link weights. However we may also take the limit of a vanishing monetary unit, in which case trade volumes become non-negative real numbers and, as we have shown, _{ij} is unchanged and the expected topology is still described by Equation (5). In the absence of topological constraints, i.e., if we imposed α_{ij} = 0, in this real-valued case the network would degenerate to a fully connected graph as in all specifications of the GM with continuous volumes [

Our results may have strong implications both for the theoretical foundations of trade models and for the resulting policy implications. It is known that the traditional GM is consistent with a number of (possibly conflicting) micro-founded model specifications [

AA and RB analyzed the data and prepared the figures. AA and DG wrote the paper. DG planned the research and supervised the project. All authors reviewed the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

AA and DG acknowledge support from the Dutch Econophysics Foundation (Stichting Econophysics, Leiden, the Netherlands). This work was also supported by the Netherlands Organization for Scientific Research (NWO/OCW).

_{1}

_{2}/

If the link weights _{ij} take non-negative real values instead of non-negative integer values, the probability

where the constraints on 〈_{ij}〉 and 〈_{ij}〉 (for all pairs _{ii} = _{ii} = 0 for all _{ij}) = _{ij} for all _{ij}〉 more naturally and to recover more general “mixed” (i.e. containing a mixture of a discrete and a continuous part) solutions for ^{*}(

Since the sets of constraints is the same as in the integer-valued case, we arrive at the same expression for ^{*}(

where we have again used the definition

Inserting Equation (30) into Equation (14) yields the following new form of

Using Equations (6) and (8), we can now calculate the connection probability and the (conditional) expected weight as

Equations (32), (33) and (34) replace Equations (16), (17) and (18) in the case of real-valued link weights. Inserting these expressions into Equation (31), we get

which replaces Equation (19) in the real-valued case and shows that _{ij} at

which allows for a fully continuous treatment. For instance, the normalization can be correctly stated as

In terms of conditional probabilities, we still find that establishing a link from country _{ij} given by Equation (5) as desired; if realized, this link acquires a weight

which is now a purely exponential distribution with (conditional) mean _{ij} and β_{ij} solving the original problem:

Note that Equation (11) holds in this case as well, as it should because it does not depend on whether link weights are taken to be integer or real. Inserting Equations (38) and (39) into Equations (36) and (37), we get the explicit form of

We have used international trade and GDP data from the database curated by Gleditsch [_{ij} (which we have symmetrized by taking the sum of _{ij} + _{ji}), yearly GDP values, and the (time-independent) distance matrix _{ij}. The number

Given a weighted undirected network with weight matrix _{ij} = Θ(_{ij}), the degree of node

the average nearest-neighbor degree of node

and the (binary) clustering coefficient of node

The average nearest neighbor strength of node

(where

The expected value (under the EGM) of each of the network properties defined above can be calculated either numerically, by averaging over many network realizations sampled independently from the probability ^{*}(_{ij} and _{ij} as follows:

where 〈_{ij}〉 = _{ij}, as given by Equation (16), and

_{ij} with _{ij} and _{ij} and _{ij}, the expected values are ultimately a function of only the GDPs and distances. In our analysis, after preliminary checking that the analytical expressions matched extremely well with the numerical averages over realizations, we have systematically adopted the analytical approach, which requires no sampling of networks and is therefore extremely efficient.

^{1}Jan Tinbergen studied physics in Leiden, where he carried out a Ph.D. under the supervision of the theoretical physicist Paul Ehrenfest. Tinbergen defended his thesis in 1929, and then became a leading economist. He was awarded the first Nobel memorial prize in economics in 1969.

^{2}Building on the hypothesis of the existence of underlying hidden metric spaces in which real-world networks are embedded, García-Pérez et al. [_{ij} appearing in the GM and measured as geodesics on our spherical tridimensional world.