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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 [ 1 — 5 ]. Moreover, in the wake of recent financial crises the interconnectedness of economies has become a matter of concern as a source of instability [ 6 ].

As the modern architecture of industrial production extends over multiple countries via geographically wider supply chains, sudden changes in the exports of a country due e. The assessment of the associated trade risks requires detailed information about the underlying network structure [ 7 ]. In general, among the possible channels of interaction among countries, trade plays a major role [ 2 — 4 ]. 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.

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 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 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. Based on this argument, emphasis has been put on explaining the expected volume of trade between two countries, given certain dyadic and country-specific macroeconomic properties. Jan Tinbergen, the physics-educated 1 Dutch economist who was awarded the first Nobel memorial prize in economics, introduced the so-called Gravity Model GM of trade [ 8 ].

The GM aims at inferring the volume of trade from the knowledge of Gross Domestic Product, mutual geographic distance, and possibly additional dyadic factors of macroeconomic relevance [ 9 , 10 ]. In the above directed specification of the GM, the flows w ij and w ji can be different. 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 [ 9 , 10 ]. Like the GDP and geographic distances, these factors can be either country-specific e. Indeed, although in this paper we focus on the GM applied to the international trade network, our discussion equally applies to many other models of socio-economic networks as well.

Our following discussion applies to both the GM and the RM, as well as any other model described by Equation 2. Similarly, it does not only apply to trade networks, since both the GM and the RM have been successfully applied to other systems as well, including mobility and traffic flows [ 11 — 14 ], communication networks [ 15 ], and migration patterns [ 16 ] the latter representing—to our knowledge—the earliest application of the GM to a socio-economic system, dating back to [ 17 ]. 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 Figure 1 we show a typical log-log plot comparing the empirical volume of the realized bilateral international trade flows with the corresponding expected values calculated under the GM as defined in Equation 1 with parameters calculated as reported in Table 1.

The figure shows the typical qualitative consistency between the GM and the empirical non-zero trade volumes. However, it should be noted that, while Equations 1 and 2 define the expected value of w 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 [ 18 , 19 ] , log-normal corresponding to multiplicative noise and requiring a linear regression of log-transformed weights [ 20 ] as we did to produce Figure 1 and Table 1 , Poisson [ 20 ], or more sophisticated [ 21 ] see [ 22 ] for a review.

The arbitrariness of the weight distribution already highlights a fundamental weakness of the traditional formulation of the model. Moreover, for both additive and multiplicative Gaussian noise, the model can produce undesired negative values. Table 1. Figure 1. Empirical non-zero trade flows vs.

Log-log plot comparing the empirical volume y -axis of all non-zero bilateral trade flows in the ITN with the corresponding expected volume x -axis predicted by the Gravity Model defined in Equation 1 , with parameters estimated as reported in Table 1. Top left: year , top right: year , bottom left: year , bottom right: year The black line is the identity line corresponding to the ideal, perfect match that would be achieved if the empirical weights were exactly equal to their expected values, i. 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 [ 22 — 24 ].

This means that the GM can be fitted only to the non-zero weights, i. If used in this way, the model effectively disregards the empirical structure of the network, both as input thus making predictions on the basis of incomplete data and as output thus failing to reproduce the topology. Operatively, the GM can be used only after the presence of a trade link has been established independently [ 22 ]. This problem is particularly critical since roughly half of the possible links are found not to be realized in the real ITN [ 25 — 28 ].

Clearly, the same problem holds for the RM and any more general model of the form specified in Equation 2. While there are variants and extensions of the GM that do generate zero weights and a realistic link density e. Indeed, even in its generalized forms, the GM predicts a largely homogeneous network structure, while the empirical topology of the ITN is much more heterogeneous and complex [ 22 , 23 ]. Established empirical atures of this heterogeneity include a broad distribution of the degree of connections and the strength total trade volume of countries [ 25 — 35 ], the rich-club phenomenon whereby well-connected countries are also connected to each other [ 36 , 37 ], strong clustering and dis assortative patterns [ 26 , 27 ].

These highly skewed structural properties are remarkably stable over time. However, they are not replicated by any current version of the GM [ 22 ]. 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 [ 24 — 28 , 38 — 44 ].

These models construct ensembles of random networks that have some desired topological property taken as input from empirical data and are maximally random otherwise. In this way the models can perfectly replicate the observed strong heterogeneity of these purely local properties, and at the same time illustrate its immediate i. In the different context of financial networks, where the main challenge is a reliable inference of the unobserved topology of a network typically of interconnected firms or banks starting from partial, node-aggregate information [ 45 ], maximum-entropy models have recently turned out to deliver the best-performing reconstruction methods so far [ 43 — 45 ].

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.

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 [ 26 — 28 ].

This is somewhat surprising, given that the economic literature largely postulates that weighted properties are per se more informative than the corresponding binary ones. 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 [ 27 , 40 , 41 ]. 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. In this model the monetary flow is balanced for each country node based on the of trade partners degree.

The model produces expectations of the GDP of countries that are consistent with real data, using both the volume of trade flows and the topology of ITN as input. These studies indicate that, in order to devise improved models of the ITN, one should include the degrees, which are purely topological properties, among the main target quantities to replicate. This is the guideline we will follow in this paper. Unlike the GM, maximum-entropy models of trade are a priori non-explanatory, i.

However, they can in fact be used to select a posteriori an explicit, empirically validated functional dependence of the structure of the ITN on underlying explanatory factors. For models with country-specific constraints, this operation can be carried out as follows. If the hidden variables are indeed at least approximately found to be functions of some country-specific factors [i.

The model, which is a reformulation of a maximum-entropy model for binary networks with given degrees, predicts that the probability of a trade connection existing from country i to country j is. The model has been tested successfully in multiple ways [ 24 , 25 , 30 , 32 , 38 ]. 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 [ 42 ]. Since, as we mentioned, a maximum-entropy model of weighted networks with given strengths and degrees [ 40 ] can correctly replicate many structural properties of the ITN [ 41 ], it makes sense to reformulate such a model as an economically inspired model of the ITN.

The resulting model is confirmed to be in good accordance with both the topology and the volumes observed in the real ITN. 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. Firstly, one of the main lessons learned from the traditional GM is that the addition of geographic distances improves the fit to the empirical volumes ificantly. Indeed, in light of the large body of knowledge accumulated in the international economics literature, it is hard to imagine a realistic and economically meaningful model of international trade that does not allow for simple pair-wise quantities controlling for trade costs and incentives, including geography [ 9 , 10 ].

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 [ 25 ] or its current weighted extension [ 42 ], allow for the inclusion of both dyadic and node-specific 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 operation ensures that, whatever the new model looks like, its predictions for the expected trade volume between connected pairs of countries remain identical to the ones proposed in more traditional macroeconomic models.

An important difference, however, is that in our model the trade volumes will be drawn from a different probability distribution. Note that, since p ij is monotonic in G , the above expression is entirely general, i. It is also worth noticing that the explanatory factors used in Equations 4 and 5 need not coincide. 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.

The distribution P W is the key quantity that fully specifies the model and determines both the topology and the link weights of the ITN. From P W , focusing on a single pair i, j of nodes and integrating out all other pairs, we can define the dyadic distribution q ij w indicating the probability mass or density that w ij takes the particular value w. 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 q ij w that enforces both Equations 4 and 5.

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