0.1.1 Binary country-product (c-p) matrix.

In order to define a suitable economic metrics to compare the trades of different countries in different products, taking into account the difference in sizes and total export, as in [2], we use Balassa’s Revealed Comparative Advantage (RCA) [18]. Using its definition [3], we consider a country to be a competitive exporter of a product if the value of its RCA for such product overcomes some minimal threshold value . We take here this value to be as in standard economics literature (see File S1 for further details on the definition of the RCA).

We can therefore construct the binary country-product matrix whose generic element is:(1)saying that country can be considered an exporter of product if and only if () . If we represent [13] countries and products as nodes of a network we can pictorially say that the node of the country is linked to the node of the product iif . Since links are not permitted between two countries or two products, the matrix defines a bipartite country-product network. This means that the nodes are divided into two sets: of nodes (countries) and of nodes (products). Connections (links) are permitted only between couples of nodes belonging to different sets.

In what follows we analyze also the effects of the possibility of including weights in the country-products matrix. In particular, this will be done by defining the weighted country-product matrix through(2)with giving the total export (e.g. in US dollars) of country for product in the considered year.

The fundamental information about the structure of the international export of products is encrypted in the matrix . It is however a matrix with some hundreds of thousands of entries and the optimal way to extract useful information on the status of the single economies is a non-trivial task. A first insight is obtained by reordering the rows and columns of the matrix respectively by the total number of exported products by each country(3)and by the number of exporting countries

(4)The quantities and are the degree or coordination numbers of the nodes and in the bipartite network and are called respectively diversification of and ubiquity of [2]. As shown by Fig. 1, through this procedure, takes a quite marked triangular structure [2], [3] which is very far from what happens for instance with the same reordering of rows and columns starting from a completely random distribution of the binary entries (compare Figs. 1 and 2). Such an organization of the international trade of products looks very far from the standard view of Ricardian or Heckscher-Ohlin theories which predict as an optimal situation a high degree of specialization of national economies for which it would be possible to rearrange rows and columns so that the matrix would result almost diagonal or block-diagonal.

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Figure 2. Graphical representation of a artificial M_cp matrix with random binary entries (same number of entries of the matrix of Fig. 1) after reordering of rows and columns by respectively decreasing K_c and K_p . It is clear that even after such a reordering the matrix does not acquire a triangular structure as instead empirical data show.

https://doi.org/10.1371/journal.pone.0070726.g002

This structure makes clear that in the international trade we find countries exporting a large fraction of all products (highly diversified countries), and some others exporting a very small fraction of products (poorly diversified countries). At the same time the products exported by a small number of countries (less ubiquitous products), which are presumably of high complexity value as produced only by few countries, are exported practically only by highly diversified countries. It is therefore plausible that such structure is related to a ranking in terms of development and competitiveness among the economies of different nations.

The fact that the matrix can be arranged to get a substantially triangular shape rather than block-diagonal, suggests that the dynamical evolution of advanced economies is quite different from the standard view: as countries evolve becoming more and more complex, they acquire a higher degree of diversification rather than specialization. This marks a sort of analogy with the evolution of biological organisms in complex and varying ecosystems. The best adaptation is achieved when organisms can rely on a broad set of resources, rather than being dependent on very specific environmental conditions. In the same way diversified nations are not dependent on very specific market conditions. Moreover the structure of the matrix suggests that the larger is the present basket of products for a given country the more likely will be in the future to make new and innovative products for it.

We argue that diversification, at least at the country level, appears to be more important than comparative advantage arguments in assessing the competitiveness of countries.

0.3.1 Unweighted vs. weighted algorithm.

In Eq. (8), in order to analyze the properties of the global market and to determine the fitness of countries and the complexity of products, we have used as matrix both the binary (unweighted) one defined in Eq. (1) and the weighted one defined in Eq. (2). Clearly using the former or the latter will give different quantitative information, even if partial and qualitatively overlapping features are present.

The choice of using the unweighted and binary version of the country-product matrix is motivated by the following consideration: we believe that it represents better than the weighted one the potential of growth of a country. For instance, if an emerging country starts the export of a new product, the information about the export given by switching from to is more important in many aspects, for the evolution of that economy, than to know the volume of the export.

On the other side, the approach based on the weighted matrix determines the effect of the information about the relative importance of the different exporters of the same tradable good. In this way it can, for instance, better detect most influent countries in the global market dynamics in different product sectors. As mentioned in Sect. 0.1 there are in principle different possible choices for the weights in the matrix .

A first possible attempt towards an extensive generalization of our metrics is represented by the direct use of the RCA matrix which is the matrix defined by the RCA coefficients. However, such RCA coefficients suffer from a number of disadvantages. In fact in order to measure a very large RCA (), a country typically must own a very large share of the export of a product and, at the same time, this product must have a much lower average share of the world wealth. This usually happens for exporters of natural resources (especially raw materials such as crude oils, metals, coal, etc.) which are in general characterized by a small diversification. As examples of such a phenomenon, we can list Chile which owns about 30% of copper export and Saudi Arabia for crude oil. On the other hand most diversified countries, which include the richest and most advanced countries, on average are characterized by a more homogeneous set of RCA values which appear to be not dominated by a single product. In this way, the choice of RCA coefficients for weighting would favor those countries with a low diversification which, by chance, have a large amount of natural resources. For such reasons RCA has been discarded for a weighted version of our method.

It is much more reasonable and effective to define a weighted country-product matrix as in Eq. (2). This is a direct generalization of the binary matrix where the entries of the matrix can assume a value ranging from 0 to 1. We want to stress that the definition adopted is still an intensive version of the matrix from the product point of view. Indeed, given a product, each exporter of this product is weighted according to the owned share of that product, however the sum over all exporters of each products is normalized to one. That is, products are considered intensively. In other words we are not taking into account that different products have in general a different share of the global export.

The reasons for such a choice are twofold. We believe that the complexity of products is intrinsically independent on the volume export. In fact by keeping products as an intensive quantity we are still able to filter purely monetary effects linked to market prices, price inefficiencies, raw materials value, out of our method. At the same time, fixed a product, we can still consider the scale of each country which export the product.

As a final remark, it has to be observed that such weighted metrics behaves as an extensive economic indicator (for instance the total GDP of a country), but it does not trivially coincide with the GDP information. Similarly the binary/unweighted case follows the behavior of a per capita indicator as shown in Fig. 5. Starting from this observation we indicate from now on as intensive fitness the one resulting from the unweighted matrix and as extensive the one from the weighted case. The intensive/extensive feature must be only referred to their different economic behavior as discussed in Fig. 5. There is no reference to the properties of the matrix adopted to estimate the two metrics.

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Figure 5. Unweighted vs. weighted metrics.

The weighted metrics behaves as an extensive economic indicator (for instance the total GDP of a country), but it does not trivially coincide with the monetary information. Similarly the binary/unweighted case follows the behavior of a per capita indicator, in that case the GDP per capita.

https://doi.org/10.1371/journal.pone.0070726.g005

This new approach, in both binary and weighted version, is very different from the original linear one presented in [2] called Method of Reflections (MR) and also the results and predictions differ greatly. Some of the main results derived by our method are discussed in the Sect. 0.4 both for the case of country competitiveness and for the analysis of the complexity of products. After that in a following sections, after having briefly described the MR, and having analyzed its ultimate mathematical meaning [13], we present a direct point by point comparison with our new non-linear method making clear the conceptual and operative advantages of this new approach.

0.3.2 Uniqueness of the metrics’ fixed point.

Before turning to the results, we now discuss the robustness of our method. Given the rather complex structure of Eq. (8) it is not immediately clear whether a non-trivial fixed point exists and, if so, under which conditions on the country-product matrix (in the trivial case of a fixed point of course exists and is given by and ).

For our purposes, being the metrics defined as the fixed point of Eq. (8), we need this fixed point not only to exist, but also to be unique, since we want our result to be independent from the choice of the initial conditions. An analytical proof, due to the strong non-linearity, to the fact that the normalization step constrains the maps to be inside an high dimensional simplex and of course to the dependency on the shape of , if at all possible, is a hard task, out of the scope of the present work. For this reason we perform a numerical analysis of the map defined by Eq. (8). Our analyses are performed for a large number of randomly generated matrices of different sizes but with a triangular shape in analogy to what is observed in the real case. In our random model, by introducing ( and are the number of countries and products respectively), the elements of the row of the matrix are defined as(9)if and(10)if , and with . The results presented here are obtained with and but changing these values even significantly doesn’t seem to change the qualitative features of the convergence to the fixed point. We choose a value for comparable to the ratio of the real matrix, i.e. but also this parameter does not seem to be relevant. We analyze a sample of 300 matrices for 5 values of , i.e: 5, 10, 75 and 150. For each matrix obtained from this model we sample uniformly the simplex where is defined and the simplex where is defined. This correspond to extract random vectors from Dirichlet Distributions of vector parameter with all unitary components and with proper dimensionality. In order to use these vectors as initial conditions for the iterations we normalize them so that and . These randomly sampled vectors are used as initial conditions for the maps defined in Eq. (8). For each realization of 1000 initial conditions are tested. Convergence is always observed to a unique fixed point, which only depends on , for all values of and for all the single initial conditions tested.

We present the example of a simple random bipartite network with with and in order to be able to visualize it. The results are qualitatively similar in all the explored combinations of parameters. In Fig. 6 the typical convergence process is shown for the corresponding particular realization of . In the vectorial space defined by the Cartesian product of the two simplexes where and are defined, the Euclidean distance , where is the order of the iteration, from the point reached at the 80th iteration is evaluated. The red line represents the convergence process with the initial conditions given by and . A subset of the paths originated from the randomly sampled initial conditions are shown in grey. All the paths converge around iteration 40 and all the oscillations are damped. As shown in the inset the convergence is exponential . The exponent depends on the size of the matrix, with bigger matrices converging faster (for , ). In order to understand the meaning of the peculiar oscillations shown in Fig. 6, we plot in Fig. 7 the bipartite network relative to that particular realization of , and, considering the trajectory highlighted in blue, we draw the nodes with size (weight) proportional to and at each iteration. The oscillations in are due to the fact that the weight is being moved from one side to the other of the bipartite network, but these oscillations are damped by the normalization. Notice that this mechanism has the ability of leading to a fixed point also the completely disconnected sub-network formed by the 5th country (in red) and the 13th product. To conclude we can state that, given the observation that the fixed point of Eq.(8) does not depend on the initial condition, the metrics proposed are measuring an intrinsic property of the matrix.

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Figure 6. Euclidean distance from the 80th iteration (fixed point) for a particular realization of M_cp with N_c = 5, N_p = 15, P_h = 0.6 and P_l = 0.05.

The red line shows the path obtained with the standard initial conditions given by and . In grey the paths of a set of randomly sampled initial condition. In blue the particular path analyzed in fig. 7. The inset shows the exponential nature of the convergence.

https://doi.org/10.1371/journal.pone.0070726.g006

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Figure 7. Representation of the non-linear iterations on a simple bipartite network with different initial conditions.

Colored nodes represent countries, grey nodes represent products. A random initial condition (top) may give rise to oscillating behaviors (blue line in fig. 6) which are dumped by the normalization step. It should be noticed that even disconnected pieces of the network (red “country” node) are brought to a fixed point. The standard uniform initial condition follows a much smoother path (red dashed line in fig. 6) and converges to the same fixed point.

https://doi.org/10.1371/journal.pone.0070726.g007

0.4.1 Country analysis: BRIC and PIIGS countries.

Different economic analyses can be carried out in the framework of our approach. In this section we propose some relevant results to show the potential applications. On one hand the two metrics introduced in the method for ranking countries and products by themselves can provide important and new information on the analysis of the growth of countries. In particular the metrics which measures products complexity permits to quantify this feature in a non-monetary way, filtering out bias such as labor cost, market speculation (i.e. raw materials and commodities), inefficiencies of prices (i.e. commodities), etc. On the other hand we argue that deviations from the standard monetary and income-based indicators are also informative, especially for the assessment of economic and financial forecast about growth and stability of countries. However, this second type of analysis goes beyond the goal of this paper and will be discussed extensively in [16].

BRIC countries, namely Brazil, Russia, India and China, are a group of countries considered as emerging economic systems which have a high rate of growth. These four countries are considered similar from a GDP point of view, i.e. in respect of their GDP growth rate. However, we argue that from a fundamental point of view these four countries undergo a very different development: while India and China appears to have a well-grounded economic development characterized by a complex basket of exports, it is not the case for Brazil and especially Russia. In fact as shown in Fig. 8 panel , by analyzing BRIC countries in standard GDP terms, we find that in the last fifteen years all these countries appear very similar and are characterized by high rate of growth of their GDP (mostly above the world growth rate). However, looking at panel of the same figure, our metrics reveals a strong heterogeneity among these four countries which a conventional analysis is not able to capture. The evolution of the fitness, which as aforementioned we interpret as the degree of competitiveness of a productive system, reveals that, while India and especially China have strongly increased their competitiveness in the global economic systems, Brazil and in particular Russia, despite a growing GDP, have lost many positions according to the fitness ranking. The economic interpretation of such difference, on the basis of our metrics, is the following:

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Figure 8. Fundamental analysis of the BRIC countries according to our metrics.

We argue that India and China undergo a real economic development characterized by accumulation of new and more and more complex capabilities. Therefore the GDP growth corresponds to a real increase of the competitiveness of these two countries. Conversely we observe that the GDP growth of Brazil and Russia appears to be mainly fueled by the price bubble of the raw material sector and these countries are not using these extra richness to develop and accumulate new capabilities in order to settle a solid basis to their productive system.

https://doi.org/10.1371/journal.pone.0070726.g008

1. India and China (IC) reflects a genuine economic and industrial development characterized by accumulation of new, more and more complex capabilities. Therefore the GDP growth corresponds to a real increase of the competitiveness of these two countries.
2. Brazil and Russia (BR) are very important raw material exporters. We therefore argue that their GDP growth is mainly fueled by the price bubble which characterizes this sector. In this sense we interpret the decreasing competitiveness of Brazil and Russia in terms of the fact that they are not using their extra richness deriving from raw materials to develop and accumulate new capabilities in order to settle a solid industrial and technological basis to their productive system.

It is worth noticing that the idea that Brazil’s GDP growth is mainly depending on commodities is becoming popular only in the last two years and the consensus on such feature is not at all uniform (see Refs. [21] and [22] as examples of two different points of view on Brazil). If one would have used the new metrics one could have seen a significative loss of complexity of Brazil economic system years in advance. In fact from 2002 there is a clear and steady decrease of the Fitness of Brazil. This anticipation of the trend is a characteristic of this innovative methodology which measures the hidden potential and not just the present status. We argue that the situation for Russia is also somewhat similar. We can therefore conclude that the development of IC countries is well-grounded from a productive point of view differently from BR countries. We believe that the most interesting result concerns Brazil, indeed its growth is usually considered of the same kind of the one of India, China and other emerging Asian countries (e.g. Vietnam, Thailand, etc). Our analysis implies instead the opposite, Brazil growth is closer to the Russian case where the development is dominated by the market price of fossil fuels. We are aware that there may also exist macro-economic and political reasons that could determine lower export for a country given a level of capabilities and therefore our method would measure a lower level of competitiveness than what expected. In fact in the case of Brazil, besides being an important raw material exporters, there exists a strong state planning of the production, sectorial incentives and a strong boost of internal production against exports. However, the great advantage of our fundamental analysis with respect to conventional ones consists in clear quantitative statements that can be extensively tested [16].

Let us now consider a different set of countries, the so-called PIIGS, i.e. Portugal, Italy, Ireland Greece and Spain. They are European developed countries which are usually considered the most fragile economies from a financial point of view among European Union. Indeed, the rating of PIIGS’ sovereign debt is on average lower than the other members of EU.

Let us move to the analysis of fitness evolution for the PIIGS as shown in Fig. 9 (as a benchmark of a non-PIIGS we choose Netherlands).

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Figure 9. Fundamental analysis of the PIIGS countries (Portugal, Italy, Ireland Greece and Spain) according to our metrics.

We find a scenario which seems to be apparently in contrast with the rating of the sovereign debt of these countries. For instance we find that Greece, Portugal have an increasing fitness and Italy is always ranked in the top 5 positions along the time period considered. The main reason of this apparent discrepancy, in our opinion, relies on the fact there exists different regimes for the economic complexity. Many different factors are responsible for the economic growth: development of capabilities, national policies, wars, geo-political instabilities, importance and development of the financial sector, etc. Our metrics assess only one of these factors, the competitiveness of the productive systems of a nation. We believe that while this aspect is the main driving force for some regimes such as the one of emerging countries, it is not the case for developed ones. In fact PIIGS are all developed countries which have saturated their phase space of products.

https://doi.org/10.1371/journal.pone.0070726.g009

The fundamental analysis of the competitiveness points out a scenario in which Greece, Portugal have an increasing fitness, Spain and Italy a stable competitiveness ranking (and a behavior very similar to Netherlands) and Italy is even always ranked in the top 5 position, very close to the level of Germany. In addition Spain, Portugal and Italy in 2010 are above the average world fitness (). We want to recall that we are considering the intensive metrics which measures the intrinsic level of complexity that each country has developed. We stress that in the weighted analysis Italy is well below China as expected. Only Ireland exhibits a decreasing fitness in the intensive scenario. We also report the evolution of Iceland’s fitness as a prototype of a developed non-PIIGS country which gets in big financial troubles in the last decade.

The reasons for this apparent discrepancy between standard rating or evaluation of these countries and our results is twofold:

1. on one side it seems that the main source of the fragility of PIIGS countries has only a financial origin independently on the competitiveness of the productive systems, except Ireland for which both analysis give similar results;
2. on the other side, the main reason, in our opinion, relies on the fact different regimes exist for the economic complexity.

On this account it is clear that different factors concur to the economic development of a nation: development of capabilities indeed, but also national policies, wars, geo-political instabilities, importance and development of financial sector, etc. In the present framework we develop a metrics to assess only one of these factors, the competitiveness of the productive systems of a nation. We believe that while this aspect is the main driving force for some regimes such as the one of emerging countries, it is not the case for developed ones. In fact PIIGS are all developed countries and somehow they almost saturated their phase space of capabilities: in fact these countries are among the most diversified ones, especially Italy, Portugal and Spain. In this sense they are in a completely different economic regime in respect of emerging countries. We therefore argue that the main driving force of the economic growth of developed countries is no more the fast development or acquisition of new capabilities and the following invasion of the product space. Instead in mature developed countries, politics, in particular economic ones, and in general non-capabilities driven features appear to dominate the growth and the evolution of these countries (see [16] for a detailed discussion on this aspect). We want to point out that this does not imply that the acquisition of new capabilities has no impact for this type of countries. Instead we believe they play a different economic role due the fact that they almost saturated the space of capabilities, hence the space of products. In fact the development of new, and generally of high technological value, capabilities in developed countries usually triggers bursts of new high complexity products on the market. However, these events tend to be rarer with respect to the acquisition of already established capabilities as it happens for emerging countries.

It is worth noticing that this second explanation calls for the concept of heterogeneity in economic growth dynamics and prediction. On this account the result discussed in [16] clearly points in this novel direction: the dynamics of the development of countries shows a high degree of heterogeneity, consequently a novel approach is required and new concepts like selective predictability must be considered.

0.4.2 Extensive vs intensive metrics.

In section 0.3.1 we have introduced a generalization of our iterative method by considering suitable weights which partially take into account the export volumes. We now want to interpret, from an economic point of view, the kind of information carried by the two cases and spot the differences of the two analyses. Let us focus on some specific countries: Germany, China, Italy, USA, United Kingdom, Austria, India and Poland. In Fig. 10 we report the evolution of the intensive fitness (panel a) and of the extensive one (panel b). Focusing first our attention on Germany, China, Italy, USA, we find that the intensive fitness ranking does not reflect the traditional monetary prediction. In fact Italy’s fitness is higher than USA’s one and almost equal to the China’s one. In 2010 Italy is the most diversified country with respect to the export basket with more than 500 products (for which the RCA coefficient is above the threshold) and our metrics correctly grasp this feature. Once the weights are taken into account, we find instead (see also Table 1 for details) a ranking closer to the one provided by GDP even if significant differences persist. For instance, from a GDP-oriented analysis China results to be the 2nd-3rd economic power, in our framework, China is already the most competitive country in extensive terms.

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Figure 10. Economic interpretation of evolution of the fitness in the intensive and extensive case.

The intensive fitness gives a medium-long term information of the development of countries, in this sense, we can consider it as informative on the growth potential of a country. On the other hand the extensive analysis complements the information carried by the intensive fitness conveying a short term perspectives and giving a stronger emphasis to the monetary aspects.

https://doi.org/10.1371/journal.pone.0070726.g010

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Table 1. Countries’ Fitnesses.

https://doi.org/10.1371/journal.pone.0070726.t001

As a second point let us compare the two pairs United Kingdom-Austria and India-Poland. From an intensive point of view these pairs appear almost degenerate while extensively we observe that, as expected, bigger countries in both pairs have larger fitness. However, it is worth noticing that even if we consider the export volumes, the weighted fitness does not simply reproduce the GDP ranking or the relative monetary distance among these countries: the fitness ratio of two countries is not trivially the ratio of their GDP. In some sense, intensive analysis is able to spot niche of competitiveness, while extensive metrics moves the focus of the analysis to the scale of the economic system.

We argue that the intensive fitness conveys long-term information of the competitiveness of a country. The intensive metrics is a measure of potential of growth (especially for emerging countries, see also [16]) and somehow a measure of resilience and recovery features of economic systems (especially for developed countries). In this sense the results of Italy in the top 3 position of the intensive fitness ranking is not surprising since historically Italy is known as a very resilient system. In the light of our fundamental analysis and neglecting specific economic policies and exogenous aspect (which could become dominant as discussed in the previous section and may enhance or contract the recovery from the recent global crisis), Italian productive system has an intrinsic strength and recovery capacity, much higher that other european countries, say Spain, Ireland, Greece.

We point out once again, that our metrics provides undoubtedly new information (for instance the Brazil analysis), but the novelty of our method relies on the fact that it gives a quantitative assessment which can be tested with respect to standard economic indicators. On the other hand the weighted fitness complements the information carried by the intensive fitness since it gives a present and short term perspectives of the country analysis giving a stronger emphasis to the monetary aspects. Russia and Brazil are paradigmatic cases in this sense. In a short term horizon or, more precisely, in the monetary horizon set by the availability of raw materials in these two countries, they are competitive (monetary information) but in terms of diversification, resilience, adaptability and, in general, competitiveness of their productive system (intensive information) they appear weak, or, at least, much weaker than other emerging countries.

0.4.3 Products.

Similarly to countries, our method defines a metrics for the complexity of products. A part from the MR of HH, this is a completely novel measure because we are not aware on the existence of economic indicators for the complexity of product which do not rely on monetary estimate. In fact a standard measure adopted is the market value of products, however, this quantity suffers from strong bias due to market speculation, labor cost, etc. While it is reasonable to believe that products characterized by a high complexity are likely to have high market prices, it is very easy to find striking counterexamples where simple products have anomalously high price, for instance the Tulip mania of XVII century. Therefore we propose the Complexity of products as a new synthetic indicator which permits to quantitatively assess the complexity of products in a non-monetary and non-market oriented way.

In this respect a large spectrum of analysis can be performed: detailed analysis of the export basket of countries, relative strength/weakness of countries with respect to export of specific products, indices to quantify the complexity of economic sectors, etc. In addition, in analogy to the evolution of country fitness, it is possible to investigate the evolution of complexity of products year by year, in such a way, in principle, we may track the evolution of the economic cycles and the development or the technological contraction of specific sectors.

As an example, in Fig. 11 we show the time evolution of the complexity for a selection of cereals from 1995 to 2010. Cereals result to be organized into two main groups: the former has an average complexity around the average complexity of all products (i.e. ), while the latter is formed of cereals whose level of sophistication is much lower than the previous as measured by our metrics (i.e. ). Given this observation, among cereals, our method reveals two different complexity regimes for cultivation. In order to verify if the two classes correspond to a real difference in the level of technology of the country exporting them we analyze the typical usage of oats and rye. Supporting the finding that these two cereals are not typical of a substance economic system, we find that they are used in livestock industry and brewed-product industry.

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Figure 11. Time evolution of the product complexity from 1995 to 2010 for a selection of cereals which result to be organized into two main groups.

The former group has an average complexity around the average complexity of all products, Q∼1, the latter one is composed of cereals whose level of sophistication is much lower than the previous as measured by our metrics, Q∼10⁻³, 10⁻⁴). By analyzing the typical typical usage of oats and rye we find that these two cereals are not typical of a substance economic system since they are used in livestock industry and brewed-product industry.

https://doi.org/10.1371/journal.pone.0070726.g011

In general, the time evolution of the product complexity must be carefully analyzed because of the specific structure of the non-linear coupled maps defining the metrics. In fact, while the country fitness is very robust with respect to errors in the database, the complexity is very sensitive to changes of the exporters of a given product, especially when the variations are due to low-fitness countries. On one hand we verified that the cleaning procedure of data is able to fix the wide anomalous oscillations of several orders of magnitude of some product complexity due to wrong custom reports - especially from small african countries. On the other hand, on average, the complexity of products shows an intrinsically higher degree of volatility with respect to fitness of countries even in a errorless dataset. In fact, given the economic assumptions underlying the metrics, a new (real and not due to errors) exporter can produce a significant variation of the complexity of a product while the addition of a product to the export basket of a country very likely will have a small effect on its fitness.

A hand-waving argument for this aspect is obtained by simply observing that since the fitness of a country is given by the sum of the complexities of its products, if we assume that products have the same degree of volatility of their complexity and are statistically independent, the volatility of the fitness of the country will by roughly times smaller. The opposite is not true because the complexity of a product is not at all the sum of the fitnesses of its producers, but a highly non-linear combination of them. For instance, for high complexity products we expect that very likely a new exporter (i.e. producer) will have a lower fitness than the typical fitness of the exporters of that product and therefore a short term decrease of complexity is, on average, expected. In this sense we argue that general trends and cycles are the meaningful analysis rather than short term variations of the level of technology in the case of products.

As a final remark, It is worth noticing that the knowledge of the intrinsic value of a product (i.e. the complexity) is critical for goods like commodities which are subject to strong speculative bubbles and whose market prices, differently from stock prices, are affected by strong inefficiencies, for instance the agricultural sector and in particular cereals. A systematic analysis of the metrics for product complexity and the features of product space will be discussed in future works.

0.5.1 Conceptual and mathematical problems.

• We observe that the nature of the and completely changes from the starting order to the following one: while the starting point of the iteration is extensive in the number of products and countries, the following order are intensive with respect to products and countries because of the average considered. This fact derives from the expression(12)considering that is the mean diversification of the countries exporting all products exported by country which therefore is a first order measure of the complexity of the product. Equation (12) makes clear a fundamental difference between our non-linear algorithm and the MR; it basically states that at first order the successfulness of a country is given by the average of the “complexity” of its products. This is very different from Eq. (8) for which instead the fitness of a country is given by the sum of the “complexity” of its products. This implies that, while in the MR two countries having the same mean complexity of the exports are supposed to have the same competitiveness independently of the relative diversification, in our method both the mean complexity of products and the diversification are, as natural, important in determining the fitness of a country in the global competition. Let us make an example to make this crucial point clear. In the HH scheme, paradoxically, a greatly diversified country (say about products given a total of about 1000 products) with average complexity of its export set equal to , whatever is , would have the same competitiveness at the following iteration step of a country exporting only one product with . Therefore the HH scheme is not consistent with respect to the assumptions underlying the capability arguments implying the importance of the concept of diversification.
• The highly non-linear (quasi-extremal) relation between competitiveness of countries and complexity of products, required by the triangular structure of the country-product matrix, cannot be implemented through an average as discussed by HH. As explained above, the triangularity of the matrix implies that the information that some countries with small competitiveness (or development) export a product must bound the complexity from below, regardless of the competitiveness of the most developed exporters. Therefore one would expect a strongly non-linear and almost extremal relation between the complexity of a product and the competitiveness of the producers. Instead in the MR model at each order the complexity of a product is given basically by the average of the of its producers, so that the information about the most complex countries exporting this product is as important as the information about the less complex ones.
• As shown in the next section through an appropriate toy model, it is simple to see that the variables describing the competitiveness of countries in the MR rapidly loose correlation with the capabilities of the countries when iterated.
• The MR changes the economic meaning of the iteration at each iteration. It can be shown that can be linearly related directly to by substituting the second equation of (11) into the first one (see File S1 for an algebraic approach). A similar argument can be made for even and for odd ones. However, it looks quite strange that in an empirical and phenomenological iterative approach to the ranking of countries and products the iterated quantities have different economic “dimensions” (averages of averages of ubiquities or diversification, respectively) depending on the odd or even order of the iteration. Even iterations with the same parity change their economic meaning throughout the iteration procedure (as the number of averages increases). The economic and statistical interpretation of these quantities is rapidly lost when increasing the order . In our framework the variables are simply the refinement of the ones of previous iteration and the iteration procedure has to be seen as an algorithm to solve the self-consistent fixed point equation.
• In [2] the authors consider at the end of the iterations for the economic analysis the rescaled quantity(13)where is the arithmetic mean of over all countries and is the standard deviation of over the same set.

By using an algebraic approach, it is possible to show (see File S1 and [13] which is, as far as we know, the first paper in which such issue is raised) that the MR makes all to converge to the same constant independent on the index , which is therefore a trivial fixed point of the transformation relating to . This is basically due to the fact that, writing in vectorial form these linear equations, the linear operator characterizing the linear transformation is the transposed of an ergodic Markov transition operator.

This explains why the authors of [2] subtract the mean value from before any economic analysis. Indeed this accounts for the subtraction of the fixed point from all . In a similar way it is possible to see that the division by the standard deviation to obtain in Eq. (13) basically accounts for the contraction factor of the distribution of the set around at increasing order due to the asymptotic convergence to such a single value for all . The fact that the authors of the MR stop the analysis at in [2] can be explained by the fact that this convergence is exponentially fast and at the value of the numerical limits of resolution of different are reached.

In an empirically defined algorithm the quantities involved in its formulation, and not to a vanishing component of them, should be directly related to observables.

Two critical issues emerges from this mathematical observation. On one hand the MR produces a shrinkage of the and distributions. Even if they are rescaled at the iteration, the behavior of the algorithm is conceptually wrong because we would expect that differences among countries are in general magnified by one iteration step and not reduced. The reason of such expected magnification is that if we compare a poorly diversified country producing ubiquitous products and a diversified ones exporting almost everything, once the information about the complexity of products is inserted in the method through the iterations, the distance between the variables measuring the successfulness of these to countries must increase.

On the other hand, the previous mathematical arguments shows that the correct way to extract the rescaled is to consider the eigenvector associated to the second largest eigenvalues of a fixed point equation (see File S1 and [13]). Even if the method is presented as an iterative method, the HH complexity index (i.e. the variables) cannot be self-consistently obtained iteratively in the form in which the MR is presented in [2] because their index is the eigenvector associated to the second largest eigenvalue of the transposed Markov operator [13].

As a final remark in [23] (pag. 24) it has been correctly noted that the iterative approach is problematic to measure the HH complexity index for countries and products, however the authors still unexplainably renormalize the variables obtained from the second eigenvector.

In summary it is possible to see that, extending the analysis in [13], the MR suffers of different critical aspects, which in our opinion make necessary a deep revision of the approach to the measure of the complexity of countries and products towards a non-linear approach. We recall that the non-linearity of the method, before even testing the metrics on economic benchmarks, is a key element to properly address the conceptual and economic consistency of a method based on the complexity/capabilities arguments which are intrinsically non-linear as extensively discussed in this paper (conceptual consistency which instead is one of missing elements of the HH scheme). We argue that a conceptual consistency of the method is even more crucial in this case because the metrics is not grounded by any economic theory.

0.6 Results IV: Comparison between our Metrics and the Method of Reflections.

In this section we provide a direct comparison between our non-linear algorithm determining the economic competitiveness of countries and the complexity of exports and the MR method.

First of all in the next Subsection 0.6.1 we show, through the use of a simple but significant toy model, that while in the MR method the correlations between the competitiveness of countries and their capabilities are rapidly lost when increasing the order of the iteration, in our method they are kept constant at all order.

In the subsequent Subsection 0.6.2 we give a direct comparison of the ranking of countries coming out from both our method and the MR. In particular we highlight the most meaningful examples of the countries with a rapid economic development as eastern Asian countries and countries whose economy is basically determined, not by a development of advanced technological capabilities, but by the monopolistic export of natural resources as oil.

0.6.1 Toy model.

It is instructive to analyze a simple toy model where we can explicitly introduce capabilities and test how the two metrics are able to extract information from . Actually, in the real world it is impossible to directly access the vector of capabilities that each country owns. Nevertheless, it is possible to study a simple model (originally proposed in [2]) in which we may explicitly define the capabilities that each country owns and how they combine to produce products. To this end we need to define two matrices already introduced in Sect. 0.2: a country-capability matrix, whose entries specify which capabilities are owned by a country, and a capability-product matrix whose elements specify which capabilities are required to make a product. The model is completed by introducing the simple rule to build the matrix: each country exports a product if and only if it has all the capabilities needed to produce it. In formulas is defined exactly as in Eq. (5).

In this way we now have access to the set of information on which the theory of hidden capabilities is based, i.e. the endowment of capabilities of a country, and we can now compare the asymptotic results of the two different iterative procedures with the real number of capabilities assigned to each country.

We implement the model by extracting random binary numbers, 0 or 1, to fill the and matrices: the entries are equal to respectively with probability and . We consider capabilities, countries and products (following exactly [2]).

In Fig. 12 we show the result of the two methods performed on the artificial matrix obtained from this toy model. Clearly, in this extremely simple framework, the best information about the capabilities is given by the diversification, which corresponds to the first order of iteration of both measures (up to a normalization factor): this is due to the fact that there is no difference whatsoever in the importance of different capabilities, and they are randomly linked to countries and products. We also show the Pearson’s correlation coefficient between the two different measures and the assigned capabilities, with respect to the iteration order. Fitness obtained by our approach correctly grasps the relevant information present in the matrix and does not significantly change with the iteration (the reason why these correlations do not improve has to be found in the relative simplicity and randomness of the model, as discussed below). Conversely obtained by the MR seems to be loosing its meaning when the equations are iterated and it is not possible to observe an asymptotic correlation value before the machine precision breaks down.

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Figure 12. Testing the capability information content measured by the two methods.

(Left) Results of the iterations on the toy model matrix. Fitness preserves the first order information, while appears to be rapidly destroying any correlation with the assigned capabilities. We plot the logarithm of Fitness and the rescaled at four different orders of iteration. (Right) Pearson’s correlation between the measures of complexity and the number of assigned capabilities vs. the iteration order. While Fitness maintains the same level of correlation of the first step, iterating the measure leads to a destruction of information. It is to be noted that in this trivial model the M_cp matrix does not contain more information than the simple diversification. Again, the logarithm of Fitness and the rescaled are considered.

https://doi.org/10.1371/journal.pone.0070726.g012

0.6.2 Economic playground: is China 2nd or 34th?

So far we tested our method and the MR with respect to theoretical aspects and toy models designed to verify the conceptual consistency of the two approaches. We now move our attention to real economic data.

A first striking observation is the anomalously low competitiveness of China in the MR scenario. Indeed MR ranks China in the 29th position in 2010 (see [23] pag. 64), just below Romania which is 27th. This result appears rather odd as it would imply that nowadays competitiveness of China is very similar to the one of Romania and far below the one of western countries. Standard economic analyses show instead that China is significantly eroding the competitiveness gap with respect to developed countries and always appears in the very top positions whatever economic indicator is adopted. Therefore, in order to test the economic consistency of the two methods, we are interested in comparing a set of countries which undergo a large variation of ranking in the two frameworks (i.e. China, India, Cyprus, Qatar, see panel a of Fig. 13). In the view of standard analysis, they represents respectively two well-established emerging countries whatever economic criterion we consider, an european country with low GDP per capita and an oil exporter.

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Figure 13. Comparison of the country ranking between our metrics and the MR.

In panel a we compare the two rankings while in panel b and c we respectively show the time evolution of the K_c and the fitness. China, India, Romania, Cyprus and Qatar are highlighted for their anomalous behavior in the MR framework and are paradigmatic with respect to the conceptual weakness of this method. In the MR, poorly diversified systems (for instance Qatar and all the oil exporters) are characterized by relatively high level of competitiveness. On other hand countries with very large diversification (China and India) are penalized and medium sized countries tend to be favored by the MR algorithm (Cyprus and Romania). This is due to the fact that the competitiveness of a country in the MR method are averages and therefore rankings are set by the average complexity of the exported products, independently on the level of diversification. Conversely, in our framework, the fitness of a country is an extensive variable with respect to the number of exported product so that the average complexity of the products and the diversification of a country are both taken into account.

https://doi.org/10.1371/journal.pone.0070726.g013

We do not make a direct comparison between our metrics and the ranking of [23] because our dataset is slightly different from the one used in [23] and for a consistent test we prefer to perform the MR on our dataset. In the present study we use the BACI dataset which is grounded on the UN Comtrade dataset. In addition we perform a further step of data cleaning. A second difference stays in the number of countries: 128 in [23], 148 in the present analysis.

In spite of some minor differences, the results of the MR on our datasets appear to be similar with respect to the one found in [23] - in fact, as shown in Fig. 13, the MR on our datasets ranks China in 33th position and Romania in 34th (compare panel b and [23] pag. 64).

The anomalous position of China is even more striking when we follow the evolution of the variable of the MR method from 1995 to 2010 (panel b of Fig. 13) where we find that the competitiveness of China follows a growth pattern which does not at all reflect the fact China is now the second GDP power behind USA. We surprisingly find that in MR framework Cyprus and Romania overcome the growth of China in the last years of our analysis. In other words, according to the MR, China, Romania and Cyprus result to be countries characterized by a very similar competitiveness and a similar pattern of growth. This scenario appears to be inconsistent with almost all economic analysis of these three countries.

Conversely our method (panel c of Fig. 13) on one side spots the spectacular growth of the chinese productive system in the last fifteen years which was ranked in 13rd position in 1995 and is now in the 2nd position just below Germany which is the country with the highest fitness. On the other hand it depicts Romania and Cyprus as economies of a completely different kind with respect to China: they are growing economies, but we do not spot, as in the chinese case, the tremendous erosion of competitiveness against most developed countries.

The economic scenario for India is even worst according to MR. India is ranked far below China, Cyprus and Romania and competes with Qatar which is a country with a very low diversification (as it happens for almost all oil exporters). Instead in our method India is an emerging country, above Romania and Cyprus, while Qatar is one of the countries with lowest fitness and with a decreasing competitiveness.

In general in the framework of MR all oil exporters (Kuwait, Saudi Arabia, Iraq, Venezuela, etc), which are paradigmatic of poorly diversified systems, are characterized by relatively high level of competitiveness and tremendous oscillations (compare in [24] for instance Kuwait ranking in 2007 and 2008. Kuwait drops in 1 year from position 66, a relatively high position for a very poorly diversified countries, to position 113. For an explanation of the instability of the HH ranking see [15]). By consequence the MR also predicts that raw materials are not among those products with very low complexity as it is expected from the observation that a country owns raw materials reserves only by a matter of chance. On the other hand countries with very large diversification are systematically penalized and medium sized countries tend to be favored by the MR algorithm. As previously observed, the reasons for such a behavior are in the fact that the variables representing the competitiveness of a country in the MR method are linear averages. It follows that the MR ranking is set by the average complexity of the products exported by a country, with an unclear dependence on the level of diversification. This explains why China and India are so poorly ranked and why poorly diversified countries are often over-ranked by the MR: even though China and India have a very diversified export basket, the average complexity of their export is very close to countries much less diversified as Romania, Cyprus and oil exporters.

Instead, in our framework, the fitness of a country is an extensive variable with respect to the number of products exported and properly takes into account both aspects: the average complexity of the products and the diversification of a country.

To sum up, the conceptual flaws of MR produce inconsistent economic results because, differently from the spirit of the theory of capabilities, in the mathematical expression of MR the diversification does not represent a competitive advantage.