For Adam Smith, wealth was related to the division of labor. As people and firms specialize in different activities, economic efficiency increases, suggesting that development is associated with an increase in the number of individual activities and with the complexity that emerges from the interactions between them. Here we develop a view of economic growth and development that gives a central role to the complexity of a country's economy by interpreting trade data as a bipartite network in which countries are connected to the products they export, and show that it is possible to quantify the complexity of a country's economy by characterizing the structure of this network. Furthermore, we show that the measures of complexity we derive are correlated with a country's level of income, and that deviations from this relationship are predictive of future growth. This suggests that countries tend to converge to the level of income dictated by the complexity of their productive structures, indicating that development efforts should focus on generating the conditions that would allow complexity to emerge to generate sustained growth and prosperity. economic development ͉ networks F or Adam Smith, the secret to the wealth of nations was related to the division of labor. As people and firms specialize in different activities, economic efficiency increases. This division of labor, however, is limited by the extent of the market: The bigger the market, the more its participants can specialize and the deeper the division of labor that can be achieved. This suggests that wealth and development are related to the complexity that emerges from the interactions between the increasing number of individual activities that conform an economy (1-3). Now, if all countries are connected to each other through a global market for inputs and outputs so that they can exploit a division of labor at the global scale, why have differences in Gross Domestic Product (GDP) per capita exploded over the past 2 centuries? (4, 5, * ) One possible answer is that some of the individual activities that arise from the division of labor described above cannot be imported, such as property rights, regulation, infrastructure, specific labor skills, etc., and so countries need to have them locally available to produce. Hence, the productivity of a country resides in the diversity of its available nontradable "capabilities," and therefore, cross-country differences in income can be explained by differences in economic complexity, as measured by the diversity of capabilities present in a country and their interactions. During the last 20 years, models of economic growth have often included the assumption that the variety of inputs that go into the production of the goods produced by a country affects that country's overall productivity (3, 6). There have been very few attempts, however, to bring this intuition to the data. In fact, the most frequently cited surveys of the empirical literature do not incorporate a single reference to any measure of diversity of inputs or complexity (7). We can create indirect measures of the capabilities available in a country by thinking of each capability as a building block or Lego piece. In this analogy, a product is equivalent to a Lego model, and a country is equivalent to a bucket of Legos. Countries will be able to make products for which they have all of the necessary capabilities, just like a child is able to produce a Lego model if the child's bucket contains all of the necessary Lego pieces. Using this analogy, the question of economic complexity is equivalent to asking whether we can infer properties such as the diversity and exclusivity of the Lego pieces inside a child's bucket by looking only at the models that a group of children, each with a different bucket of Legos, can make. Here we show that this is possible if we interpret data connecting countries to the products they export as a bipartite network and assume that this network is the result of a larger, tripartite network, connecting countries to the capabilities they have and products to the capabilities they require (Fig. 1A) . Hence, connections between countries and products signal the availability of capabilities in a country just like the creation of a model by a child signals the availability of a specific set of Lego pieces. Note that this interpretation says nothing of the processes whereby countries accumulate capabilities and the characteristics of an economy that might affect them. It just attempts to develop measures of the complexity of a country's economy at a point in time. However, the approach presented here can be seen as a building block of a theory that accounts for the process by which countries accumulate capabilities. A detailed analysis of capability accumulation is beyond the scope of this article but the implications of our approach will be discussed briefly in Discussion. In this article we develop a method to characterize the structure of bipartite networks, which we call the Method of Reflections, and apply it to trade data to illustrate how it can be used to extract relevant information about the availability of capabilities in a country. We interpret the variables produced by the Method of Reflections as indicators of economic complexity and show that the complexity of a country's economy is correlated with income and that deviations from this relationship are predictive of future growth, suggesting that countries tend to approach the level of income associated with the capability set available in them. We validate our measures of the capabilities available in a country by introducing a model and by showing empirically that our metrics are strongly correlated with the diversity of the labor inputs used in the production of a country's goods, approximated by using data on the use of labor inputs in the United States. Finally, we show that the level of complexity of a country's economy predicts the types of products that countries will be able to develop in the future, suggesting that the new products that a country develops depend substantially on the capabilities already available in that country. We look at country product associations by using international trade data with products disaggregated according to 3 alternative data sources and classifications: First, the Standard International Trade Classification (SITC) revision 4 at the 4-digit level (see ref. 8; the data are available at www.nber.org/data, http://cid.econ. udavis.edu/data/undata/undata.html, and www.chidalgo.com/ productspace/data.html); second, the COMTRADE Harmonized System at the 4-digit level; and third, the North American Industry Classification System (NAICS) at the 6-digit level (SI Appendix, Section 1). We interpret these data as bipartite networks in which countries are connected to the products they export (Fig. 1B) . Mathematically, we represent this network using the adjacency matrix M cp , where M cp ϭ 1 if country c is a significant exporter of product p and 0 otherwise. We consider country c to be a significant exporter of product p if its Revealed Comparative Advantage (RCA) (the share of product p in the export basket of country c to the share of product p in world trade) is greater than some threshold value, which we take as 1 in this exercise (RCA cp Ն 1) (see SI Appendix, Section 2). We characterize countries and products by introducing a family of variables capturing the structure of the network defined by M cp (SI Appendix, Section 3). Because of the symmetry of the bipartite network, we refer to this technique as the ''Method of Reflections,'' as the method produces a symmetric set of variables for the 2 types of nodes in the network (countries and products). The Method of Reflections consists of iteratively calculating the average value of the previous-level properties of a node's neighbors and is defined as the set of observables: for N Ն 1. With initial conditions given by the degree, or number of links, of countries and products: k c,0 and k p,0 represent, respectively, the observed levels of diversification of a country (the number of products exported by that country), and the ubiquity of a product (the number of countries exporting that product). Hence, we characterize each country through the vector k ជ c ϭ (k c,0 , k c,1 , k c,2 . . . k c,N ) and each product by the vector k ជ p ϭ (k p,0 ,k p,1 ,k p,2 , . . . ,k p,N ). For countries, even variables (k c,0 ,k c,2 ,k c,4 , . . . ) are generalized measures of diversification, whereas odd variables (k c,1 ,k c,3 ,k c,5 , . . . ) are generalized measures of the ubiquity of their exports. For products, even variables are related to their ubiquity and the ubiquity of other related products, whereas odd variables are related to the diversification of countries exporting those products. In network terms, k c,1 and k p,1 are known as the average nearest neighbor degree (9,10). Higher order variables, however, (N Ͼ 1) can be interpreted as a linear combination of the properties of all of the nodes in the network with coefficients given by the probability that a random walker that started at a given node ends up at another node after N steps (see SI Appendix, Section 4). We can begin understanding the type of information about countries captured by the Method of Reflections by looking at where countries are located in the space defined by the first two sets of variables produced by our method: k c,0 and k c,1 . Fig. 1C shows that there is a strong negative correlation between k c,0 and k c,1 (10, 11), meaning that diversified countries tend to export less ubiquitous products. Deviations from this behavior, however, are informative. For example, whereas Malaysia and Pakistan export the same  number of products, the products exported by Malaysia (k MYS,0 ϭ 104, k MYS,1 ϭ 18) are exported by fewer countries than those exported by Pakistan (k PAK,0 ϭ 104, k PAK,1 ϭ 27.5). Combining this fact with our third level of analysis, we see that Malaysian products are exported by more diversified countries than the exports of Pakistan (k MYS,2 ϭ 163 k PAK,2 ϭ 142, SI Appendix, Section 8). This suggests that the productive structure of Malaysia is more complex than that of Pakistan, due, as we will show shortly, to a larger number of capabilities available in Malaysia than in Pakistan. In SI Appendix we show that the negative relationship presented in the k c,0 -k c,1 diagram is not a consequence of variations in the level of diversification of countries and in the ubiquity of products. We prove this by creating 4 null models (11) that control, with increasing stringency, for the diversification of countries and the ubiquity of products and show that these distributions, per se, are not responsible for the negative relationship observed in the data (see SI Appendix, section 6). Minimalistic Model. We show that the location of countries in the k c,0 -k c,1 diagram is informative about the capabilities available in a country by introducing a simple model based on the assumption that country c will be able to produce product p if it has all of the required capabilities ( Fig. 2A) . We implement this model by considering a fixed number of capabilities in each country and represent this by using a matrix C ca , that is equal to 1 if country c has capability a and 0 otherwise. We represent the relationship between capabilities and the products that require them by a matrix ⌸ pa whose elements are equal to 1 if product p requires capability a and 0 otherwise. Using the notation introduced above, together with our only assumption, we can model the structure of the M cp matrix as: The simplest implementation of this model is to consider C ca ϭ 1 with probability r and 0 with probability 1 Ϫ r and ⌸ pa ϭ 1 with probability q and 0 with probability 1 Ϫ q. An emergent property of the matrix resulting from this model is that the average ubiquity of a country's products tends to decrease with its level of diversification for a wide range of parameters (Fig.
In REF , the authors present a simple method to infer the relative number of inputs available in a country from trade data connecting countries to the products they export.
The building blocks of economic complexity
Proceedings of the National Academy of Sciences of the United States of America
Economics Physics Medicine