# Maximum-Entropy Tools for Economic Fitness and Complexity

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## Abstract

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## 1. Introduction

#### 1.1. Economic Complexity

#### 1.2. Revealed Comparative Advantage: Current Practice and Flaws

- The RCA as a null model represents a fully connected or very dense network, as by the definition in Equation (6) it has a non-zero value for each i and $\alpha $ that have a non-zero total trade. In practice, this is the case for almost all links. In contrast, the world trade network is quite sparse with only 2–4% of all potential links realized throughout the analyzed years.
- The current definition of the RCA only applies to the bipartite network of countries and commodities, while the original world trade network contains another dimension of information, being the receiving importing country. Keeping in mind that a null model should mimic the original network, this importer dimension should also be represented in any appropriate null model—especially so, because the trade weight that the RCA null model would expect does not depend at all on the receiving country, while in reality this is of course of major importance (one would expect more trade to a country with a lot of incoming trade).
- Most importantly, the current methodology does not take into account the statistical significance of the filtered values. An RCA of over 1 could signify an important export product of a country, but could just as well be due to a statistical fluctuation through the years. This flaw is something that Tacchella et al. also partially realized (see supplement of [7]), leading them to develop a hidden Markov model approach to binarize the country-commodity matrix to reduce this noise. We choose a different path, keeping to the original data and performing a statistical analysis to keep noise at bay.

## 2. Methodology

#### 2.1. The Extended RCA

#### 2.2. Extension to the Multiplex Network

#### 2.3. Link Weight Probability Distribution

- Directed binary configuration model (DBCM),
- Multiplex directed binary configuration model (MDBCM),
- Strength-replaced MDBCM.

#### 2.4. The Directed Binary Configuration Model

**G**exists maximizes the Shannon–Gibbs entropy, subject to the constraint defined above (Equation (16)) and enforcing a normalized probability distribution:

**G***is to tune the Lagrange multipliers so that the likelihood of retrieving that original graph is maximized. Following [14], this is achieved by setting

#### 2.5. Multiplex Directed Binary Configuration Model

#### 2.6. Strength-Replaced MDBCM

#### 2.7. The Weight Unit Probability

#### 2.8. Statistical Significance

#### 2.9. Practical Implementation

- Calculate $\langle {w}_{i,j}^{\alpha}\rangle $ for each link using the extended RCA as defined in Equation (7).
- Find the hidden variables—and with that, ${p}_{i,j}^{\alpha}$—applying either the DBCM or the (regular or strength-replaced) MDBCM, by solving Equations (29) or (34), respectively.
- Combine $\langle {w}_{i,j}^{\alpha}\rangle $ and ${p}_{i,j}^{\alpha}$ in Equation (43) to find the variance on each link $i,j,\alpha $.
- Use Equation (40) to find the z-score of each country-commodity pair $i,\alpha $ and filter all links in the bipartite network using a threshold on the z-score (typically $z\ge 1$, $z\ge 2$ or similar).
- Apply the fitness and complexity algorithm as developed Tacchella et al. in [2].

## 3. Results

#### 3.1. Comparison with Previous Results

#### 3.1.1. Evolution of Fitness and Complexity

#### 3.1.2. Ranking Countries by Fitness

#### 3.1.3. Correlation with GDP per Capita

#### 3.2. Results with Filtering on Higher Statistical Significance

#### 3.3. Additional Results: z-Score Spectra

## 4. Discussion

## 5. Materials and Methods

#### 5.1. Data

#### 5.2. Numerical Methods

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

WTN | World Trade Network |

RCA | Revealed Comparative Advantage |

DBCM | Directed Binary Configuration Model |

MDBCM | Multiplex Directed Binary Configuration Model |

## References

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**Figure 1.**Evolution of country fitnesses throughout the iterations of the fitness and complexity algorithm both clearly show a convergence to a fixed point (using $z\ge 0$ for exact replication of revealed comparative advantage (RCA) filtering).

**Figure 2.**The normalized gross domestic product (GDP) per capita and fitness in 1995 are correlated with a correlation coefficient of 0.64 in the reproduction of the original (RCA filtered) results, with a standard error of 0.052 (which is substantial on this scale).

**Figure 3.**The inverse cumulative degree distribution of exporting countries in 2012 after filtering with different z-score thresholds clearly exposes the increased sparseness of the network after filtering.

**Figure 4.**Normalized GDP per capita versus fitness (1995) plots show a similar correlation for higher z-score filtering thresholds. The standard error of the correlation decreases remarkably with higher thresholds (compared to 0.052 for $z\ge 0$). (

**a**) $z\ge 1$ filtered, correlation coefficient: 0.66, standard error: 0.034; (

**b**) $z\ge 2$ filtered, correlation coefficient: 0.64, standard error: 0.028.

**Figure 5.**Commodity complexity spectra, showing the export product baskets of the USA and China in terms of their complexity and z-score. (

**a**) USA in 1995, with a relatively large number of high complexity commodities with $z\ge 2$; (

**b**) China in 1995, with a most high complexity commodities just under $z=0$.

Tacchella et al. (2010) | Replication (2012) |
---|---|

Germany | Germany |

China | China |

Italy | Italy |

Japan | Japan |

USA | USA |

France | Belgium |

UK | France |

Austria | Netherlands |

Spain | India |

Belgium | UK |

**Table 2.**Some examples of commodities with $0\le z\le 1$ for the Netherlands in 2004. These are allowed to pass through the original revealed comparative advantage (RCA) filter, but are denied by any filter with $z\ge 1$.

Fish (fresh or chilled) |

Peas |

Rubber inner tyre tubes |

Sacks and bags of jute |

Compacting machinery |

Resistance welding machines |

Electric lamps and light fittings |

z ≥ 0 | z ≥ 1 | z ≥ 2 |
---|---|---|

China | China | China |

Germany | Germany | Germany |

USA | USA | Italy |

Japan | Japan | Japan |

Italy | Italy | USA |

India | Belgium | Belgium |

Belgium | India | India |

France | France | France |

Netherlands | Netherlands | Netherlands |

Spain | Spain | UK |

UK | UK | Spain |

Hong Kong | Hong Kong | Switzerland |

Switzerland | Switzerland | Hong Kong |

Czech Republic | Austria | Austria |

Austria | Czech Republic | Czech Republic |

South Korea | South Korea | South Korea |

Sweden | Sweden | Sweden |

Poland | Turkey | Thailand |

Turkey | Thailand | Denmark |

Denmark | Malaysia | Turkey |

Thailand | Denmark | Singapore |

Malaysia | Poland | Malaysia |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Krantz, R.; Gemmetto, V.; Garlaschelli, D.
Maximum-Entropy Tools for Economic Fitness and Complexity. *Entropy* **2018**, *20*, 743.
https://doi.org/10.3390/e20100743

**AMA Style**

Krantz R, Gemmetto V, Garlaschelli D.
Maximum-Entropy Tools for Economic Fitness and Complexity. *Entropy*. 2018; 20(10):743.
https://doi.org/10.3390/e20100743

**Chicago/Turabian Style**

Krantz, Ruben, Valerio Gemmetto, and Diego Garlaschelli.
2018. "Maximum-Entropy Tools for Economic Fitness and Complexity" *Entropy* 20, no. 10: 743.
https://doi.org/10.3390/e20100743