# Unfolding the Complexity of the Global Value Chain: Strength and Entropy in the Single-Layer, Multiplex, and Multi-Layer International Trade Networks

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Data Set

#### 2.2. Building the Single-Layer, Multiplex, and Multi-Layer Networks

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Allard, A.; Serrano, M.Á.; García-Pérez, G.; Boguñá, M. The geometric nature of weights in real complex networks. Nat. Commun.
**2017**, 8, 14103. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Caldarelli, G.; Cristelli, M.; Gabrielli, A.; Pietronero, L.; Scala, A.; Tacchella, A. A network analysis of countries’ export flows: Firm grounds for the building blocks of the economy. PLoS ONE
**2012**, 7, e47278. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Caldarelli, G.; Wolf, S.; Moreno, Y. Physics of humans, physics for society. Nat. Phys.
**2018**, 14, 870. [Google Scholar] [CrossRef] - Cingolani, I.; Panzarasa, P.; Tajoli, L. Countries’ positions in the international global value networks: Centrality and economic performance. Appl. Netw. Sci.
**2017**, 2, 21. [Google Scholar] [CrossRef] [PubMed] - Cristelli, M.; Tacchella, A.; Pietronero, L. The heterogeneous dynamics of economic complexity. PLoS ONE
**2015**, 10, e0117174. [Google Scholar] [CrossRef] [PubMed] - De Benedictis, L.; Tajoli, L. The world trade network. World Econ.
**2011**, 34, 1417–1454. [Google Scholar] [CrossRef] - Fagiolo, G.; Reyes, J.; Schiavo, S. World-trade web: Topological properties, dynamics, and evolution. Phys. Rev. E
**2009**, 79, 036115. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Formichini, M.; Cimini, G.; Pugliese, E.; Gabrielli, A. Measuring the Impact of Technological Innovations on Industrial Products through a Multilayer Network Approach. Preprints
**2018**. [Google Scholar] [CrossRef] - Garlaschelli, D.; Loffredo, M.I. Fitness-dependent topological properties of the world trade web. Phys. Rev. Lett.
**2004**, 93, 188701. [Google Scholar] [CrossRef] [PubMed] - Garlaschelli, D.; Loffredo, M.I. Structure and evolution of the world trade network. Phys. A Stat. Mech. Appl.
**2005**, 355, 138–144. [Google Scholar] [CrossRef] [Green Version] - He, J.; Deem, M.W. Structure and response in the world trade network. Phys. Rev. Lett.
**2010**, 105, 198701. [Google Scholar] [CrossRef] [PubMed] - Hidalgo, C.A.; Klinger, B.; Barabási, A.L.; Hausmann, R. The product space conditions the development of nations. Science
**2007**, 317, 482–487. [Google Scholar] [CrossRef] [PubMed] - Hidalgo, C.A.; Hausmann, R. The building blocks of economic complexity. Proc. Natl. Acad. Sci. USA
**2009**, 106, 10570–10575. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Serrano, M.A.; Boguná, M. Topology of the world trade web. Phys. Rev. E
**2003**, 68, 015101. [Google Scholar] [CrossRef] [PubMed] - Tacchella, A.; Cristelli, M.; Caldarelli, G.; Gabrielli, A.; Pietronero, L. A new metrics for countries’ fitness and products’ complexity. Sci. Rep.
**2012**, 2, 723. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Schweitzer, F.; Fagiolo, G.; Sornette, D.; Vega-Redondo, F.; Vespignani, A.; White, D.R. Economic networks: The new challenges. Science
**2009**, 325, 422–425. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Barigozzi, M.; Fagiolo, G.; Mangioni, G. Community structure in the multi-network of international trade. In Complex Networks; Springer: Berlin, Germany, 2011; pp. 163–175. [Google Scholar]
- Barigozzi, M.; Fagiolo, G.; Mangioni, G. Identifying the community structure of the international-trade multi-network. Phys. A Stat. Mech. Appl.
**2011**, 390, 2051–2066. [Google Scholar] [CrossRef] [Green Version] - Piccardi, C.; Tajoli, L. Existence and significance of communities in the world trade web. Phys. Rev. E
**2012**, 85, 066119. [Google Scholar] [CrossRef] [PubMed] - Newman, M. Networks; Oxford University Press: Oxford, UK, 2018. [Google Scholar]
- Saracco, F.; Di Clemente, R.; Gabrielli, A.; Squartini, T. Randomizing bipartite networks: The case of the World Trade Web. Sci. Rep.
**2015**, 5, 10595. [Google Scholar] [CrossRef] [PubMed] - Cristelli, M.; Gabrielli, A.; Tacchella, A.; Caldarelli, G.; Pietronero, L. Measuring the intangibles: A metrics for the economic complexity of countries and products. PLoS ONE
**2013**, 8, e70726. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Battiston, F.; Nicosia, V.; Latora, V. Structural measures for multiplex networks. Phys. Rev. E
**2014**, 89, 032804. [Google Scholar] [CrossRef] [PubMed] - Kivelä, M.; Arenas, A.; Barthelemy, M.; Gleeson, J.P.; Moreno, Y.; Porter, M.A. Multilayer networks. J. Complex Netw.
**2014**, 2, 203–271. [Google Scholar] [CrossRef] [Green Version] - Mastrandrea, R.; Squartini, T.; Fagiolo, G.; Garlaschelli, D. Reconstructing the world trade multiplex: The role of intensive and extensive biases. Phys. Rev. E
**2014**, 90, 062804. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lee, K.M.; Goh, K.I. Strength of weak layers in cascading failures on multiplex networks: Case of the international trade network. Sci. Rep.
**2016**, 6, 26346. [Google Scholar] [CrossRef] [PubMed] - Ghariblou, S.; Salehi, M.; Magnani, M.; Jalili, M. Shortest paths in multiplex networks. Sci. Rep.
**2017**, 7, 2142. [Google Scholar] [CrossRef] [PubMed] - Alves, L.G.A.; Mangioni, G.; Cingolani, I.; Rodrigues, F.A.; Panzarasa, P.; Moreno, Y. The nested structural organization of the worldwide trade multi-layer network. arXiv, 2018; arXiv:1803.02872. [Google Scholar]
- Timmer, M.P.; Dietzenbacher, E.; Los, B.; Stehrer, R.; Vries, G.J. An illustrated user guide to the world input-output database: The case of global automotive production. Rev. Int. Econ.
**2015**, 23, 575–605. [Google Scholar] [CrossRef] - Aleta, A.; Moreno, Y. Multilayer Networks in a Nutshell. arXiv, 2018; arXiv:1804.03488. [Google Scholar] [CrossRef]
- Saracco, F.; Di Clemente, R.; Gabrielli, A.; Squartini, T. Detecting early signs of the 2007–2008 crisis in the world trade. Sci. Rep.
**2016**, 6, 30286. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Squartini, T.; Van Lelyveld, I.; Garlaschelli, D. Early-warning signals of topological collapse in interbank networks. Sci. Rep.
**2013**, 3, 3357. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Network representations of international trade. (

**A**) the single-layer network of transactions among five countries. Each node ${c}_{i}$ represents a country, and the links (solid lines) refer to the interactions among countries. The weight ${w}_{{c}_{i}{c}_{j}}$ of each link (represented by the width of the line) from country ${c}_{i}$ to country ${c}_{j}$ is proportional to the sum of the value (US dollars) of all transactions of products and services exchanged from country ${c}_{i}$ to country ${c}_{j}$. Self-loops refer to economic transactions of a country with itself; (

**B**) the multiplex network of transactions among countries. Each of the three layers is associated with an industry, and all layers are populated by the same nodes, each representing a country involved in transactions in the corresponding industry. In this representation, the directed connections within each layer (solid lines), i.e., the intra-layer connections, convey information regarding the amount of value exchanged from one country to another (or to itself) in the corresponding industry. Notice that the weights from the single-layer representation are distributed among layers in the multiplex network, and are represented by the different widths of the lines. Each cross-layer connection (dashed line) links each country to itself across layers and has no economic interpretation; (

**C**) the multi-layer network of transactions among countries. This is a generalization of the two previous cases. Each layer represents an industry and all layers are populated by the same nodes, each representing a country. The intra-layer directed connections (solid lines) in each layer represent transactions from one country to another (or to itself) within the corresponding industry, while the cross-layer connections (dashed lines) represent transactions from one country to another (or to itself) across different industries. In the multi-layer network, the weights of the links from the multiplex network are re-distributed among connections both within and between layers, and are again represented by the different widths of the lines.

**Figure 2.**Strength distributions of each network in the period from 2000 to 2014. The color varies from dark blue (year = 2000) to light green (year = 2014). (

**A**) from left to right, the upper panels show the in-strength distributions for the single-layer, multiplex, and multi-layer networks, whereas the bottom panels (

**B**) show the out-degree distributions for the single-layer, multiplex, and multi-layer networks.

**Figure 3.**Entropy of node strength in the worldwide single-layer, multiplex, and multi-layer networks in 2000. (

**A**) As we increase the complexity of the trade structure from the single-layer to the multi-layer network, strength is re-distributed less uniformly across transactions and stages of production, and new heterogeneous market structures and dominance positions are uncovered; (

**B**) entropy of the in- and out- strengths associated distinctively with intra- and cross-layer connections in the multi-layer network. Cross-layer connections are characterized by a lower value of entropy, thus suggesting a higher concentration of value in production stages that span different industries than in production stages that occur within industries.

**Figure 4.**Evolution of entropy of strength in the international trade network in each network representation. (

**A**) the upper panels show the evolution of the entropy of in-strength in the single-layer (left-hand column), multiplex (middle column), and multi-layer (right-hand column) networks; (

**B**) the bottom panels show the evolution of the entropy of out-degree strength in the three respective networks.

**Figure 5.**Evolution of the entropy of in- and out-strengths related to the intra- and cross-layer connections in the multi-layer network. (

**A**) The upper panels show the evolution of entropy of in-strength related to intra-layer (left-hand column) and cross-layer (right-hand column) connections. (

**B**) The bottom panels show the evolution of entropy of out-strength associated with the two types of connections.

© 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/).

## Share and Cite

**MDPI and ACS Style**

A. Alves, L.G.; Mangioni, G.; Rodrigues, F.A.; Panzarasa, P.; Moreno, Y.
Unfolding the Complexity of the Global Value Chain: Strength and Entropy in the Single-Layer, Multiplex, and Multi-Layer International Trade Networks. *Entropy* **2018**, *20*, 909.
https://doi.org/10.3390/e20120909

**AMA Style**

A. Alves LG, Mangioni G, Rodrigues FA, Panzarasa P, Moreno Y.
Unfolding the Complexity of the Global Value Chain: Strength and Entropy in the Single-Layer, Multiplex, and Multi-Layer International Trade Networks. *Entropy*. 2018; 20(12):909.
https://doi.org/10.3390/e20120909

**Chicago/Turabian Style**

A. Alves, Luiz G., Giuseppe Mangioni, Francisco A. Rodrigues, Pietro Panzarasa, and Yamir Moreno.
2018. "Unfolding the Complexity of the Global Value Chain: Strength and Entropy in the Single-Layer, Multiplex, and Multi-Layer International Trade Networks" *Entropy* 20, no. 12: 909.
https://doi.org/10.3390/e20120909