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.