Reconstructing the world trade multiplex: The role of intensive and extensive biases

In economic and financial networks, the strength of each node has always an important economic meaning, such as the size of supply and demand, import and export, or financial exposure. Constructing null models of networks matching the observed strengths of all nodes is crucial in order to either detect interesting deviations of an empirical network from economically meaningful benchmarks or reconstruct the most likely structure of an economic network when the latter is unknown. However, several studies have proved that real economic networks and multiplexes topologically differ from configurations inferred only from node strengths. Here we provide a detailed analysis of the world trade multiplex by comparing it to an enhanced null model that simultaneously reproduces the strength and the degree of each node. We study several temporal snapshots and almost 100 layers (commodity classes) of the multiplex and find that the observed properties are systematically well reproduced by our model. Our formalism allows us to introduce the (static) concept of extensive and intensive bias, defined as a measurable tendency of the network to prefer either the formation of extra links or the reinforcement of link weights, with respect to a reference case where only strengths are enforced. Our findings complement the existing economic literature on (dynamic) intensive and extensive trade margins. More generally, they show that real-world multiplexes can be strongly shaped by layer-specific local constraints.

Figure 1

Comparison between the observed undirected binary and weighted properties (red points) and the corresponding ensemble averages of the WCM (green points) and the ECM (blue points) for the aggregated WTW in the 2002 snapshot. Top left: average nearest neighbor degree $k_i^{nn}$ versus degree $k_i$. Top right: binary clustering coefficient $c_i$ versus degree $k_i$. Bottom left: average nearest neighbor strength $s_i^{nn}$ versus strength $s_i$. Bottom right: weighted clustering coefficient $c_i^w$ versus strength $s_i$.

Figure 2

Temporal evolution of the average nearest neighbor degree $\left(k_i^{nn}\right)$ from 1992 to 2002 and comparison with the corresponding maximum-entropy ensembles with specified degrees and strengths (ECM). (a) Average of $k_i^{nn}$ across all vertices (red, obs.; blue, randomized); (b) standard deviation of $k_i^{nn}$ across all vertices; (c) correlation coefficient between $k_i^{nn}$ and $k_i$; (d) correlation coefficient between $k_i^{nn}$ and $\langle k_i^{nn}\rangle$. Red, observed values; blue, expected values. The $95\%$ confidence intervals of all quantities are represented as vertical bars.

Figure 3

Temporal evolution of the binary clustering coefficient $\left(c_i\right)$ from 1992 to 2002 and comparison with the corresponding maximum-entropy ensembles with specified degrees and strengths (ECM). (a) average of $c_i$ across all vertices (red, obs.; blue, randomized); (b) standard deviation of $c_i$ across all vertices; (c) correlation coefficient between $c_i$ and $k_i$; (d) correlation coefficient between $c_i$ and $\langle c_i\rangle$. Red, observed values; blue, expected values. The $95\%$ confidence intervals of all quantities are represented as vertical bars.

Figure 4

Temporal evolution of the properties of the ANNS $s_i^{nn}$ in the 1992–2002 snapshots of the observed undirected WTW and of the corresponding maximum-entropy ensembles with specified degrees and strengths: (a) average of $s_i^{nn}$ across all vertices (red, obs.; blue, randomized); (b) standard deviation of $s_i^{nn}$ across all vertices; (c) correlation coefficient between $s_i^{nn}$ and $s_i$; (d) correlation coefficient between $s_i^{nn}$ and $\langle s_i^{nn}\rangle$. Red points stands for observed values, blue for the randomized ones; the $95\%$ confidence intervals of all quantities are represented as vertical bars.

Figure 5

Temporal evolution of the properties of the WCC $c_i^w$ in the 1992–2002 snapshots of the observed undirected WTW and of the corresponding maximum-entropy ensembles with specified degrees and strengths: (a) average of $c_i^w$ across all vertices (red, obs.; blue, randomized); (b) standard deviation of $c_i^w$ across all vertices; (c) correlation coefficient between $c_i^w$ and $s_i$; (d) correlation coefficient between $c_i^w$ and $\langle c_i^w\rangle$. Red points denote observed values and blue the randomized ones; the $95\%$ confidence intervals of all quantities are represented as vertical bars.

Figure 6

Average nearest neighbor degree $\left(k_i^{nn}\right)$ versus node degree $\left(k_i\right)$ in the 2002 snapshots of the commodity-specific (disaggregated) versions of the observed binary undirected WTW (red points) and corresponding average over the maximum entropy ensemble with specified degrees and strengths (blue points): (a) commodity 93, (b) commodity 09, (c) commodity 39, (d) commodity 90, (e) commodity 84, and (f) aggregation of the top 14 commodities (see Table 1 for details). From (a) to (f), the intensity of trade and level of aggregation increases.

Figure 7

Binary clustering coefficient $\left(c_i\right)$ versus node degree $\left(k_i\right)$ in the 2002 snapshots of the commodity-specific (disaggregated) versions of the observed binary undirected WTW (red points) and corresponding average over the maximum entropy ensemble with specified degrees and strengths (blue points): (a) commodity 93, (b) commodity 09, (c) commodity 39, (d) commodity 90, (e) commodity 84, and (f) aggregation of the top 14 commodities (see Table 1 for details). From (a) to (f), the intensity of trade and level of aggregation increases.

Figure 8

Average nearest neighbor strength $\left(s_i^{nn}\right)$ versus node strength $\left(s_i\right)$ in the 2002 snapshots of the commodity-specific (disaggregated) versions of the observed binary undirected WTW (red points) and corresponding average over the maximum entropy ensemble with specified degrees and strengths (blue points): (a) commodity 93, (b) commodity 09, (c) commodity 39, (d) commodity 90, (e) commodity 84, and (f) aggregation of the top 14 commodities (see Table 1 for details). From (a) to (f), the intensity of trade and level of aggregation increases.

Figure 9

Weighted clustering coefficient $\left(c_i^w\right)$ versus node strength $\left(s_i\right)$ in the 2002 snapshots of the commodity-specific (disaggregated) versions of the observed binary undirected WTW (red points) and corresponding average over the maximum entropy ensemble with specified degrees and strengths (blue points): (a) commodity 93, (b) commodity 09, (c) commodity 39, (d) commodity 90, (e) commodity 84, and (f) aggregation of the top 14 commodities (see Table 1 for details). From (a) to (f), the intensity of trade and level of aggregation increases.