Randomizing world trade. II. A weighted network analysis

Based on the misleading expectation that weighted network properties always offer a more complete description than purely topological ones, current economic models of the International Trade Network (ITN) generally aim at explaining local weighted properties, not local binary ones. Here we complement our analysis of the binary projections of the ITN by considering its weighted representations. We show that, unlike the binary case, all possible weighted representations of the ITN (directed and undirected, aggregated and disaggregated) cannot be traced back to local country-specific properties, which are therefore of limited informativeness. Our two papers show that traditional macroeconomic approaches systematically fail to capture the key properties of the ITN. In the binary case, they do not focus on the degree sequence and hence cannot characterize or replicate higher-order properties. In the weighted case, they generally focus on the strength sequence, but the knowledge of the latter is not enough in order to understand or reproduce indirect effects.

Figure 1

Edge weights in the weighted undirected ITN. (a) Cumulative distributions of edge weights, for all years from 1992 (top) to 2002 (bottom). Orange (upper curves): real network; blue (lower curves): expectation for the maximum-entropy null model with specified strengths. (b) Same as the previous panel, but excluding zero weights (missing links). Orange (dark gray): real network; green (light gray): null model. (c) Percentage of missing links as a function of time. Red (upper points): real network; blue (lower points): null model.

Figure 2

Average nearest-neighbor strength ${\tilde{s}}_i^{nn}$ versus strength ${\tilde{s}}_i$ in the 2002 snapshot of the real weighted undirected ITN (red points), and corresponding average over the maximum-entropy ensemble with specified strengths (blue solid curve).

Figure 3

Weighted clustering coefficient ${\tilde{c}}_i$ versus strength ${\tilde{s}}_i$ in the 2002 snapshot of the real weighted undirected ITN (red points), and corresponding average over the maximum-entropy ensemble with specified strengths (blue solid curve).

Figure 4

Temporal evolution of the properties of the (rescaled) average nearest-neighbor strength ${\tilde{s}}_i^{nn}$ in the 1992–2002 snapshots of the real weighted undirected ITN and of the corresponding null model with specified strengths. (a) Average of ${\tilde{s}}_i^{nn}$ across all vertices (red, upper symbols: real data; blue, lower symbols: null model). (b) Standard deviation of ${\tilde{s}}_i^{nn}$ across all vertices (red, upper symbols: real data; blue, lower symbols: null model). (c) Correlation coefficient between ${\tilde{s}}_i^{nn}$ and ${\tilde{s}}_i$ (red, upper symbols: real data; blue, lower symbols: null model). (d) Correlation coefficient between ${\tilde{s}}_i^{nn}$ and $\langle {\tilde{s}}_i^{nn}\rangle$. The $95\%$ confidence intervals of all quantities are represented as vertical bars.

Figure 5

Temporal evolution of the properties of the (rescaled) weighted clustering coefficient ${\tilde{c}}_i$ in the 1992–2002 snapshots of the real weighted undirected ITN and of the corresponding null model with specified strengths. (a) Average of ${\tilde{c}}_i$ across all vertices (red, upper symbols: real data; blue, lower symbols: null model). (b) Standard deviation of ${\tilde{c}}_i$ across all vertices (red, upper symbols: real data; blue, lower symbols: null model). (c) Correlation coefficient between ${\tilde{c}}_i$ and ${\tilde{s}}_i$ (red, initially lower symbols: real data; blue, initially upper symbols: null model). (d) Correlation coefficient between ${\tilde{c}}_i$ and $\langle {\tilde{c}}_i\rangle$. The $95\%$ confidence intervals of all quantities are represented as vertical bars.

Figure 6

Average nearest-neighbor strength ${\tilde{s}}_i^{nn}$ versus strength ${\tilde{s}}_i$ in the 2002 snapshots of the commodity-specific (disaggregated) versions of the real weighted undirected ITN (red points), and corresponding average over the maximum-entropy ensemble with specified strengths (blue solid curves). (a) Commodity 93; (b) commodity 09; (c) commodity 39; (d) commodity 90; (e) commodity 84; (f) aggregation of the top 14 commodities (see Ref. [1] for details). From (a) to (f), the intensity of trade and level of aggregation increases.

Figure 7

Weighted clustering coefficient ${\tilde{c}}_i$ versus strength ${\tilde{s}}_i$ in the 2002 snapshots of the commodity-specific (disaggregated) versions of the real weighted undirected ITN (red points), and corresponding average over the maximum-entropy ensemble with specified strengths (blue solid curves). (a) Commodity 93; (b) commodity 09; (c) commodity 39; (d) commodity 90; (e) commodity 84; (f) aggregation of the top 14 commodities (see Ref. [1] for details). From (a) to (f), the intensity of trade and level of aggregation increases.

Figure 8

Edge weights in the weighted directed ITN. Red (light gray, upper curves and points): real network; blue (dark gray, lower curves and points): expectation for the maximum-entropy ensemble with specified out-strengths and in-strengths. (a) Cumulative distributions of edge weights in all years from 1992 (top) to 2002 (bottom). (b) Same as the previous panel, but excluding zero weights (missing links). (c) Percentage of missing links as a function of time.

Figure 9

Total average nearest-neighbor strength ${\tilde{s}}_i^{\text{tot/tot}}$ versus total strength ${\tilde{s}}_i^{\text{tot}}$ in the 2002 snapshot of the real weighted directed ITN (red, upper points), and corresponding average over the null model with specified out-strengths and in-strengths (blue, lower points).

Figure 10

Directed average nearest-neighbor strengths versus vertex strengths in the 2002 snapshot of the real weighted directed ITN (red points), and corresponding averages over the null model with specified out-strengths and in-strengths (blue solid curves). (a) ${\tilde{s}}_i^{\text{in/in}}$ versus ${\tilde{s}}_i^{\text{in}}$; (b) ${\tilde{s}}_i^{\text{in/out}}$ versus ${\tilde{s}}_i^{\text{in}}$; (c) ${\tilde{s}}_i^{\text{out/in}}$ versus ${\tilde{s}}_i^{\text{out}}$; (d) ${\tilde{s}}_i^{\text{out/out}}$ versus ${\tilde{s}}_i^{\text{out}}$.

Figure 11

Total weighted clustering coefficient ${\tilde{c}}_i^{\text{tot}}$ versus total strength ${\tilde{s}}_i^{\text{tot}}$ in the 2002 snapshot of the real weighted directed ITN (red, lighter points), and corresponding average over the null model with specified out-strengths and in-strengths (blue, darker points).

Figure 12

Weighted clustering coefficients versus vertex strengths in the 2002 snapshot of the real weighted directed ITN (red points), and corresponding averages over the null model with specified out-strengths and in-strengths (blue solid curves). (a) ${\tilde{c}}_i^{\text{in}}$ versus ${\tilde{s}}_i^{\text{in}}$; (b) ${\tilde{c}}_i^{\text{out}}$ versus ${\tilde{s}}_i^{\text{out}}$; (c) ${\tilde{c}}_i^{\text{cyc}}$ versus ${\tilde{s}}_i^{\text{in}}·{\tilde{s}}_i^{\text{out}}$; (d) ${\tilde{c}}_i^{\text{mid}}$ versus ${\tilde{s}}_i^{\text{in}}·{\tilde{s}}_i^{\text{out}}$.

Figure 13

Temporal evolution of the properties of the (rescaled) total average nearest-neighbor strength ${\tilde{s}}_i^{\text{tot/tot}}$ in the 1992–2002 snapshots of the real weighted directed ITN and of the corresponding null model with specified out-strengths and in-strengths. (a) Average of ${\tilde{s}}_i^{\text{tot/tot}}$ across all vertices (red, upper symbols: real data; blue, lower symbols: null model). (b) Standard deviation of ${\tilde{s}}_i^{\text{tot/tot}}$ across all vertices (red, upper symbols: real data; blue, lower symbols: null model). (c) Correlation coefficient between ${\tilde{s}}_i^{\text{tot/tot}}$ and ${\tilde{s}}_i^{\text{tot}}$ (red: real data; blue: null model, indistinguishable from real data). (d) Correlation coefficient between ${\tilde{s}}_i^{\text{tot/tot}}$ and $\langle {\tilde{s}}_i^{\text{tot/tot}}\rangle$. Vertical bars are $95\%$ confidence intervals.

Figure 14

Averages and their $95\%$ confidence intervals (across all vertices) of the directed average nearest-neighbor strengths in the 1992–2002 snapshots of the real weighted directed ITN (red, upper symbols), and corresponding averages over the maximum-entropy ensemble with specified out-strengths and in-strengths (blue, lower symbols). (a) ${\tilde{s}}_i^{\text{in/in}}$; (b) ${\tilde{s}}_i^{\text{in/out}}$; (c) ${\tilde{s}}_i^{\text{out/in}}$; (d) ${\tilde{s}}_i^{\text{out/out}}$.

Figure 15

Temporal evolution of the properties of the (rescaled) total weighted clustering coefficient ${\tilde{c}}_i^{\text{tot}}$ in the 1992–2002 snapshots of the real weighted directed ITN and of the corresponding null model with specified out-strengths and in-strengths. (a) Average of ${\tilde{c}}_i^{\text{tot}}$ across all vertices (red, upper symbols: real data; blue, lower symbols: null model). (b) Standard deviation of ${\tilde{c}}_i^{\text{tot}}$ across all vertices (red, upper symbols: real data; blue, lower symbols: null model). (c) Correlation coefficient between ${\tilde{c}}_i^{\text{tot}}$ and ${\tilde{s}}_i^{\text{tot}}$ (red, upper symbols: real data; blue, lower symbols: null model). (d) Correlation coefficient between ${\tilde{c}}_i^{\text{tot}}$ and $\langle {\tilde{c}}_i^{\text{tot}}\rangle$. The $95\%$ confidence intervals of all quantities are represented as vertical bars.

Figure 16

Averages and their $95\%$ confidence intervals (across all vertices) of the directed weighted clustering coefficients in the 1992–2002 snapshots of the real weighted directed ITN (red, upper symbols), and corresponding averages over the null model with specified out-strengths and in-strengths (blue, lower symbols). (a) ${\tilde{c}}_i^{\text{in}}$; (b) ${\tilde{c}}_i^{\text{out}}$; (c) ${\tilde{c}}_i^{\text{cyc}}$; (d) ${\tilde{c}}_i^{\text{mid}}$.

Figure 17

Total average nearest-neighbor strength ${\tilde{s}}_i^{\text{tot/tot}}$ versus total strength ${\tilde{s}}_i^{\text{tot}}$ in the 2002 snapshots of the commodity-specific (disaggregated) versions of the real weighted directed ITN (red, upper points), and corresponding average over the maximum-entropy ensemble with specified out-strengths and in-strengths (blue, lower points). (a) Commodity 93; (b) commodity 09; (c) commodity 39; (d) commodity 90; (e) commodity 84; (f) aggregation of the top 14 commodities (see Ref. [1] for details). From (a) to (f), the intensity of trade and level of aggregation increases.

Figure 18

Total weighted clustering coefficient ${\tilde{c}}_i^{\text{tot}}$ versus total strength ${\tilde{s}}_i^{\text{tot}}$ in the 2002 snapshots of the commodity-specific (disaggregated) versions of the real weighted directed ITN (red, upper points), and corresponding average over the maximum-entropy ensemble with specified our-strengths and in-strengths (blue, lower points). (a) Commodity 93; (b) commodity 09; (c) commodity 39; (d) commodity 90; (e) commodity 84; (f) aggregation of the top 14 commodities (see Ref. [1] for details). From (a) to (f) the intensity of trade and level of aggregation increases.