Factores de riesgo de la obesidad infantil

We now perform a weighted analysis of the World Trade Multiplex, by taking into account the values of import and export observed between countries.

In Figure 2.5(a) and (b) we show the color-coded weighted directed multi- plexity matrix Mw and the distribution of its off-diagonal entries. We clearly

see that, even though several pairs of commodities are still strongly overlapping, the multiplexity distribution is concentrated over a range of significantly smaller values with respect to the corresponding binary distribution. Indeed, the notion of weighted multiplexity, by involving the minimum of the weights of two recip- rocated links, provides a stricter criterion with respect to the unweighted case. In particular, for any pair of nodes and any pair of layers, it is more unlikely to achieve the maximum weighted value min{wα_ij, wβ_ij} than the maximum binary

2.3 Empirical analysis of the World Trade Multiplex 0.1 0.2 0.3 0.4 _m0.5 0.6 0.7 0.8 0.9 bin 0 5 10 15 20 25 30 35 40 z(m ) (a) 0.1 0.2 0.3 0.4_r 0.5 0.6 0.7 0.8 bin 0 5 10 15 20 z(r ) (b) 2 4 6 8 10 12 14 16 18 20 z(r) 0 5 10 15 20 25 30 35 40 z(m ) (c)

Figure 2.4: Analysis of significance of the values of binary multiplexity and multireciprocity for the WTM.Left: binary transformed multiplexity µαβ_b versus its corresponding z-score z(mαβ_b ); center: binary transformed multi- reciprocityραβ_b versus its correspondingz-score z(rαβ_b ); right: z(rαβ_b )vsz(mαβ_b ). Only off-diagonal values are reported.

value min{aα ij, a

β

ij}. Lower values of multiplexity with respect to Figure 2.1(a)

are therefore expected. We also expect to find a similar reduction for the multi- reciprocity later on.

In Figure 2.5(c) and (d) we report the color-coded weighted rescaled multi- plexity matrix and the corresponding distribution of off-diagonal entriesµαβ

w . The

fact that many values are now mapped to zero means that a significant compo- nent of the overlap between commodities can be explained simply in terms of the correlated strength sequences of the various layers. Importantly, we see that some pairs of layers actually exhibit negative rescaled multiplexity, even though the distribution is far from symmetric. This result, which is only visible in the weighted analysis, means that there are pairs of commodities for which the ob- served trade multiplexity is actuallylower than expected under the null model: these commodities prefer ‘not to be traded together’.

We then analyze the weighted multireciprocity of the WTM. Recently, it has been shown that the aggregated version of the network has a strong weighted reci- procity [11], a result that we can now complement with the analysis of the disag- gregated multiplex. In Figure 2.6(a) and (b) we report the color-coded weighted multireciprocity matrixRw, along with the distribution of its off-diagonal entries.

In analogy with the binary case, we see that the aggregated network exhibits a reciprocity which is significantly higher than the multireciprocity associated to any individual pair of layers. Yet several pairs of commodities are characterized by a substantial level of multireciprocity. In Figure 2.6(c) and (d) we show the

Directed multiplex networks Agriculture, farming Chemicals Textiles Metals Heavy, precision industry

Figure 2.5: Analysis of the weighted multiplexity between layers of the WTM.Top panels: color-coded weighted multiplexity matrix Mw (a) and cor-

responding distribution of off-diagonal multiplexity valuesmαβ_w (withα6=β) (b). Bottom panels: same as for the top panels, but with raw weighted multiplexity mαβw replaced by rescaled weighted multiplexity µαβw . Note that, in panel (c),

white entries represent negative values.

corresponding results for the rescaled weighted multireciprocityρα,β_w . We see that many values become close to zero and some become negative, in analogy with the behaviour of the multiplexity. The identification of pairs of layers with negative rescaled multireciprocity indicates that the corresponding commodities ‘prefer not to be traded in opposite directions’, in contrast with the results we found in the binary analysis.

In Figure 2.7 we compare the weighted multireciprocity and the weighted mul- tiplexity. When we consider the raw values (a), we observe a clear linear trend (although more scattered than in the corresponding unweighted case). The trend becomes even more robust, and less noisy, for the filtered values, as shown in (b). In both panels, the most significant commodities (both in terms of trade volumes and economic relevance) mainly lie along the diagonal, while the outliers represent

2.3 Empirical analysis of the World Trade Multiplex Agriculture, farming Chemicals Textiles Metals Heavy, precision industry

Figure 2.6: Analysis of the weighted multireciprocity between layers of the WTM. Top panels: color-coded weighted multireciprocity matrix Rw (a)

and corresponding distribution of off-diagonal multireciprocity valuesrαβ_w (with α6=β) (b). Bottom panels: same as for the top panels, but with raw weighted multireciprocity rwαβ replaced by rescaled weighted multireciprocity ραβw . The

dashed lines represent the value of (raw and rescaled) weighted reciprocityrwmono

and ρmonow of the aggregated monoplex network. Note that, in panel (c), white

entries represent negative values.

less relevant products (for instance, some textiles or less traded craft goods). We also see pairs of commodities whose multireciprocity is similar to the reciprocity of the aggregate trade network. These commodities, such as cereals and heavy industry products, are not necessarily the most traded ones, still they better rep- resent the reciprocity patterns of total trade among countries, possibly because they give the main contribution to the reciprocity of the aggregated network.

Quantitatively, another important difference between the binary and the weighted approach lies in the statistical significance of the values of multiplexity and mul- tireciprocity, as we can see from the analysis of thez-scores (Figure 2.8). Indeed, in the unweighted case we found that even the smallest values of µαβ_b and ραβ_b

Directed multiplex networks 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 rw 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 mw (a) 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 DWCM 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 DW CM (b)

Figure 2.7: Relation between the values of weighted multiplexity and multireciprocity for the WTM.Scatter plots of off-diagonal weighted multi- reciprocity values versus off-diagonal weighted directed multiplexity values. Left: raw values (rαβ

w vsmαβw ); right: rescaled values (ραβw vsµαβw ).

are significant, as the correspondingz-scores are larger than the critical value zc.

Instead, here we observe almost no correlation (except for the aforementioned sign concordance) between weighted multiplexity or multireciprocity and the cor- responding z-scores (see Fig. 2.8(a) and 2.8(b) respectively). Indeed, the same value ofµαβ_b orραβ_b may even correspond toz-scores with different orders of mag- nitude. This means that, even for two pairs of layers with the same observed value of weighted multiplexity or multireciprocity, the statistical significance of the inter-layer coupling can be very different. Moreover, the absolute value of many weighted z-scores is found below the significance threshold zc = 2, identi-

fying pairs of uncorrelated layers (a result that is unobserved in the binary case). Finally, many pairs of commodities have a negativez-score below−zcfor the mul-

tiplexity and/or multireciprocity. For these pairs, the tendencynot to be traded in the same direction and/or in opposite direction is statistically validated and confirms a difference with respect to the binary case.

As a final result, in Figure 2.8(c) we show the relation existing betweenz mαβ_w andz rαβ

w

. We find an overall level of correlation which however leaves room for a significant scatter of points around the identity line. This scatter is big enough to imply that, for a given significance thresholdzc, the pairs of commodities can

be partitioned in the following five classes:

2.3 Empirical analysis of the World Trade Multiplex

(z(mαβ

w )> zc) but not in opposite directions (z(rwαβ)<−zc): examples are

apparel articles vs ships and boats; food industry residues, prepared animal feed vs ores, slag and ash;

2. a few pairs of commodities that tend to be traded in opposite directions (z(rαβw )> zc) but not in the same direction (z(mαβw )<−zc): examples are

ores, slag and ash vs footwear and gaiters; apparel articles vs ores, slag and ash;

3. a moderately-sized group of pairs of commodities that tend to be traded neither in the same direction (z(mαβ

w )<−zc) nor in opposite ones (z(rαβw )< −zc): examples are raw hides and skins vs arms and ammunitions; tobacco

vs ships and boats;

4. a large group of pairs of commodities for which there is no statistically significant tendency in at least one of the two directions (|z(mαβ

w )| < zc

and/or|z(rαβ

w )|< zc): examples are tobacco vs inorganic chemicals; explo-

sives, pyrotechnic products vs vehicles (note that this class can be further split in sub-classes where commodities are uncorrelated in one direction but correlated in different ways in the other direction);

5. a very large group of pairs of commodities that tend to be traded both in the same direction (z(mαβ

w )> zc) and in opposite ones (z(rαβw )> zc): examples

are sugar vs cocoa; soap, waxes, candles vs sugar.

It should be noted that, in contrast with the above classification, the binary anal- ysis concluded that all pairs of commodities belong to the last class only.

Directed multiplex networks 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 DWCM 10 5 0 5 10 15 20 z( m ) (a) 0.10 0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 DWCM 10 5 0 5 10 15 20 z( r) (b) 10 5 0 5 10 15 20 z(r) 10 5 0 5 10 15 20 z( m ) (c) 5 1 3 2 4

Figure 2.8: Analysis of significance of the values of weighted multiplexity and multireciprocity for the WTM.Left: weighted transformed multiplexity µαβ

w versus its correspondingz-scorez(mαβw ); center: weighted transformed multi-

reciprocityραβ

w versus its correspondingz-score z(rαβw ); right: z(rαβw )vsz(mαβw ).

In panel (c), numbered circles correspond to the bullet points reported in Sec. 2.3.3. Only off-diagonal values are reported.