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Link activationContingency tables corresponding to the three cases described in the “Methods” section are shown in Table 1. This Table is quite revealing in several ways. The most interesting aspect is that the highest probability of link establishment occurs when an agreement is activated (Operational Activation in t).Table 1 Contingency tables.Full size tableIn this case, the probability of activation of a new link is 8.8%—namely, the ratio of new activation 7.3% to the total number of links that were not active at year t-1 (82.6%)—which is significantly higher than in the case of links not covered by a commercial agreement (No Trade Agreement), amounting to 1.4%.Therefore, the findings show that operational activation is associated with creating new trade relations between two particular countries. The third set, which considers links where a trade agreement exists in both years (t-1) and t (Trade Agreement in t-1 and t), also shows a consistent activation probability of 6%. This result confirms the assumption that the coverage of a commercial agreement, and not only its implementation, encourages the genesis of new links.Moreover, Table 1 suggests some interesting considerations on trade persistence. To establish these probabilities, we focus on the row totals in which a trade relationship is present at year (t-1), i.e., 28.8% in the case Trade Agreement in t-1 and t. The presence of an agreement influences in a positive way the probability of maintaining a trade relationship. In fact, when a trade agreement is present in both years, (t-1) and t, the probability of preserving the trade relationship is 87.1% ((frac{25.1}{28.8}times {100})), while when a trade agreement is activated at year t, the probability slightly decreases to 81.6%. In cases where trade agreements are missing (No Trade Agreement in t) we observe the probability of retaining a relationship decreases to 77.3%.Another interesting aspect concerns the probability of link deactivation. Once more, the coverage of a trade agreement favors a lower likelihood of deactivation of existing links. The ratio of the percentage of links that were active at year (t-1) and are no more active at year t to the total is 22.7% ((frac{1}{4.4}times {100})) in the case of a lack of agreement. This probability decreases to 18.4% ((frac{3.2}{17.4}times {100})) if we consider only the year of activation of the agreement (Operational Activation), and drops to 12.8% ((frac{3.7}{28.8}times {100})) when looking at agreements present in both years.Together, these results provide insights into the role of trade agreements in the network topology of cereal trade. While the establishment of a trade agreement promotes the potential for new trade links, the presence of the agreement in two consecutive years allows both to maintain an existing relationship and reduce the likelihood of link shutdowns.Flow variationsIn this second part, we study the impact of trade agreements on existing trade flows, analyzing the relationship between the flows at time t and the flows at time (t-1) in each of the three cases described in the “Methods” section—i.e., No trade agreements, Operational Activation in t, and Trade agreement in t-1 and t—measured in US$, Kcal and m(^3) of virtual water.Figure 3Kernel Density scatterplot between trade flows of cereals at time t (on the y-axis) and time (t-1) (on the x-axis) for the three different sets: No trade agreements (column a), Operational Activation in t (b), and Trade agreement in (t-1) and t (c). Panels in the first, second and third row refer to flows in US$, Kcal, and virtual water (m(^3)), respectively. Flow values are shown on a logarithmic scale. The color bar indicates probability densities, and the bisector is highlighted. Notice (i) the higher volumes in the case of flows covered by trade agreement and (ii) a a less relevant increase in volume when the flows are seen in the virtual water lens.Full size imageFigure 3 shows three different scatterplots for each unit of measure (US$ and Kcal and m(^3)). The scatterplots are colored by Kernel Density Estimation (KDE), a non-parametric technique for probability density functions. KDE aims to take a finite sample of data and infer the underlying probability density function. Figure 3 relates the flows at time (t-1) with the flows at time t, both reported on a logarithmic scale since the quantities span several orders of magnitude. Let’s start focusing on flows in terms of dollars and kilocalories. What stands out from the figure is the displacement of the flows toward higher values when they are covered by trade agreements (Trade Agreement in t-1 and t), compared to the case where flows have no trade agreement.We have quantitative evidence of this result by looking at Table 2 where the average flows in both years are shown. The average values of flows in both US$ and Kcal are much higher when there is a trade agreement over time (Trade agreement in t-1 and t). Flows have an average value of (6.13times 10^{7})$, larger than the mean of (3.05times 10^{7})$ achieved by flows not covered by a trade agreement. By comparing the distributions of the two distinct sets with different dimensions by applying the non-parametric Mann-Whitney test, we stand to evaluate this result as extremely significant (p-value approximately 0).Table 2 Average values of trade flows and flow variation index (rho _{ij}) for each of the three sets, in US$ (a), Kcal (b), and Virtual water (VW, m(^3)). The bar indicates the average operator.Full size tableAlso, while operational activation plays a crucial role in creating new links in the global cereal trade, it does not appear to hold central importance in driving flow increases. The average value of flows in both years (t-1) and t are, in fact, smaller than those not covered by trade agreements.The view appears slightly different when we look at the values in terms of virtual water (VW, m(^3)), i.e., the sum of the blue and green components. Flows with a commercial agreement show higher averages values than those not covered by agreements (see panel (c) of Table 2), but the increase is significantly lower than the one recorded in the other two units (US$ and Kcal). The increase recorded in dollars is about 100%, while in terms of virtual water this increase is less than 30%. In the next subsection, we will focus on this peculiar behavior, which reveals a different water content of the goods traded along links covered or not by agreements.Another significant result that emerges from Fig. 3 is the smaller amplitude (around the bisector) of the cloud in the case of link covered by agreements in both years (t-1) and t. This is confirmed by comparing the weighted average of the absolute value of the inter-annual flow variation index (overline{rho _{ij}}_{w}) (weights are the flows traded in the year (t-1)). The index (rho _{ij}) is used to highlight cases where the activation or the presence of the agreement generates a significant flow increase.Larger (rho _{ij}) values correspond to larger average variations from year (t-1) to year t. Accordingly, we observe that in the presence of trade agreement at time (t-1) and t a smaller (rho _{ij}) value of 24.79 percentage points (p.p) is found (see panel (a) of Table 2).Considering all the units (US$, Kcal, and m(^3)), this value is about half of the average inter-annual variation that occurs when there is no trade agreement. Hence, the presence of a commercial agreement over time reduces large fluctuations, stabilizing the year-to-year variations.To shed light on the response of water flows to the occurrence of the agreement, we refer to water productivity (WP)34, both in economic and nutritional terms. Table 3 shows that the Nutritional WP for the total virtual water is, on average, 35% higher in the flows under a trade agreement than in flows that are not under any treaty, while the Economic WP is 62% higher. We also analyze the two virtual water components, blue and green, separately.Interestingly, for blue water in the presence of a trade agreement, the Nutritional WP and the Economic WP for the flows covered by trade agreement are, on average, 68% and 93% higher than for the flows not covered by agreements. In other words, for one cubic meter of water used for grain production, more kilo-calories and dollars are exchanged when an agreement is in place, and this difference is even more significant in terms of blue water.Table 3 Average of nutritional ((mathrm {kcal/m^3})) and economic ((mathrm {US$/m^3})) water productivity (WP) for the total, blue and green virtual water.Full size tableWe also investigate in detail which products contribute most to the imbalance between flows in terms of kcal or water. To this aim, Fig. 4 reports the nutritional WP for each grain item distinguishing whether or not there is a commercial agreement (similar results occur if the economic WP is considered).The figure highlights that the nutritional WP is generally higher in the case where flows are covered by trade agreements (green bars). The most noticeable cases are Maize and Wheat, which are also the most traded products: the value of nutritional WP increases from 1978 (mathrm {kcal/m^3}) (No trade agreement) to 2851 (mathrm {kcal/m^3}) in case of a trade agreement for Wheat, and from 4471 (mathrm {kcal/m^3}) to 5026 for Maize.Figure 4The bar chart shows the nutritional WP for each cereal product in the two sets of Trade agreement in t-1 and t (in green) and No trade agreement (in red). The number over the bars represents the percentage of kcal traded for each product compared to the total kcal of all cereals. Note that green bars are higher than the red ones in 80% of cases.Full size imageA few products have a higher nutritional WP value when the flows are not involved in any treaty, e.g., Rye. This behavior can be traced back to a few flows that dominate the market between countries not linked by trade agreements. For example, trade in Rye in 2014 is attributable to just two major flows in terms of caloric intake relative to water quantity (notably, one between Germany and Japan, the other between Russia and Turkey).Figure 4 clearly shows that grains characterized by greater water efficiency generally move along the links covered by agreements.Performance of trade agreements in increasing flowOur results show that links covered by agreements exhibit larger flows than links not covered by treaties. We also intend to obtain information about the possible flow increase under a specific agreement.As mentioned in the “Methods” section, we selected only those operating links when the agreement came into force to evaluate the variation index ((rho _a)) under a specific treaty. Consequently, since there are trade agreements that came into force before the time interval considered, these are excluded from this analysis. As a result, the total number of agreements selected for this analysis is 99, 61 of which show an increase (positive (rho _{a}) values), while the remaining 38 exhibits a decrease in the flux intensities compared to the overall global trend. We present in Table 5 the results for positive (rho _{a}) variations, while trade agreements with negative (rho _{a}) values are reported in Supplementary Material (5). We provide this analysis in terms of economic flows (US$), but very similar results are obtained if calories (kcal) or virtual water (m(^3)) are chosen as the unit of measure.Table 4 Flow values in millions of dollars in year t and percent changes (rho _{a}) from (t-1) to t for each trade agreement.Full size tableWhat stands out in Table 4 is that most of the positive percentage changes occur in Europe and Central Asia regions. This may be due to long-term commercial activities in Europe, which are supported by the geographical proximity of the countries, as well as the wide variety of political and economic treaties among them. Europe, in fact, is characterized by a fourfold increase in cereal production since the 1960s due to the adoption of the Common Agricultural Policy, which has intensified trade in Europe and towards external markets30.A closer inspection of Table 4 shows that among the agreements with the most significant flows that showed the greatest increases, we find EEA (European Economic Area) in Europe and Central Asia, Japan-ASEAN in East Asia and Pacific, and COMESA in Sub-Saharan Africa.With lower flow values but large increases ((rho _{a})) due to the entry into force of trade agreements, the India-Sri Lanka agreement in South Asia stands out above all others. Also, the treaty signed in 2013 between EU-Colombia and Peru shows significant variations in terms of the percentage of flow increase, but the volume of the corresponding flow is inferior when compared with other trade agreements. On the other hand, the North American Free Trade Agreement (NAFTA), which became effective in 1994, has a lower (rho _{a}) value, but the flows on which the variation is calculated are significantly higher. More

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