# COVID-19's Impact on International Trade
## 2 Datasets and Google Matrix Algorithms
Analogously to the PageRank and CheiRank probabilities, it is possible to define the ImportRank and the ExportRank probabilities \(\hat{P}_{c}\) and \(\hat{P}_{c}^{*}\) for the country \(c\), and \(\hat{P}_{p}\), and \(\hat{P}_{p}^{*}\) for the product \(p\). Furthermore, we define \(\hat{K}\) and \(\hat{K}^{*}\) as the ImportRank and the ExportRank node index, and the other indexes \(\hat{K}_{c}\), \(\hat{K}_{c}^{*}\), \(\hat{K}_{p}\) and \(\hat{K}_{p}^{*}\). Qualitatively, the PageRank probabilities are proportional to the volumes of ingoing trade flow and the CheiRank probabilities to the volume of outgoing flow. Thus, approximately, products and countries subject to high importations (exportations) have a high PageRank (CheiRank) probability. Let us note that, in contrast to the usual import-export description--i.e., the one obtained with the ImportRank and the ExportRank, which takes only account of bilateral direct trade exchanges--the PageRank and CheiRank description takes into account the multiplicity of all the cascades of transactions encoded in the WTN. Following [26; 28], we determine the PageRank-CheiRank trade balance of a given country \(c\) as \(B_{c}=(P_{c}^{*}-P_{c})/(P_{c}^{*}+P_{c})\). In a similar way, the ImportRank-ExportRank trade balance is \(\hat{B}_{c}=(\hat{P}_{c}^{*}-\hat{P}_{c})/(\hat{P}_{c}^{*}+\hat{P}_{c})\). The sensitivity of the trade balance \(B_{c}\) to the price of a given product \(p\) is obtained by multiplying the corresponding money matrix elements \(\left\{M_{c^{\prime}c^{\prime\prime}}^{p}\right\}_{\forall c^{\prime},c^{ \prime\prime}}\), by a factor \((1+\delta)\), where \(\delta\) is an infinitesimal number. The PageRank-CheiRank trade balance sensitivity is then defined as \(dB_{c}/d\delta\). It is also useful to determine the PageRank-CheiRank product \(p\) trade balance, defined as \(B_{p}=(P_{p}^{*}-P_{p})/(P_{p}^{*}+P_{p})\), which is, a priori, non zero, in contrast to the ImportRank-ExportRank product \(p\) trade balance \(\hat{B}_{p}=(\hat{P}_{p}^{*}-\hat{P}_{p})/(\hat{P}_{p}^{*}+\hat{P}_{p})\), which is equal to zero due to the existing symmetry of product \(p\) transactions \(\hat{P}_{p}=\hat{P}_{p}^{*}\). Finally, the product \(p\) trade balances of a given country \(c\) are defined as \(B_{pc}=(P_{pc}^{*}-P_{pc})/(P_{p}^{*}+P_{p})\) and \(\hat{B}_{pc}=(\hat{P}_{pc}^{*}-\hat{P}_{pc})/(\hat{P}_{p}^{*}+\hat{P}_{p})\). We also use the REGOMAX algorithm [28; 30; 31] to obtain the reduced Google matrix \(G_{\text{R}}\) associated with a selected subset of \(N_{r}\) WTN nodes (with \(N_{r}\ll N\)). This algorithm allows us to incorporate all the direct and indirect transaction pathways connecting any couples of nodes among the \(N_{r}\) nodes of interest. The reduced Google matrix \(G_{\text{R}}\), obtained from the full WTN Google matrix \(G\), is used to construct a reduced network summarizing the strongest trade interactions between a selection of nodes representing countries and products. More detailed descriptions of the Google matrix based algorithms and the related numerical methods can be found in [25; 26; 28; 30; 31]. Let us note that the above-depicted algorithms have been used in [42; 43] to analyze the multiproduct WTN for years before 2019, and hence, the effects of COVID-19 on the WTN have not been captured there.