# COVID-19's Impact on International Trade
## 2 Datasets and Google Matrix Algorithms  
We use the UN Comtrade data [13] for the years 2018, 2019 and 2020 (collected in September 2021) to construct the trade flows of the multiproduct WTN following [26; 28], giving a detailed description of the applied methods. For each year, a money matrix, \(M^{p}_{cc^{\prime}}\), gives the export flow of product \(p\) from country \(c^{\prime}\) to country \(c\) (transactions are expressed in USD of the current year). The money matrix element, \(M^{p}_{cc^{\prime}}\), is taken as the maximum between the reported volume of product \(p\) imported by the country \(c\) from the country \(c^{\prime}\) and the reported volume of product \(p\) exported from the country \(c^{\prime}\) to the country \(c\). Although these volumes should be of the same order, this procedure allows us to reduce the effect of the absence or delay of reporting by certain countries. The data set concerns \(N_{c}=194\) countries and territories and \(N_{p}=10\) principal groups of products, as reported in Table 1. Thus, the total Google matrix \(G\) size is \(N=N_{c}N_{p}=1940\) corresponding to the number of couples (product, country). The Google matrix elements \(G_{ij}\) of direct trade flows are constructed in the manner described in detail in [26; 28]. The monetary values of products traded from a node \(j\) to the other nodes \(i\), renormalized in such a way that their sum is equal to one, constitute the elements \(S_{ij}\) of the matrix \(S\), which represents Markov trade transitions. Each column associated with a dangling node--i.e., a node from which no trade flows emanate--is replaced by a column with all elements equal to \(1/N\). A personalization vector \(v\) encodes the relative weight of each product in the global trade volume, and all the countries are treated on equal grounds following the main UN principle. The Google matrix elements are \(G_{ij}=\alpha S_{ij}+(1-\alpha)v_{i}\), where \(v_{i}\) is the personalization vector component of node \(i\) (\(v_{i}>0\), \(\forall i\) and \(\sum_{i=1}^{N}v_{i}=1\)). Here, we use the damping factor value \(\alpha=0.5\). Similarly, we also construct the Google matrix \(G^{*}\) from the money matrix with inverted trade directions. The stationary probability distribution on the WTN is given by the PageRank vector \(P\) defined by the eigenproblem \(GP=P\)[14; 15; 18]. For the WTN network with inverted trade flows, described by \(G^{*}\), the stationary probability distribution is given by the CheiRank vector \(P^{*}\), defined as \(G^{*}P^{*}=P^{*}\). The PageRank node index \(K\) and the CheiRank node index \(K^{*}\) are obtained by sorting in descending order of the components of the PageRank vector \(P\) and of the CheiRank vector \(P^{*}\), respectively. For the WTN, each node corresponds to a given couple \((p,c)\) of a country \(c\) and product \(p\). Hence, the PageRank (CheiRank) component \(P_{pc}\) (\(P_{pc}^{*}\)) measures the ability of a country \(c\) to import (export) a product \(p\). Following [26; 28], the sums over all the product types \(p\) give the PageRank probability \(P_{c}=\sum_{p}P_{pc}\) and the CheiRank probability \(P_{c}^{*}=\sum_{p}P_{pc}^{*}\) of a given country \(c\). Similarly, the sums over the countries provide the PageRank and CheiRank probabilities \(P_{p}=\sum_{c}P_{pc}\) and \(P_{p}^{*}=\sum_{c}P_{pc}^{*}\) of a product \(p\). Sorting these probabilities in descending order allows us to define the related indexes \(K_{c}\), \(K_{c}^{*}\), \(K_{p}\) and \(K_{p}^{*}\). From the import and export volumes, it is possible to define, for a couple \((p,c)\), an ImportRank probability \(\hat{P}_{pc}=V^{-1}\sum_{c}M_{cc^{\prime}}^{p}\) and an ExportRank probability \(\hat{P}_{pc}^{*}=V^{-1}\sum_{c}M_{cc^{\prime}}^{p}\), where \(V=\sum_{p,c,c^{\prime}}M_{cc^{\prime}}^{p}\) is the total volume of the WTN.