In this section, we briefly introduce the definitions of the centrality measures used later for the econometric analysis. We then describe the methodology that we apply in the paper. In particular, we use the community detection method based on the Estrada communicability distance, recently proposed in Bartesaghi et al. (2020). We present the main steps of the methodology and we refer the reader to the cited reference for a detailed description.

The application of centrality measures, as well as the study of the network topology of the WTN, are useful in explaining the initial diffusion of the pandemic, as explained in “Related literature” section. The origins of COVID-19 are still largely uncertain, as well as the very early stages of its spread outside China. However, it can be supposed that, during this first period, the chances of “importing” the SARS-COV-2 virus were not the same for all countries. On the contrary, the established trade routes for the circulation of commodities and the mobility of people have probably driven the direct or the indirect import of the SARS-COV-2 virus inside national borders, determining significant initial differences in the early contagion rates between entire clusters of countries.

These reasons motivate the use of centrality measures and of the Estrada communicability distance in explaining the first wave of the contagion. On the one hand, the Estrada communicability distance allows us to highlight the strategic commercial links, which can survive beyond a global shock. On the other hand, suitable centrality measures quantify specific factors, such as the number and volume of trades, the power of a country in commercial framework and triadic relations between countries, which are non-negligible in studying the initial diffusion of the pandemic.

From now on, we consider a simple weighted undirected network \(G=(V,E),\) where *V* is the nodes set with \(|V| = n\) and *E* is the set of links. The unweighted and weighted adjacency relations are represented by matrices \({\mathbf {A}}\) and \({\mathbf {W}}\), respectively.

### Centrality measures

Many centrality measures have been proposed in the literature. Among them, some measures highlight various characteristics of the WTN.

The first measure we use in the analysis is the most intuitive one, i.e. the degree centrality. This measure counts the number of links incident on a vertex. For weighted networks, the corresponding measure is the strength centrality. In the WTN, these measures quantify how much a country directly trades, in terms of number and volume of trades.

The eigenvector centrality [see Bonacich (1972)] is formally represented by the *i*-th component of the principal eigenvector of the adjacency matrix. Since it quantifies the connections of a vertex with its neighbours that are themselves central, it can be interpreted as a measure of the power of a country in the trading scenario. The extension to the weighted case is immediate, as the weighted adjacency matrix \({\mathbf {W}}\) preserves all characteristics of \({\mathbf {A}}\).

The betweenness centrality of a node is the fraction of the shortest paths between pairs of nodes passing through it. With reference to the WTN, with trade many other elements are transferred between countries. In this perspective, this measure quantifies the influence that a country has in spreading information within the network.

We also consider the local clustering coefficient, which measures the tendency to which nodes in a network tend to cluster together. Since the WTN is represented by an indirect and weighted network (see “Data, samples and variables” section for a detailed description of the network), we focus on the local weighted coefficient proposed by Onnela et al. (2005):

$$\begin{aligned} C_i(\tilde {{\mathbf{W}}}) = \frac{\sum _{j} \sum _{j\ne k} {\tilde{w}}_{ij}^{1/3} {\tilde{w}}_{jk}^{1/3} {\tilde{w}}_{ki}^{1/3}}{d_i(d_i -1)} \end{aligned}$$

(1)

where \(d_i\) is the degree of node *i* and \(\tilde {{\mathbf{W}}}\) is the weighted adjacency matrix obtained by normalizing the entries \(w_{ij}\) of \({\mathbf {W}}\) as \({\tilde{w}}_{ij} = \frac{w_{ij}}{\max (w_{ij})}\) \(\forall i,j\). Notice that \(C_i(\tilde {{\mathbf{W}}})=C_i\) represents the geometric mean of the links weights incident to the node *i*, divided by the number of potential triangles \(d_i(d_i-1)\) centred on it. The main idea is to replace the total number of triangles in which a node *i* belongs with the “intensity” of the triangle, defined here as the geometric mean of its weights. Since it is a measure of how many nodes are locally clustered, the clustering coefficient is extremely interesting to investigate in the context of international trade. Indeed, trade relationships induce a dependency between countries, as two nodes that are both trading partners of a node are likely to trade themselves. From this perspective, it is interesting to investigate how countries are reciprocally dependent, that is, how nodes are clustered together.

### Community detection based on communicability distance

The main idea is to detect communities by optimising a quality function that exploits the additional information contained in a metric structure based on the Estrada communicability. At first, we recall the definition of the Estrada communicability (simply, communicability) between two nodes *i* and *j* (see Estrada and Hatano (2008)):

$$\begin{aligned} G_{ij}=\sum _{k=0}^{+\infty }\frac{1}{k!}[\mathbf{A }^k]_{ij}=\left[ e^{\mathbf{A }} \right] _{ij}. \end{aligned}$$

(2)

As the *ij*-entry of the *k*-power of \(\mathbf{A }\) provides the number of walks of length *k* starting at *i* and ending at *j*, \(G_{ij}\) accounts for all channels of communication between two nodes, giving more weight to the shortest routes connecting them. The elements \(G_{ii}\), \(i=1,\ldots ,n\) are known in the literature as subgraph centrality (Estrada and Rodriguez-Velazquez 2005). The communicability matrix is, then, the exponential of the matrix \({\mathbf {A}}\), simply denoted by \(\mathbf{G }\).

In the case of a weighted network, the weighted communicability function is defined as

$$\begin{aligned} G_{ij}=\sum _{k=0}^{+\infty }\frac{1}{k!}[(\mathbf{S }^{-{\frac{1}{2}}}\mathbf{W }\mathbf{S }^{-{\frac{1}{2}}})^k]_{i j}=\left[ e^{(\mathbf{S }^{-{\frac{1}{2}}}\mathbf{W }\mathbf{S }^{-{\frac{1}{2}}})} \right] _{ij} \end{aligned}$$

(3)

where \(\mathbf{S }\) is the diagonal matrix whose diagonal entries are the strengths of the nodes. Following Crofts and Higham (2009), the matrix \(\mathbf{W }\) in Formula (3) has been normalized to avoid the excessive influence of links with higher weights in the network.

Using the communicability, a meaningful distance metric \(\xi _{ij}\) can be constructed, as defined in (Estrada 2012):

$$\begin{aligned} \xi _{i j}=G_{ii}-2G_{ij}+G_{jj}. \end{aligned}$$

(4)

By definition, communicability measures the amount of information transmitted from node *i* to *j*. On the other hand, \(G_{ii}\) measures the importance of a node according to its participation in all closed walks to which it belongs. Hence, in terms of information diffusion, \(G_{ii}\) is the amount of information that, after flowing along closed walks, returns to node *i*.

Thus, the quantity \(\xi _{ij}\) accounts for the difference in the amount of information that returns to nodes *i* and *j* and the amount of information exchanged between them. The greater is \(G_{ij}\), the larger is the information exchanged and the nearer are the nodes; the greater are \(G_{ii}\) or \(G_{jj}\), the larger is the information that comes back to the nodes and the farther are the nodes. Since \(\xi _{i j}\) is a metric, then \(G_{ii}+G_{jj}\ge 2G_{ij}\), i.e. no matter what the structure of the network is, the amount of information absorbed by a pair of nodes is always larger than or equal to the amount of information transmitted between them.

This metric is meaningful if we apply it to the WTN. Indeed, network flows along links measure how well two countries communicate in terms of commercial exchanges. For instance, the link between two nodes may be identified with the total trade or money flow between two countries.

We assume that two nodes are considered members of the same community if their mutual distance \(\xi _{ij}\) is lower than a threshold \(\xi _0 \in [\xi _{min},\xi _{max}]\). In particular, we construct a new community graph with adjacency matrix \(\mathbf{M }=[m_{ij}]\) given by:

$$\begin{aligned} m_{ij}= \left\{ \begin{array}{ll} 1 &{} \ \mathrm{if}\ \xi _{ij}\le \xi _0 \\ 0 &{} \ \mathrm{otherwise} \\ \end{array} \right. \end{aligned}$$

(5)

In this way, clustered groups of nodes that ’strongly communicate’ emerge, varying the threshold \(\xi _0\).

As well explained in Bartesaghi et al. (2020), \(\xi _0\) is not arbitrarily chosen but is obtained by solving the following optimisation problem:

$$\begin{aligned} \xi _0 \in \arg \max Q. \end{aligned}$$

The objective function *Q* is

$$\begin{aligned} Q = \sum _{i,j} \gamma _{ij}x_{ij}, \end{aligned}$$

(6)

where \(x_{ij}\) is a binary variable equal to 1 if nodes *i* and *j* belong to the same community and 0 otherwise. \(\gamma _{ij}\) is a function measuring the cohesion between nodes *i* and *j*. Originally proposed in Chang et al. (2016), it is defined in Bartesaghi et al. (2020) as follows:

$$\begin{aligned} \gamma _{ij} = ({\bar{\xi }}_j - {\bar{\xi }}) - (\xi _{ij}- {\bar{\xi }}_i), \end{aligned}$$

(7)

where \({\bar{\xi }}_{j}\) is the average distance between node *j* and nodes other than *j* and \({\bar{\xi }}\) is the average distance over the whole network.

Since two nodes are cohesive (and incohesive, respectively) if \(\gamma _{ij}\ge 0\) \((\gamma _{ij}\le 0)\), in terms of distance, they are cohesive if they are close to each other and, on average, they are both far away from the other nodes.

From this perspective, \(\gamma _{ij}\) can be seen as the ’gain’ if positive or the ’cost’ if negative of grouping two nodes *i* and *j* in the same community. The applied methodology will allow us to discover communities in the WTN based on all the possible channels of interactions and exchanges between countries.

### Econometric model

#### Baseline model

In what follows, we want to assess the role of the WTN in the evolution of the pandemic in the five weeks between March 11th and April 21st, 2020. At the same time, we want to control for additional socio-economic factors that can have an impact on the diffusion of the pandemic. To avoid the possibility that, in turn, these factors might be affected by COVID-19 diffusion, we include them as referring to 2019.

The baseline model that we adopt to test for the role that network centrality has played in explaining the number of infections (INF) and deaths (DEATH) in the first wave of the COVID-19 outbreak (i.e. between March 11st, 2020 and April 21st, 2020) is the following:

$$\begin{aligned} Y_{it}=\beta _0 + \beta _{1,i} TNC_i+{\mathbf {Z}}'_i{\varvec{\beta }}_Z+ \gamma _t + \epsilon _{it} \end{aligned}$$

(8)

where \(Y_{it}\) is either the number of COVID-19 infections (INF) or the number of deaths (DEATH) in country *i* and week *t*. The variable \(TNC_{i}\) stands for trade network centrality and represents a given centrality measure^{Footnote 1} (respectively: degree, strength, weighted eigenvector and weighted clustering coefficient) measured in 2019; \({\mathbf {Z}}\) is a vector of additional regressors that can explain the number of infections and fatalities due to COVID-19, namely GDP per capita (GDPPC, at constant 2010 US$), total resident population (POP), the share of elderly population (POP65+), the number of hospital beds per 1,000 inhabitants (HBEDS) and the average temperature in February and March (TEMP) in degrees Celsius, all measured in 2019. The term \(\gamma _t\) is a series of five week-specific dummies that capture the trend in the dynamics of COVID-19 infections and fatalities for all our countries,^{Footnote 2} while \(\epsilon _{it}\) is the stochastic error component with zero mean and finite variance \(\sigma ^2_{\epsilon }\). To control for the unobserved arbitrary within-group correlation of our observations, we cluster the standard errors at the country level.

Since \(Y_{it}\) is a count variable, and our regressors are time-invariant because they are all measured in 2019, we estimate Eq. (8) using a pooled negative binomial regression model. As is common for count-data models, we test for the overdispersion of our data, that is, for the fact that the conditional mean can be lower than the conditional variance, typically due to the presence of unobserved factors than can affect the number of COVID-19 infections or deaths. In such a case, the main assumption for the use of the Poisson model is violated, and the negative binomial model fits the data better.

We also check for the presence of potential multicollinearity by re-estimating Eq. (8) through a linear regression model and using a variance inflation factor (VIF) statistic. ^{Footnote 3} Multicollinearity can be considered an issue if the VIF statistic takes a value higher than the commonly accepted threshold of 5. To check which of the proposed trade network centrality measures provides the highest explanatory power in predicting \(Y_{it}\), we use the Akaike information criterion (AIC) and Bayesian information criterion (BIC).

To compare the magnitude of the estimated coefficients, we standardise all the regressors by subtracting their mean and dividing by their standard deviation. For each variable, we report the incidence rate ratio (IRR), which measures the impact of a unit increase of the regressor on the risk of contagion (mortality) from COVID-19, computed as the ratio between the number of infected (deceased) individuals and the number of non-infected (surviving) individuals. In this respect, the IRR of a regressor is easier to interpret than the corresponding estimated coefficient, since the latter measures the impact of a unit increase in the regressor itself on the log of the expected number of infections or deaths. We also test for the validity of our negative binomial regression mode in two ways. First, we estimate Eq. (8) using a Poisson model, and we use the Pearson goodness of fit test, where a significant \(\chi ^2\)-distributed statistic would reveal that, because of overdispersion in the data, the Poisson regression model is not appropriate, and a negative binomial specification should be preferred. Second, after estimating Eq. (8), we compute the average predicted probabilities and we compare the observed number of infections and deaths with the number predicted by our negative binomial regression model.

#### Econometric model considering WTN mesoscale structure

To check whether the WTN community structure had an impact on COVID-19 diffusion during the first wave, we re-estimate Eq. (8) using an averaged local clustering coefficient of network communities detected with the methodology described in “Community detection based on communicability distance” section. We then compare the IRRs with those estimated for the local clustering coefficient (as in Eq. (1)). Specifically, for each community, we compute the average of the clustering coefficients \({\bar{C}}\) of the countries therein. Therefore, each country in community *k* has a new clustering coefficient equal to \({\bar{C}}_{k}\), defined by

$$\begin{aligned} {\bar{C}}_{k} = \frac{1}{n_k} \sum _{j=1}^{n_k} C_{j} \end{aligned}$$

(9)

where \(n_k\) is the size of community *k* and \(C_{i},\) is the local clustering coefficient of node *i* as in Eq. (1).

Coefficient \({\bar{C}}\) has two properties: on the one hand, it still reflects the country’s centrality within all its triadic relations expressed by the local clustering coefficient in Eq. (1). On the other hand, \({\bar{C}}\) takes into account the mesoscale structure of the WTN based on communicability. In other words, with this new coefficient, we capture the impact of a country’s centrality in a subset of the WTN, where nodes strongly exchange trade-related information that can be directly observable (such as merchandise trade) or indirectly observable (such as the interactions characterising the supply chain of a good).

We then re-estimate Eq. (8) using as a network centrality measure the average community local clustering \({\bar{C}}_k,\) as in Eq. (9), and we compare the newly estimated IRR with that of the local clustering coefficient of each country. We also provide a series of robustness tests in which we re-estimate Eq. (8) week by week, dropping the term \(\gamma _t\) and using a series of five distinct cross-sectional negative binomial regression models for each of the two dependent variables, INF and DEATH, respectively. In this way, we can observe whether, and to what extent, the estimated IRRs vary along the first wave of the COVID-19 pandemic, and test for the stability of the IRRs for the country-specific network centrality measures, as compared with the corresponding community-level measures.