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The multilayer architecture of the global input-output network and its properties
The network architecture plays a central role in explaining the propagation of shocks and the robustness of \added {different financial and economic systems}. \added{The 2007-2009 global recession and the subsequent European debt crisis have shown that the distress of some financial institutions in a particular country could be easily transmitted across borders into other countries, because of international inter-dependencies and the very high degree of financial integration Cetorelli_Goldberg_2011, Shin_2012, Stefanie_et_al_2013, Bostandzic_et_al_2018, Hale_et_al_2019, Park_Shin_2020.} From the sphere of real economic activities, in an economic system of heterogeneous \added{and} interdependent agents (e.g., industries or firms), \added{the understanding of the structure of their interactions } plays a crucial role in exploring how shocks of a specific agent can be propagated to the others \added{possibly leading} to a large aggregate fluctuation or a systemic failure in the economy Acemoglu_et_al_2012, Carvalho_2014, Acemoglu_et_al_2016, Acemoglu_et_al_2017, Atalay_2017, Eppinger_et_al_2020. \added{It has been also suggested that such understanding at an international scale is useful in examining the cross-country transmission and the propagation of local shocks Carvalho_et_al_2016, Acemoglu_et_al_2016, Luu_et_al_2018b, Luu_et_al_2018a, Boehm_et_al_2019, Eppinger_et_al_2020, Giammetti_et_al_2020.} In summary, through input-output linkages in a production network, shocks to a particular node may have two potential effects on other nodes: (i) supply-side shocks generate propagation to the downstream customers, and (ii) demand-side shocks generate effects on upstream suppliers Acemoglu_et_al_2016. Furthermore, direct and indirect propagations capture both the first-order impact on the “nearest \added{neighbors}” of the affected industry as well as the higher-order impacts on those who are the customers of the customers or the suppliers of the suppliers, and so on. As shown in Acemoglu_et_al_2012, not only lower-order structural properties among sectors, \deleted{such as the node degrees or strengths,} but also more complex properties of the economic network can play a defining role in explaining cascade effects in the whole system. More specifically, even if two different networks have the same first-order characteristics (e.g. same degree or strength sequence) different higher-order interactions among industries may lead to different severity levels of aggregate downturns Acemoglu_et_al_2012.\footnote{Different higher-order of connectivity coefficients can be defined to capture more complex patterns of cascades in an input-output network. For example, at the third-order level of inter-connection, higher aggregate fluctuations could be observed if in the network the sectors having high strengths also share many common suppliers Acemoglu_et_al_2012.}
More in general, once the network structure is taken into consideration, the understanding of topological properties, from a microscopic to a macroscopic perspective, is crucial \added{to assess }how a local shock or disruption to a specific node or cluster of nodes can be propagated and amplified to the rest of the system. \added{This is why the understanding of the higher-order topological properties \added{(such as those at the mesoscale level like the community structure studied in the present work)} may have meaningful implications in managing complex economic systems.} In contrast to homogeneous random \added{and regular} graphs, it is often observed that real networks properties are more heterogeneous Newman_2003. For example, some nodes may have intensive interactions with many other nodes while some others only concentrate on few partners (heterogeneity in the degree and strength sequences). Furthermore, at a broader scale, nodes and edges may fall into different \added{sets} such that the internal interactions within these groups are stronger than those between different \added{ones} (heterogeneity in clustering \added{behaviors}). The latter feature is often referred to as the community structure, which has been intensively studied in the literature for a wide range of networked systems FORTUNATO_2010.
\added{From a perspective of policy implications, } the identification of such a mesoscale structure can be successfully utilized to assess the functioning, the stability and the evolution of networks. Let us consider an economic system of interacting agents as an example. First, if agents can be classified into different blocks, one could examine whether a local shock \added{or a “communication"} will spend most of the time or will even be trapped in a particular block or will be spread widely into the whole system. \added{However, in the presence of large blocks (e.g. referring to the present work, large international trading communities), the diffusion of a shock may have a pervasive effect on many other internal nodes.}
\added{Moreover,} further analysis of the importance and position of nodes within each of these groups may also provide meaningful implications, e.g. (i) central nodes that have many links to the other partners may play a crucial role in the functioning of that group, and (ii) boundary nodes that are mutually connected with those from different blocks may play an important role in blocks' communication and act as the “gatekeepers” or “middle men” that spread shocks \added{and carry on them from a block to another one}.
On top of this, such an identification can be used to explore the network origin of the business cycle co-movements among sectors or countries, since agents who belong to a tightly connected cluster \added{(or community)} may tend to synchronize more their economic activities Long_Plosser_1983, Burstein_et_al_2008, diGiovanni_Levchenko_2010, Johnson_2014, diGiovanni_et_al_2018. For example, in a micro-founded model of business cycles, Long_Plosser_1983 show that input-output interdependencies could be an important factor responsible for the co-movements among the outputs of different industries in the economy. At the international scale, Burstein_et_al_2008 suggest that countries more engaged in sharing international productions exhibit a higher level of co-movement of business cycles. \added{In another study, diGiovanni_Levchenko_2010 find that nations with strongly trading relations among themselves also often show a higher level of business cycle synchronization. In a similar vein, from a micro perspective, diGiovanni_et_al_2018 report that French firms, that have stronger trade linkages with a particular foreign country, also tend to have with it a higher correlation. } \added{Among different observed economic networks, perhaps the global trade system has received a more remarkable attention, due to its economic importance and data availability. It has been widely suggested that the world trade system exhibits a heterogeneous structure at different orders of interconnection. For example, using the methods of network analysis and analyzing the network of merchandise trade imbalances between countries over the period from 1948 to 2000, Serrano_et_al_2007 find the presence of both local heterogeneity (when only few trade linkages possess a large percentage of the country’s total flow) and global heterogeneity (when only a small proportion of all the trade linkages in the whole network carries most of its total flow). Fagiolo_et_al_2008, carrying out a weighted network analysis over the period 1981-2000, show that many weak trade linkages coexist with some much stronger ones, and countries with intense trade relationships tend to be more clustered together. Benedictis_Tajoli_2009 analyze the World Trade Network (WTN) over the years from 1950 to 2000 and confirm the presence of trading blocks with a strong heterogeneity in the way countries select their trade partners. Focusing on the mesoscale structure of the WTN, Bartesaghi_et_al_2020 show that the global trade system can be decomposed into different communities in which the internal interactions among members in each group are more intensive than the external ones. In addition, within each group, there exist some “main, central" countries “surrounded" by the other “satellite" members. The persistence of communities in the global trade system has been also investigated during the COVID-19 pandemic, focusing on centrality measures that are meaningful in capturing possible structural modifications Antonietti2022.} \added{Furthermore, although the global trade system has a certain degree of persistence over a definite time horizon, if we investigate deeper the temporal dynamics of different structural properties of the network, we can observe an evolution over time Serrano_et_al_2007, Benedictis_Tajoli_2009. The driving forces of such temporal dynamics can be traced back to several concepts and factors suggested by trade theories. For example, due to the extensive margin effects, the system can expand if new trade linkages are established. In contrast, with the intensive margin effects, the network dynamics can be driven by changes in the strength of the trade relationships over time}\footnote{In this case, the binary links already exist but the weights can become more (or less) intensive over time.} Felbermayr_Kohler_2006, Helpman_et_al_2008. \added{There could be other factors governing the evolution of the global trade system, such as changes in the country fitness Anderson_1979, Garlaschelli_Loffredo_2005, new partnerships of bilateral and multi-bilateral trade agreements or currency unions Subramanian_Wei_2007, Helpman_et_al_2008, Benedictis_Tajoli_2009. Mundt2021 proposes an empirical model of network formation that takes into account the endogeneity of structural network characteristics. Applied to the Input-Output data, the model shows significant fluctuations of network ties over time, inducing changes also in the aggregate network structural properties.} While much of empirical analysis of economic networks has so far either focused on single types of relations separately or combined all of them together into an aggregated one, less attention has been paid to a much richer structure of these networks, i.e. the so-called multilayer architecture Biancon_multilayer_book_2018, Battiston_et_al_2018. In fact, in many real networks, various types of interactions may coexist: linkages among different nodes of the same type, linkages among different nodes from different types, and those between nodes with their copies (replicas). \added{For example, in the financial system, banks often interact among themselves in different layers via different channels and various markets Carlos_et_al_2014, Poledna_et_al_2015, Bargigli_et_al_2015, Luu_Lux_2019. In a similar vein, the global trade system exhibits a multilayer \added{structure}, which should be considered as one of the critical factors for the emergence of multiple channels of cross-border propagation of economic crises Lee_Goh_2016. In the increasing integration of global economies, each country may simultaneously trade with other countries on many different commodities-based layers Barigozzi_et_al_2010, Gemmetto_Garlaschelli_2014, Gemmetto_et_al_2016. Barigozzi_et_al_2010 study the properties of the multilayer trade network where each of 97 layers represents trade relations among 162 countries in a particular commodity over the 1992-2003 period. Interestingly, their findings suggest that the distributions of link-weights (trade intensities), the averages of connectivity, the clustering coefficients and centrality levels vary across commodity-based layers and are very different from those of the aggregated trade network. The heterogeneity and dissimilarity among layers may reflect the increasing specialization process in the global trade network when countries tend to focus on certain products. Gemmetto_Garlaschelli_2014 and Gemmetto_et_al_2016 apply a statistical network approach to examine the role of the intrinsic heterogeneity in the local constraints like the degree sequences and/or strength sequences in explaining the overlaps and correlations among layers in the undirected and the directed versions of the international trade network. The network is comprised of more than 200 nodes (countries) with nearly 100 layers representing different traded commodities. Comparing across layers (traded commodities), the weighted analysis shows that while many pairs of layers display only a negligible degree of similarities between themselves, few layers exhibit a high level of overlaps, implying that some countries tend to export or import more with a similar set of trading partners in some particular commodities. These studies also suggest an important role of the distribution of hubs (i.e., countries with a high level network centrality) in explaining the relations between layers.}
\added{Empirical analyses, with a focus on the global production system that comprised supply-chain or input-output trade linkages among industries from different countries, have been received less attention due to fact that more comprehensive, global data only becomes available only recently. As pointed out in Baldwin_et_al_2015, international production relations have been flourishing over the last few decades and they still continue to evolve. The authors also suggest that more work is needed to uncover the complexity and hidden patterns embedded inside the global production system, as the importance and the degree of integration or dependency varies across countries, sectors, types of intermediate and final goods in such system. To our best knowledge, one of the first attempts to analyze the topological properties of the world input-output network of the individual industries from different countries was the study of Cerina_et_al_2015. Using the World Input Output Database (WIOD, 2013 release), Cerina_et_al_2015 study different properties from the local to the global level of the world input-output network where nodes are the individual sectors while weighted links represent the monetary trading flows between them. Their results suggest that, overall, the binary links are very dense but the interaction intensities are highly asymmetric. At the local level, the authors show that the network analysis, based on different centrality and community coreness measures, can provide deeper insights into identification of the key sectors. In a recent work, focusing on the production network among industries from 20 country members in the European Monetary Union (EMU), Luu_et_al_2018b show that the external (cross-border) input-output linkages exhibit a hierarchical structure in which a small number of industries from few country members trade more intensely among themselves and form a cohesive core. In contrast, less active “satellite" industries mostly trade with the key sectors in the core. From a network perspective, these key sectors, indeed, play as hubs bridging industries from different country members of the EMU. As shown in Luu_et_al_2018a, such an asymmetric network architecture has crucial implications for the understanding of the pathway on how a local shock to a particular sector in a country member of the EMU can be transmitted to those in other countries.} \added{Using the methods of the multilayer analysis, Russo_et_al_2022 investigate the evolution of the international trading network of different components and parts of the automotive industry over the period 1993-2018. According to their study, several denser and more internally cohesive sub-networks can be identified like a cluster comprised by trade relations of Germany with some Central Eastern European countries, another cluster formed by the US and its partners in the preferential trade agreements such as Canada and Mexico, and another one with a prominent role of Japan and China. Russo_et_al_2022 also show that the shape as well as the composition of clusters and the relative importance of countries also change over the years, with a remarkable rise of China after its accession to WTO.}
\added{The better understanding of the production networks can, of course, be useful for conducting effective industrial policies. For instance, successful experience and lessons from Japan, South Korea, Taiwan, and other “Newly Industrialized Country" (NIC) have pointed out that industrial policies\added{,} that select some “strategic" sectors to promote\added{,} can play a prominent role in facilitating the industrialization process Antonio_2002, WB_Industrial_Policy_Survey_2006, Liu_2019, Antras_Gortari_2020. However, under the expansion of the global production network in which we often observe such a hierarchical structure dominated by few countries and their industries, the potential role of industrial policies (especially for less developed countries or for infant industries) might be more limited than what was traditionally supposed WB_Industrial_Policy_Survey_2006.}
The main contribution of our study is to provide an analysis of the global trade network from a different perspective, i.e. the multilayer architecture of the input-output interdependencies among industries and countries. \added{In our multilayer network countries are nodes and sectors represent layers and we focus} on the mesoscale structure and related network properties spanning from nodes to nodes across layers. We consider a comprehensive structure of the network in which different types of connections (i.e. intralayer as well as interlayer ones) do exist all together. In addition, after classifying different clusters of nodes and associated layers, we further investigate the internal structural properties of each cluster and then compare with those of the other clusters. \added{We also examine the role of individual countries and industries in forming and functioning different large international clusters. Through these analyses,} we aim to uncover hidden structures and complexities inside the detected communities \added{and shed light on the diversification and specialization patterns of countries when they trade their inputs and outputs in and between different industries-based layers}. To these ends, we extend various network metrics and community detection methods used for single layer networks to those applicable for complex networks with a much richer multilayer architecture. In the next steps, we apply them to analyse the input-output relations between different sectors in different countries, using the world input-output database (WIOD, 2016 release; see Timmer2015, Timmer_et_al_2016).
Our results show that the multilayer analysis is able to catch more peculiar trade relationships that cannot be detected at the \added{mono-layer} aggregate trade network. Among others, we find that, in the weighted version, while many layers have dissimilar internal structures, few others are somewhat more overlapped or correlated. On top of that, the interlayer interactions are also highly heterogeneous. Digging deeper at a broader mesoscale structure, we can identify several large international communities in which some countries trade more intensively in certain specific industry-based layers. However, it is worth to emphasize that such a mesoscale structure does evolve over time, which may reflect the competition as well as cooperation dynamics among industries and among nations. For example, in 2000, the first largest international community was comprised by sectors from the US, Japan together with several sectors from China, while the second largest cluster mainly consists of industries from the former Eastern Bloc's countries. In contrast, as for the data in 2014, we identify the first largest international community where relevant sectors of main Asian players (China, India, Japan, South Korea) together with those from Australia are clustered together. On the other hand, sectors from countries involved in the North American Free Trade Agreement (Canada, Mexico and USA) belong to the second largest one.
The remainder of this paper is organized as follows. In Section (ref), we describe the multilayer representation of the global trade system as well as the \added{proposed methodology used} to analyze such a network. Section (ref) summarizes \added{ preliminary analysis of data and the main results with related economic interpretation}. \added{A focus on temporal evolution is also provided}. We conclude in Section (ref). Further information on the dataset and additional results are left to the Appendices.
\added{We shall now introduce the general mathematical representation of a multilayer trade network. Formally, to deal with a multilayer network, one of the main approaches}\footnote{\added{An alternative approach consists in using tensors. For the sake of brevity, we refer here only to supra-adjacency representation. For further details of the tensor approach, we refer the reader to DeDomenico_et_al_2013 and BCG2022.}}\added{ is to represent the trade interdependencies among $L$ sectors from $N$ different countries as a block matrix, with $L\times L$ blocks, each one of order $N$, called the weighted supra-adjacency matrix $W^{supra}$ with size $N_{supra}\times N_{supra}$, where $N_{supra}=NL$ Biancon_multilayer_book_2018:}
\begin {equation} W^{supra}=\left[
\right]. \end {equation}
Each $W^{[\alpha, \alpha]}$ is a $N$-square block matrix representing the intralayer interactions among $N$ nodes in layer $\alpha$, while when $\alpha \neq \beta$, each $W^{[\alpha, \beta]}$ with size $N\times N$ is a weighted (adjacency) block matrix capturing the interlayer interactions between nodes in layer $\alpha$ and nodes in layer $\beta$.
Its binary version, \added{indicating the existence of links}, is characterized by a supra-adjacency matrix $A^{supra}$ with the same size, \added{where the block matrix $W^{[\alpha, \beta]}$ is replaced by the binary matrix $A^{[\alpha, \beta]}$.}
The supra weighted matrix $W^{supra}$ can be further decomposed into two distinct parts $W^{supra}=W^{intra}+ W^{inter}$, in which the first part $W^{intra}$ consists of only intralayer linkages and the second part $ W^{inter}$ consists of only external interlayer linkages: \begin {equation} W^{intra}= \left[
\right], \end {equation} and \begin {equation} W^{inter}= \left[
\right], \end {equation} where $O$ is the $N$-square matrix whose elements are zero.
Similarly, in the binary version, we can also define $A^{intra}$ and $A^{inter}$ as the intralayer and interlayer adjacency matrices, respectively.
\added{We can extend the standard vertex centrality measures in the multilayer context. Since each country or sector in the global trade network can be both the buyer and the seller at the same time, it is necessary to distinguish between the incoming and outgoing linkages with their partners. Given the matrix $W^{supra}$ showing the interaction intensities, and the matrix $A^{supra}$ representing the existent interactions, one can compute the different types of degrees and strengths of nodes (countries) across layers (sectors). We refer the reader to the (ref) for the formal definitions of in- and out strengths (in and out-degree, respectively).}
We recall that $A^{[\alpha, \beta]}$ and $W^{[\alpha, \beta]}$ capture the interdependencies between nodes in layer $\alpha$ with nodes in layer $\beta$ in the binary and weighted versions, respectively. Hence, to examine how strong the interaction between two layers $\alpha$ and $\beta$ is, we measure the average connectivity and intensity based on the elements of these two matrices. Let us define the average link and weight between two layers $\alpha$ and $\beta$ as
and
The average strength $\langle w^{[\alpha, \beta]} \rangle$ can be further normalized by $w^*$, \added{that is} the largest element of the weighted \added{supra-adjacency} matrix $W^{supra}$:
which will lead to $0\leq \langle w^{[\alpha, \beta]} \rangle_{norm} \leq 1$.
In the following, we will briefly explain the methods used to measure the overall overlaps and correlations between layers. Following Gemmetto_Garlaschelli_2014, Gemmetto_et_al_2016, Luu_Lux_2019, we define the overall (normalized) degree of overlaps between every pair of layers $\alpha$ and $\beta$ in the binary version as
Analogously, for the weighted version, the overall (normalized) level of overlaps between two layers $\alpha$ and $\beta$ is given by
It can be easily shown that $O^{[\alpha, \beta]}_{bin}$ ranges in $[0, 1]$, with $O^{[\alpha, \beta]}_{bin}=0$ if and only if two layers $\alpha$ and $\beta$ have no overlaps at all, while $O^{[\alpha, \beta]}_{bin}=1$ if the two adjacency matrices are identical. Similar interpretations apply to $O^{[\alpha, \beta]}_{w}$ used in the weighted version.
Alternatively, the overall degree of similarity between two layers $\alpha$ and $\beta$ \added{could be measured by} the Pearson correlation coefficient (element by element) between $A^{[\alpha, \alpha]}_{ij}$ and $A^{[\beta, \beta]}_{ij}$ or between $W^{[\alpha, \alpha]}_{ij}$ and $W^{[\beta, \beta]}_{ij}$ over all possible pairs of node indices $(i,j)$:
where in general, $\langle X \rangle$ and $\sigma [X]$ are the notations for the mean and standard deviation of $X$.\\
\added {It should be emphasized that} while the average connectivity and the average intensity \added {defined in Eqs. ((ref)) and ((ref))} are based on the interlayer linkages, the level of overlaps and the degree of similarity between two layers $\alpha$ and $\beta$ only depend on the intralayer linkages. \added {In the context of the global input-output analyzed in our current work, the average connectivity and the average intensity for each pair of layers (sectors) $\alpha$ and $\beta$ indicate whether, over all different $N$ countries, these two sectors strongly interact or not. The level of overlap and of degree similarity, in contrast, returns the extent to which trade relations among $N$ countries in a particular industry $\alpha$ are also similar to trade relations among $N$ countries in another industry $\beta$.}
The communicability between every pair of nodes quantifies the number of all possible connection paths between them ESTRADA_et_al_2012. It should be emphasized that the basic network metrics such as degrees or strengths can only capture the second order structural interdependencies (from all other nodes to a node or from a node to all other nodes). Even the clustering coefficients, which are often used to analyse the clustering \added{behaviors} in complex networks (e.g. see Luu_et_al_2017 and the literature therein), can explain the third order of the structural correlations among three nodes alone. In contrast, the communicability is able to catch richer information about the direct as well as indirect pathways associated with different orders of interconnectedness between nodes.
\added{The communicability among nodes has already been used to detect the community structure in the context of monoplex networks Estrada_Hatano_2008, Bartesaghi_et_al_2020.} As a natural extension, the multilayer communicability indicates the number of paths through both possible intralayer and interlayer links that connect a given node in a particular layer to the other nodes in the multilayer architecture. In the context of global input-output network, communicability indicates the number of different upstream and downstream propagation channels between sectors and between countries, via both \added{immediate} and \added{not immediate} pathways. In fact, it can be somewhat related to the concept of the average propagation length often used to measure the economic distances between industries in the literature related to input-output analysis Dietzenbacher_et_al_2005, Miller_Blair_2009.
In the binary version, given a supra-adjacency matrix $A^{supra}$, the communicability matrix of the multilayer network is \begin {equation} G_{bin}=I + \frac {(A^{supra})^1}{1!}+ \frac {(A^{supra})^2} {2!}+...+ \frac {(A^{supra})^k}{k!}+...=\exp(A^{supra}). \end {equation} Note that $G_{bin}$ can be further expressed in the form of a supra matrix as
\begin {equation} G_{bin}=\exp(A^{supra})= \left[
\right], \end {equation}
where in general $G^{[\alpha, \beta]}$ (size $N\times N$) is the matrix containing the communicability between pairs of nodes belonging to layer $\alpha$ and layer $\beta$ Biancon_multilayer_book_2018. As a special case, when $\alpha=\beta$, $G^{[\alpha, \alpha]}$ represents communicability among nodes within layer $\alpha$. However, it is worth to mention that $G^{[\alpha, \alpha]}$ may differ from $\exp(A^{[\alpha, \alpha]})$ if there exist couplings among replicas and/or interlayer connections.
It is worth noting the similarity between the communicability matrix and the Leontief inverse matrix. They are both matrix functions obtained as a sum of a power series expansion. The Leontief inverse matrix is a resolvent-type matrix and unravels the technological interdependence within the productive systems in a given economy Silva2017. Then, the Leontief inverse and the communicability matrix are related measures of aggregate fluctuation capacity and they differ in the way long walks in the network are penalized\footnote{Notice that, as for the Leontief matrix, the input-output interdependencies play a key role in the communicability matrix. Therefore, a similar interpretation based on micro-founded models as in Long_Plosser_1983 can be further investigated}. While in the Leontief inverse equal weights are assigned to walks of different length, in the communicability matrix a walk of length $k$ is penalized\footnote{Notice that a similar assignment of decreasing weight on walks can be given representing the influence structure induced by a Neumann series.} in the sum by a factor $1/k!$. This choice accounts, in a more realistic way, for the fact that a shock originated in a node will have a reduced impact on the latter nodes in the production chain.
Exploiting the communicability matrix we can define suitable centrality measures for the nodes in the multilayer. In particular, in the directed version, for a node $i$ on layer $\alpha$, it is necessary to distinguish the incoming paths from all nodes in a layer $\beta$ to $i$ and the outgoing paths from $i$ to all nodes in a layer $\beta$. Hence, we define the receive (communicability) centrality $\text{rc}_{i}^{[\alpha \leftarrow \beta]}$ and the broadcast (communicability) centrality $\text{bc}_{i}^{[\alpha \rightarrow \beta]}$ as follows:
The total communicability centralities of node $i$ in layer $\alpha$ from/to all nodes in all $L$ layers are then given by
It is straightforward to extend the communicability matrix and the \added{definitions of centrality measures} to their weighed counterparts. In the next step, we can define the communicability matrix $G_w$ for the weighted version as \begin {equation} G_w=I + \frac { \bar {W}^{supra}}{1!}+ \frac { (\bar {W}^{supra})^2}{2!}+...+ \frac {(\bar {W}^{supra})^k}{k!}+...=\exp( \bar {W}^{supra}). \end {equation}
where $ \bar {W}^{supra}$ represents the weighted supra-matrix after a suitable normalization. The normalization helps to avoid the excessive influence of links with higher weights in the network \added{as pointed out in Crofts_Higham_2009 and Estrada_book_2011. The element $\bar{W}^{[\alpha, \beta]}_{ij}$ of the normalised matrix $\bar{W}^{supra}$ represents the incidence of the trade between sector $\alpha$ in economy $i$ and sector $\beta$ in economy $j$ with respect to the average volumes \added{they trade}.}
Note that the communicability matrix, considering all possible walks between two nodes, accounts also for the incidence of the trade between sector $\alpha$ in $i$ and sector $\beta$ in $j$ through intermediate sectors/nodes. As for the inverse of the Leontief matrix, $G_w$ takes into consideration both the effects of indirect and direct trades, but communicability assigns a decreasing weight to indirect connections when their length (i.e., the length of the walk between two nodes) increases.
Based on the weighted communicability matrix $G_w$, one can also easily derive the weighted receive (communicability) centralities and broadcast (communicability) centralities, both for layer-layer communicability centralities and for the total ones, similar to the binary counterparts defined in Eqs. ((ref)) and ((ref)).
In the following, we will explain how to apply communicability centralities to identify communities in a multilayer network. We start by introducing on the network a suitable distance based on the idea of communicability. \added{To ensure that the introduced distance is well defined, we need to work with a symmetrized matrix. To this end, we first symmetrize the matrix $W^{supra}$ by constructing $W^{symm}=\frac{1}{2}({W}^{supra}+{({W}^{supra}})^T)$. To avoid the excessive influence of links with higher weights in the network, we then normalize the new matrix $W^{symm}$ by $\bar{W}^{symm}=S^{-\frac{1}{2}}W^{symm}S^{-\frac{1}{2}}$, where $S$ is the (supra) diagonal matrix of the strengths of each node in each level.} As expressed by the definition, the communicability between two nodes is a weighted sum of the number of all walks connecting the pair (see ESTRADA_et_al_2012). Indeed, the fact that two nodes can be connected by means of all possible walks, and not only by shortest paths, is implicit in the idea of communicability.
In general, the communicability-based distance $\xi_{ij}^{[\alpha, \beta]}$ between node $i$ in layer $\alpha$ and node $j$ in layer $\beta$ is defined as
where $G$ is computed as in Eq. ((ref)) in the binary version and as in Eq. ((ref)) in the weighted one \added{by replacing $\bar{W}^{supra}$ with $\bar{W}^{symm}$}. Let us also emphasize that we are assigning the same meaning to distances between nodes in the same layer, distances between versions of the same node in different layers, and distances between different nodes in different layers.
Following Chang_et_al_2016 and Bartesaghi_et_al_2020 we can compute the cohesion function $\gamma_{ij}^{[\alpha,\beta]}$, between any couple of nodes, as
where $\bar{\xi}_{ii}^{[\alpha,\alpha]}$ is the average communicability distance of node $i$ belonging to layer $\alpha$ from all the other nodes in all layers and $\bar{\xi}$ is the average communicability distance over the whole network.
The cohesion function $ \gamma_{ij}^{[\alpha,\beta]}$ can be interpreted as a cohesion measure between the two nodes. Specifically, when positive (respectively, negative), it represents the gain (respectively, the cost) of grouping nodes $i$ in layer $\alpha$ and $j$ in layer $\beta$ in the same community of a given partition. We assume then to maximize the global cohesion function ${\cal Q}$:
where $x_{[i,\alpha],[j,\beta]}$ is the Kronecker delta function, which is equal to $1$ if two nodes $[i, \alpha]$ and $[j,\beta]$ are in the same cluster and $0$ otherwise. Hence, we obtain the best possible partition via maximization of the function ${\cal Q}$ defined in ((ref)).
More specifically, the network partition that allows to identify clusters and the consequent optimal partition are established according to the steps of the following algorithm.
\added{It is worth stressing} few key points of the presented methodology. We aim at clustering nodes and layers $[i,\alpha]$ where $i$ are countries and $\alpha$ are sectors on the basis of a specific communicability distance. \added{Our procedure is based on the selection of the optimal threshold. The value of the threshold allows indeed to disentangle the role of very tight relationships between couple of nodes. For instance, a very low threshold leads to a great number of isolated nodes, while higher values produce instead larger communities. In the proposed approach, the value of the threshold is not selected as a prior, but it is computed by the procedure assuring the optimal partition obtained from the maximization of the cohesion function ${\cal Q}$ defined in formula ((ref)).}
We \added{analyze} the mesoscale structure and the related topological properties of the global trade network from the perspective of the multilayer architecture, using an updated version of the world input-output database (WIOD, 2016 release).\footnote {The database is described in detail by Timmer_et_al_2016. It is publicly available at: http://www.wiod.org/database/wiots16.} The updated version released in 2016 is the second wave of WIOD data, providing yearly information for trade in input-output among 56 different sectors in 44 countries and the aggregate of the rest-of-the-world \added {(ROW)} over the period from 2000 to 2014 (see Tables (ref) and (ref) in the (ref) for a list of countries and sectors). \added {In total, we have $56 \times 44 = 2464$ industries worldwide. Traded amounts (weights) among them are expressed in millions of dollars. We provide more detailed descriptive statistics of the weights in Table (ref).} An initial analysis of the topology of the network has been developed. Since it includes standard techniques of network analysis, for the sake of brevity, we refer the reader to the (ref). \added{The results naturally lead to the following related question from a topological perspective at a broader scale of the global trade system in input-output.} Are there clusters of countries whose members interact stronger among themselves simultaneously in different layers associated with different industries? To answer this question, we conduct an analysis of a multilayer community structure in the global input-output network.
In this section, we report main results obtained by applying the procedure based on multilayer communicability defined in Section (ref). The proposed approach has been applied directly to the multilayer network, which has been preliminarily transformed in an undirected one \added{by} substituting each pair of bilateral directed links with \added{an} undirected link, with a weight equal to the average weight. \\ Starting from a supra-adjacency matrix with 2464 nodes, given by 44 countries and 56 sectors and using \added{volumes of trade} in 2014, the methodology provides 117 communities (except some isolated nodes). Notice that clustered groups of nodes are located both intralayers and interlayers. This means that we can find countries \added{that are} members of the same community in a specific sector, as well as countries sharing the same community in more sectors. In particular, two large international communities \added{(namely Community 1 and Community 2)} are detected, with 443 and 318 nodes, respectively. The other ten communities have instead a smaller size, being between 30 and 50 nodes. A graphical representation of all members of the top twelve communities, in terms of number of nodes, can be found in Figure (ref), (ref).
Looking at the clusters obtained, some noticeable elements can be identified. First of all, we observe an Australian-Asian community, where all relevant sectors of main Asian players (China, India, Japan, South Korea) are clustered together. On the other hand, countries involved in the North American Free Trade Agreement (Canada, Mexico and USA) belong to the second community. \\ Furthermore, it is interesting to explore the clustering \added{behavior} of European countries that are “in the middle” between these two relevant communities. Indeed, we find that almost all of them are clustered in Community 1 for some sectors and in Community 2 for other sectors (see, for instance, Germany and Great Britain). To further explain this result, we \added{compute} for each country the number of sectors that belong to these two communities. We also compute\footnote{\added{Detailed results are reported in Table (ref) in (ref).}} the Gini heterogeneity index\footnote{In particular, since we deal with nominal variables, we compute the index $H_{j}$ to quantify the heterogeneity of a country $j$ as $H_{j}=1 - \sum_{i=1}^{c}p^{2}_{i,j}$ where $c$ is the total number of clusters detected by the procedure explained in Section (ref) and $p_{i,j}$ is the proportion of sectors of country $j$ in the cluster $i$ such that $\sum_{i=1}^{c} p_{i,j}=1$. This formula is also known in the literature as the Gini-Simpson index.} to give an indication of the dispersion of sectors for each country along the communities. \added {A higher value of this index for a particular country means that, overall, its sectors are more spread out between different communities.}
As regard to European countries, \added{we notice} how some of them (Bulgaria, Germany, Greece, Norway, Portugal) show a behavior similar to the Australian-Asian countries, being these countries concentrated in the first community for most trade of sectors. However, looking at the sectors, this participation occurs with a different degree of heterogeneity, as indicated by the Gini index. For instance, Germany has 37 sectors in Community 1 and only 6 sectors in Community 2. In contrast, Czech Republic, France, Great Britain, Ireland, Poland, Russia are instead more concentrated in Community 2. The remaining European countries have only a limited number of sectors belonging to the top two communities, while the other sectors are concentrated in specific groups.\\ Moreover, it is worth pointing out the presence of countries with a lower level of heterogeneity in the community location of their sectors, \added{characterized} by a concentration in the same community for almost all sectors (except isolated ones). Indeed, we observe that besides the top largest international clusters, the next large communities are \added{mainly} formed by different sectors of the same country. These are, for example, the cases of Spain (Community 3), Italy (Community 4), Brazil (Community 5), Finland (Community 6), Croatia (Community 7), Turkey (Community 8), Indonesia (Community 9). \added{This behavior can be explained by stronger domestic trades with respect to relations with sectors of other countries.}
\added{The presence of large international communities together with that of smaller ones leaning to domestic or regional input-output relations have been also found in the other studies such as Baldwin_et_al_2015, Cerina_et_al_2015, Luu_et_al_2018a, Luu_et_al_2018b. Indeed, industries of different countries seem to have different degrees of internationalization. Some of them have been more engaged in cross-border trade linkages and, hence, can contribute to or even lead the functioning of regional or international clusters. It should be emphasized that this is not necessary related to the size of the country (or the size of domestic market), as we do observe that the largest international communities are comprised of both sectors from large economies like the US and China and those from smaller ones like Taiwan, Ireland or Czech Republic and ROW.} \footnote{\added{Note that ROW consists of different nations not listed in the database, in which important members are, for example, those of OPEC nations and ASEAN 6.}} \added{Among possible explanations, perhaps the relations between sectors from “headquarter economies" and sectors from “factory economies" play a prominent role Baldwin_2006, Baldwin_et_al_2015. In line with the findings by Benedictis_Tajoli_2009 for the World Trade network, our conjecture is that the selection of the trading partners does matters for input-output relations. As discussed in Baldwin_et_al_2015, “headquarter economies" with more advanced technology and higher wages (e.g., the US, Japan, Germany) tend to offshore certain stages of production to low-wage nations (the factory economies like Mexico, Poland and Czech Republic, Taiwan), which may lead to the creation of regional or international input-output chains across the world such as “Factory Asia", “Factory North America" and “Factory Europe". In between these two distinct types of countries, the “hybrid economies" can simultaneously play a certain role as a headquarter economy and a certain role as a factory one.\\ Furthermore, some industries from “headquarter economies" act as hubs with many linkages with different industries from different nations in the global input-output network. In contrast, those from “factory economies" tend to rely more on few key players from “headquarter economies". This is also, indeed, inline with the results from basic network analysis in the (ref), where we find in the weighted version of the network that some nodes have a much higher level of the incoming and outgoing strengths.}
Now, if we look at each single sector-based layer, we can further explore how countries in the same layer are classified in communities\footnote{\added{Detailed results can be found in Table (ref), (ref)}}. Overall, we find that the analysis for sectors shows a higher level of heterogeneity (as shown by the Gini index), \added{compared to that for countries}. This implies that, in the same \added {considered} layer (industry), countries are on average split in several different communities. However, except for sectors “T" and “U" where countries act almost always as isolated communities, \added{the highest} number of countries in each sector belongs to one of the \added {two largest international} communities. It is also interesting to note that sectors with a lower heterogeneity (as \lq\lq Mining and quarrying (B)\rq\rq and \lq\lq Manufacture of coke and refined petroleum products (C19)\rq\rq) are \added{characterized} by a concentration of countries in Community 1.
To explore the similarity between sectors, we computed the Jaccard similarity index (see Figure (ref)), displaying only coefficients higher than $0.7$. In this way, we emphasize couple of sectors \added{characterized} by a similar classification of countries in communities. We observe \added{a highest similarity between} the sectors \lq\lq Crop and animal production, hunting and related activities (A01)\rq\rq and \lq\lq Manufacture of food products, beverages and tobacco products (C10-C12)\rq\rq. We have indeed that all countries (except Ireland) have been classified in the same way in these two sectors. This result can be easily explained by the fact that activities involved in these two sectors are closely related \added{(see also the analysis of the interrelations between layer-based sectors in the (ref))}. Another interesting case is represented by the sectors F and L68 (Construction and Real Estate) that show 41 countries analogously classified (except Malta, Mexico and Ireland). The same \added{behavior} is also observed for the other two couples G47-L68 (Retail Trade - Real Estate) and G47-K64 (Retail Trade - Financial Service).
So far, we have focused on the analysis of the mesoscale structure in the multilayer version of the global input-output network, with $44$ nodes and $56$ layers. In the next step, we \added{shall} compare previous results with the community structure detected from the mono-layer aggregate version, where each node is a country and a link now considers the total trade between a couple of countries. For consistency, we apply the communicability-based approach proposed in Bartesaghi_et_al_2020 to the mono-layer aggregate network. We obtain seven communities (except five isolated nodes) and report their composition in Table (ref). It is noteworthy that two big communities are detected also in this case but the composition of large communities is mainly driven by geographical patterns. This is in line with the results already detected in the literature for aggregate trade network (see Barigozzi_et_al_2010, Bartesaghi_et_al_2020). It is interesting to note that the first community includes almost all European Countries. Only some well-known couples are clustered alone (see for instance Spain and Portugal in community 4 or Great Britain and Ireland in community 5). The second community groups instead together North American and Asian players. \\ \added{However, the analysis of the mono-layer aggregate version may lead to an oversimplified conclusion that the input-output network among countries was simply composed by regional blocks. Indeed, we do observe the presence of large international communities in the multilayer version. The aggregate version also ignores the fact that trade inter-dependencies among countries do vary across industry-based slices. All taken together, the results allow us to emphasize how the multilayer analysis, that takes into consideration distinction among $56$ industry-based layers, is able to catch more peculiar relationships between countries that cannot be detected at the aggregate level.}
\added{We further investigate the internal topological properties of the communities detected in the multilayer version of the network}\footnote{Detailed results can be found in Figure (ref) and Table (ref) in (ref)}\added{. Our main purpose here is to identify the “key actors" in these communities. To this end,} we start selecting countries and sectors that belong to each identified community and we build a sub-network. More specifically, only the links among members of the community are maintained and analyzed, while those externally built with nodes in the other communities are discarded. In particular, we focus on the two most representative communities, i.e. the Community 1 and 2, since these two largest clusters consist of different sectors from various countries. Figure (ref) shows the ranking of the top sectors in each community based on the (total) node strengths. The results again confirm the dominance of China in the first largest community and that of USA in the second community at the industry levels as many of the influential sectors in the communities 1 and 2 are actually from these two countries. Looking at the industry codes of the most influential sectors, we also observe an interesting difference in economic and \added {industrial} structure between the two communities and hence between the two \added {nations}. While influential sectors from China are mostly manufacturing-related industries, those coming from the US are more dominated by real-estate, finance, technology or service-related sectors.
\added{Notice that for the sake of illustration, in Section (ref) we have focused on mesoscale structure of the considered network in the selected year 2014, which provides the latest available information we can have from the WIOD database.} \added{We shall now investigate the temporal evolution of the international communities. To this end, we considered the period 2000-2014 and we applied the method based on the multilayer communicability distance described in Section (ref) to extract different clusters for each of these selected years. Remarkably, we find evidence that the clustering behaviors of trades in input-output among industries and countries do restructure over the period from 2000 to 2014. In general, not only the community composition changes, but the centrality rankings of the members inside each community is also reordered, with the diminishing of some industries and the emergence of new “key players”. \\ In particular, we start analyzing the similarity between the communities obtained over time. To do this, we computed the Adjusted Rand Index (ARI) between clusters for each couple of years (see Hubert and Rand). This index falls in the interval $[0,1]$ and it is equal to one only if two partitions are completely identical. Numerous measures for comparing clusterings have been proposed in the literature, however the ARI}\footnote{\added{Given a set of elements and two different partitions, we compute the sum of the number of pairs of elements that are in the same subset in both partitions and the number of pairs of elements that are in the different subsets in both partitions. The ratio between this value and the total number of pairs is the Rand Index and gives a frequency of occurrence of agreements over the total pairs. ARI is the corrected-for-chance version of the Rand index and it gives the overall concordance of two methods taking into account that the agreement between partitions could arise by chance alone.}} \added{remains the most well-known and widely used (Steinley). We display in Figure (ref) values of the index for each pair of years. We often find that while the mesoscale structure exhibits a certain level of persistence when two subsequent years are compared, it becomes less and less similar after longer time periods. Such temporal behavior can be related to substitution and margin effects from international trade theories and the basic network properties that we have analyzed in the (ref). Qualitative substitution and extensive margin effects, based on countries and industries that change their trade partners or even obtain new ones, are indeed not very important, since the binary version of the network in the period 2000-2014 is always (and already) very dense. Hence the network can display a certain degree of persistence. Nevertheless, similarly to what has recently been found for the World Trade network Felbermayr_Kohler_2006, Helpman_et_al_2008, Benedictis_Tajoli_2009, intensive margin effects play a more important role in explaining the evolution of the global input-output network. Intuitively, strengthening trade relations may take time and lead to the structural change in the whole network only once the magnitude of traded amount has been sufficiently increased. It should be also emphasized that not all trade relations are simultaneously intensified, as different sectors and countries can play different roles in such a hierarchical system. Furthermore, exogenous shocks and events such as the formation or break of trade agreements, economic zones, monetary unions may also lead to disruptive changes in the community structure of the network.}
\added{To provide deeper insights into the changes in the internal structure of the communities, we shall now analyze their composition over time}\footnote{For the sake of illustration and conciseness, we only report the results related to some selected years before 2014. Results for other years are available upon request to the authors.}. As shown in Figure (ref) (panel a) starting in 2000, the first largest community, with 571 members, is mainly composed by sectors from the US, Japan and interestingly several ones from China. In contrast, the second largest cluster (with 167 members) mainly consists of industries from the former Eastern Bloc's countries, led by those from Russia and Poland\footnote{See also Figure (ref) in the (ref) for additional results.}
Over the next few consecutive years, first major changes are observed in 2002 (see Figure (ref) (panel b)). We have indeed an important reduction of the total number of communities and a higher concentration in the largest group with 895 members. In particular, Community 1 is enriched by the inclusion of several sectors from European countries: Austria, Belgium, Germany and Great Britain join the other European countries (Italy, France, Spain, Ireland) in this big community. On the other hand, the presence of Eastern Bloc's country in Community 2 is reducing. The composition of Community 2 is indeed affected by the effects of the disintegration of the Eastern Bloc with the dissolution of the Warsaw Pact and Comecon. We find that the former members of the Warsaw Pact, that joined NATO (as Poland, Hungary and Czech Republic) in 1999, moved from Community 2 (including Russia) to Community 1 (including United States)\footnote{See also Figure (ref) in the (ref) for additional results.}. \added{A relevant aspect that could lead to major changes in the mesoscale structure of the global input-output network from 2002 is the entrance of China in World Trade Organization (WTO) in December 2001}.\footnote{See, for example, Subramanian_Wei_2007 for further discussions on the impacts of WTO on trade.} \added{China's accession to the WTO brought a number of benefits to both China as well as the world. Reduced barriers to trade and larger foreign direct investment inflows boosted export as well as import markets of this country.} \\
\added{Notice that the composition of the first largest community remains almost the same in the next few consecutive years (as can be seen from Figure (ref)). However, some Chinese sectors become more and more important.} For example, the role of Chinese manufacturing industries assumes in 2004 a higher relevance in this community (see also Figure (ref) (a) in the (ref)).
\added{ This result is in line with Chen and Benedictis_Tajoli_2009 that show an important increase of labor-intensive manufactured goods' exportation and of importation of raw materials that come from China and the other developing Asian economies (aggregated in ROW) during that period. In contrast, looking at the internal structure of the second largest community in 2004, we find that sectors from the former Eastern Bloc's countries play a less important role, but those from Belgium and France become more dominant (for details see also Figure (ref) in the (ref))}.
Overall, we observe that the structure of the largest international communities still continues to evolve in the subsequent years. For example, as for the data in 2009, a number of influential industries from China no longer stay in the first largest community with those from US but emerge as the “key players” in the second largest one. In particular, as displayed in Figure (ref) (d), a large community dominated by Asian countries is noticeable (in particular formed by sectors of South Korea, China and Taiwan)\footnote{Additional results can be found in Figure (ref), (ref).}. The role of Asian countries \added{still continue} to increase over time, leading China in 2014 \added{(as already reported and discussed in Section (ref))} to form the largest international community together with some sectors from the other Asian countries such as South Korea, Taiwan, India, Japan. \added{Altogether, the aforementioned observations may be the signals that the global trade in input-output system has been moving from a former US-Russia bipolar system to another bipolar (or multipolar) one, with the rising role of sectors from China in the recent years.}
\added{To bring our temporal analysis of the mesoscale structure of the global trade network in input-output to a close, we report in Figure (ref) the Gini index, computed for each year, at either country or sector level.}\footnote{\added{Notice that we have computed the Gini index for each country as well as for each sector in 2014 (Tables (ref) and (ref) in (ref), last column). Then we compute the index for each country and sector in the other years, to examine its evolution over time from 2000 to 2014. As explained previously, a higher value of this index for a country means that on average its sectors are more spread out between different communities. In contrast, a higher value of this index for a sector (layer) implies that, in the same traded layer, countries are on average split in various communities.}}
\added{We notice, on average, a lower level of heterogeneity for countries, confirming the presence of relevant domestic trades \added{(see Figure (ref), panel (a)). As reported and discussed in Section (ref), industries from a number of countries engage less actively in the global input-output network, and many of them still mainly rely on their domestic counterparts. The Gini indices for these countries, therefore, are relatively lower. On the other hand, a higher level of heterogeneity \added{is observed for several Eastern European and Mediterranean Sea countries}. We argue that for these countries, a relatively stronger international diversification occurs primarily at the (quantitative) intensive margin effects rather than the extensive ones, since the binary version of the network capturing the existence of trade links is already very dense and does not change very much.} Indeed, these countries derived great benefits from their proximity to the European Union, large amounts of investment from their Western European \added{neighbors} and the far-ranging domestic reforms on which their accession was conditional (see for details Ait). \added{In a similar vein,} some Asian developing countries benefited from a strong export orientation and the increased intra-regional integration, also due to the proximity to the Chinese growth pole. Relevant players (as United States, Germany, Japan and China) show instead a moderate level of heterogeneity due to a greater balance between domestic and international trade linkages.}
\added{Concerning sectors (see Figure (ref) panel (b)), we notice relatively higher values for their Gini indices, with few exceptions. As already observed for 2014 in Section (ref), the two sectors “Mining and quarrying (B)" and “Manufacture of coke and refined petroleum products (C19)" also show systematically a lower heterogeneity and are mainly involved in Community 1 in the previous years. This is not very surprising since from a perspective of the factors of production, over different countries worldwide, these two industries are very closely connected to each other.}
\added{In addition, major changes in the mesoscale structure of the global input-output network are again confirmed by a significant drop in the Gini index in 2002 for all sectors, where a very high concentration of countries and sectors in the largest community is observed. As discussed before, the restructure of trade relations among European countries, the disintegration of the Eastern Bloc, the rising role of sectors from China after its accession to WTO, etc. may explain such noticeable changes in the network in that year.}
This work analyses the global trade network in input-output, using the recently released WIOD database. We show that, by viewing the network through the lens of a multilayer architecture, we are able to extract richer information on interdependencies between countries and industries around the world that cannot be easily detected at the aggregate mono-layer level.
We first analyse the heterogeneity and the diversification in input-output relationships and find that although the list of trading partners is generally broad, some sectors trade more intensively with a number of other sectors in different countries. We then view countries as nodes and industries as layers and explore the similarities and interactions among the layers. Again, in the weighted version, we observe that the similarity levels and the interaction strengths are varied across pairs of layers and that few couples of layers tend to be more overlapped and/or more strongly interacting with each other. Such first insights motivate us to examine the multilayer architecture at a broader scale rather the node-node or layer-layer interrelations alone.
As investigated in our work, at the mesoscale level there exist several large international communities in which some countries trade their inputs and outputs more intensively in some specific industry-based layers. In such a network structure, the world somewhat seems to have bipolar or multipolar trade system with different clusters that are more internally cohesive.
\added{Furthermore,} interestingly, we also observe that the clustering \added{behaviors} of trades in input-output among industries and countries restructure over the period from 2000 to 2014. In general, not only the internal composition changes, but the centrality rankings of the members inside each community also reorder, with the diminishing role of industries from some countries and the growing importance of those from some other countries.\\ As in 2000, the first largest community is still mainly comprised by sectors from the US, Japan together with several sectors from China. In contrast, the second largest cluster consists of industries mostly from the former Eastern Bloc's countries (e.g. Russia and Poland). As in the data for 2014, the most recent year available in the WIOD database, we identify an Australian-Asian community, where relevant industries of main Asian players (China, India, Japan, South Korea) and Australia are clustered together. On the other hand, those from countries involved in the North American Free Trade Agreement (Canada, Mexico and USA) belongs to the next largest community.
\added{The empirical results obtained from our present work provide several meaningful implications. First, it is worth mentioning that large international clusters comprised by different industries from different nations detected in our present work may pave the potential pathways for large cascading failures. In particular, a shock to a “key" sector (hub) of a “key" country in one of these clusters could be transmitted and propagated to many other foreign sectors located in the same cluster via cross-border input-output relations, and possibly lead to a large downturn in the global trade. As shown during the time of the Covid pandemics, the shortages of inputs provided by some industries can lead to disruptions in a number of subsequent dependent industries in other countries Eppinger_et_al_2020.} \added{In addition, the presence of such large international clusters, in which members of each cluster interact more strongly among themselves, may also imply meaningful network origins (i.e., those related to international input-output relations) of business cycle synchronizations among different countries in the world Burstein_et_al_2008, Johnson_2014. Moreover, as pointed out in our present work, the multilayer analysis helps to uncover the complexity of the relations among countries when they trade within and between different layers-based industrial sectors. The proposed approach also allows us to explore the interdependencies among layers rather than just the node-node pairwise relations in the aggregate mono-layer analysis. As shown in Lee_Goh_2016, Korniyenko_et_al_2018 and in Tzavellas_2022, more severe cascading effects can occur in an architecture of multiple layers. This is mainly because the simultaneous presence of intralayer as well as interlayer linkages in such an architecture can lead to collective dynamics of interdependent layers and create multiple potential channels of the diffusion of shocks among countries that are ignored by the analysis of the aggregate version or the analysis of single layers alone.}
Our present work opens several directions for future research. First, the findings on the emergence of several international communities and their structural changes over time need to be investigated further to determine the underlying economic and other potential mechanisms. Second, since in this study we focus on the mesoscale structure and the related properties of the global input-output network, in our future work, we plan to extend our analysis to study and quantify the other important network measures and properties such as the multilayer clustering coefficients and the multilayer centrality DeDomenico_et_al_2013, BCG2022. Third, we believe that incorporating additional layers representing other economic relations such as financial links, trade in final goods and services on top of the input-output interrelations among countries would be able to give a more comprehensive and richer analysis of the multilayer architecture of the world-wide economic network. \added{Such an analysis certainly provides useful inputs to the study of different potential channels of cascading failures and transmission of shocks in the global economy Lee_Goh_2016, Korniyenko_et_al_2018}.
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