The multilayer architecture of the global input-output network

and its properties

Rosanna Grassi1, Paolo Bartesaghi1, Clemente Gian Paolo2, and Duc Thi Luu3

1Department of Statistics and Quantitative Methods, University of Milano - Bicocca,Via Bicocca degli Arcimboldi 8, 20126, Milano, Italy

2Department of Mathematics for Economics, Financial and Actuarial Sciences,Universita Cattolica del Sacro Cuore, Largo Gemelli 1, 20123, Milano

3Department of Economics, University of Kiel, Germany

September 8, 2021

Abstract

We analyse the multilayer architecture of the global input-output network using sectoraltrade data (WIOD, 2016 release). With a focus on the mesoscale structure and relatedproperties, we find that the multilayer analysis that takes into consideration the splittinginto industry-based layers is able to catch more peculiar relationships between countries thatcannot be detected from the analysis of the single-layer aggregated network. We can identifyseveral large international communities in which some countries trade more intensively insome specific layers. However, interestingly, our results show that these clusters can restruc-ture and evolve over time. In general, not only their internal composition changes, but thecentrality rankings of the members inside are also reordered, with the diminishing role ofindustries from some countries and the growing importance of those from some other coun-tries. These changes in the large international clusters may reflect the outcomes and thedynamics of cooperation as well as competition among industries and among countries inthe global input-output network.

KEY WORDS: Input-output linkages, global trade, international trade clusters, mesoscalestructure, multilayer architecture, layer-layer interdependencies.

JEL CLASSIFICATION: C67, F10, F40

1

arX

iv:2

109.

0294

6v1

[ec

on.E

M]

7 S

ep 2

021

1 Introduction

The network architecture plays a central role in explaining the propagation of shocks and therobustness of economic and financial systems. From the financial aspect of the economy, the ex-perience of the 2007-2009 global recession and the subsequent European debt crisis have shownthat, due to international inter-dependencies and the very high degree of financial integration,the financial distress of some financial institutions in a particular country could be easily andquickly transmitted across borders into other countries (e.g. see Cetorelli and Goldberg, 2011;Shin, 2012; Kleimeier et al., 2013; Bostandzic and Weiß, 2018; Hale et al., 2019; Park and Shin,2020). From the sphere of real economic activities, in an economic system of heterogeneous,interdependent agents (e.g. industries or firms), the understanding of the topological propertiesof the interactions among them plays a crucial role in exploring how shocks to a specific agentcan be propagated to the others and possibly lead to a large aggregate fluctuation or a systemicfailure (Acemoglu et al., 2012; Carvalho, 2014; Carvalho et al., 2016; Acemoglu et al., 2016,2017; Atalay, 2017; Luu et al., 2018a,b; Boehm et al., 2019; Eppinger et al., 2020). In summary,through input-output linkages in a production network, shocks to a particular node may havetwo potential upstream and downstream effects on other nodes: (i) supply-side shocks generatepropagation to the downstream customers, and (ii) demand-side shocks generate effects on up-stream suppliers (e.g. see Acemoglu et al., 2016).Furthermore, direct and indirect propagations capture both the first-order impact on the “near-est neighbours” of the affected industry as well as the higher-order impacts on those who are thecustomers of the customers or the suppliers of the suppliers, and so on. As shown in Acemogluet al. (2012), not only lower-order structural properties among sectors such as the node degreesor strengths but also more complex properties of the economic network can play a defining rolein explaining cascade effects in the whole system. More in general, once the network structureis taken into consideration, the understanding of topological properties, from a microscopic to amacroscopic perspective, is crucial to diagnose the pathway of how a local shock or disruptionto a specific node or cluster of nodes can be propagated and amplified to the rest of the system.In contrast to homogeneous random graphs, it is often observed that real networks propertiesare more heterogeneous (e.g. see Newman, 2003). For example, some nodes may have intensiveinteractions with many other nodes while some others only concentrate on few partners (hetero-geneity in the degree and strength sequences). Furthermore, at a broader scale, nodes and edgesmay fall into different groups such that the internal interactions within these groups are strongerthan those between different groups (heterogeneity in clustering behaviours). The latter featureis often referred to as the community structure, which has been intensively studied in literaturefor a wide range of networked systems (e.g. see Fortunato, 2010, and the literature therein forthe detection of this structure in various networks from different fields). The identification ofsuch a mesoscale structure can be usefully utilized to assess the functioning, the stability andthe evolution of networks. To name a few, let us take an economic system of interacting agentsas an example. First, if agents can be classified into different blocks, one could examine whethera local shock will be spent most of the time or even trapped in a particular block, region orcan be spread widely into the whole system. On top of this, such an identification can be usedto explore the network origin of the business cycle co-movements, since agents who belong toa tightly connected cluster may tend to synchronize more their economic activities (e.g. see

2

Burstein et al., 2008; Di Giovanni and Levchenko, 2010; Johnson, 2014; Di Giovanni et al.,2018). Further analysis of the importance and position of nodes within each of these groupsmay also provide meaningful implications, e.g. (i) central nodes that have many links to theother partners may play a crucial role in the functioning of that group, and (ii) boundary nodesthat are mutually connected with those from different blocks may play an important role inblocks’ communication and act as the “gatekeepers” or “middle men” that spread shocks fromone to another block.

While much of analysis of economic and financial networks has so far either focused onsingle types of relations separately or combined all of them together into an aggregated one,less attention has been, however, paid to a much richer structure of these networks, i.e. theso-called multilayer architecture (e.g. Bianconi, 2018; Battiston et al., 2018). In fact, in manyreal economic and financial networks, various types of interactions may coexist: linkages amongdifferent nodes of the same type, linkages among different nodes from different types, and thosebetween nodes with their copies (replicas). For example, in the financial system, banks ofteninteract among themselves via different channels and various markets (e.g. see Leon et al.,2014; Poledna et al., 2015; Bargigli et al., 2015; Luu and Lux, 2019). It has been also shownin recent studies that focusing on a single layer or on a combined layer may lead to a wrongassessment of systemic risk or underestimate the importance of the inter-plays among differentpropagation channels of shocks (Kok and Montagna, 2013; Poledna et al., 2015). In a similarvein, the global trade system does exhibit a multilayer architecture, which should be consideredas one of the critical factors for the emergence of multiple channels of cross-border propagationof economic crises. In the increasing of integration of global economies, each country may tradewith other countries on many different commodities-based layers (e.g. see Barigozzi et al., 2010;Gemmetto and Garlaschelli, 2014; Gemmetto et al., 2016). At the sectoral level, industriesare also interrelated to both domestic and external partners (e.g. see Cerina et al., 2015; delRıo-Chanona et al., 2017; Alves et al., 2018; Luu et al., 2018b). Recent literature on cascadeeffects in the global trade network also suggests that the cascade effects could be severely andsystematically underestimated either when we analyse the system in the aggregated single-layerframework or when we analyse different types of trade relations separately (e.g. Lee and Goh,2016; Luu et al., 2018a).

The main contribution of our study is to provide an analysis of the global trade networkfrom a different perspective, i.e. the multilayer architecture of the input-output interdependen-cies among industries and countries, focusing on the mesoscale structure and related networkproperties spanning from nodes to nodes across layers. We consider a comprehensive structureof 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 associatedlayers, we further investigate the internal structural properties of each cluster and then com-pare with those of the other clusters. Through this, we aim to uncover hidden structures andcomplexities inside the detected communities. To these ends, first, we extend various networkmetrics and community detection methods used for single layer networks to those applicable forcomplex networks with a much richer multilayer architecture. In the next steps, we apply themto analyse the input-output relations between different sectors in different countries, using theworld input-output database (WIOD, 2016 release; see Timmer et al. (2015, 2016)).

Our results show that the multilayer analysis, in which countries are nodes and sectors

3

represent layers, is able to catch more peculiar trade relationships that cannot be detected atthe single-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 overlappedor correlated. On top of that, the interlayer interactions are also highly heterogeneous.

Digging deeper at a broader mesoscale structure, we can identify several large internationalcommunities in which some countries trade more intensively in certain specific industry-basedlayers. However, it is worth to emphasize that such a mesoscale structure of the global tradenetwork in input-output does evolve over time, which may reflect the competition as well ascooperation dynamics among industries and among nations.

For example, in 2000, the first largest international community is comprised by sectors fromthe US, Japan together with several sectors from China, while the second largest cluster mainlyconsists of industries from the former Eastern Bloc’s countries. In contrast, as for the data in2014, we identify the first largest international community where relevant sectors of main Asianplayers (China, India, Japan, South Korea) together with those from Australia are clusteredtogether. On the other hand, sectors from countries involved in the North American Free TradeAgreement (Canada, Mexico and USA) belong to the second largest one.

The remainder of this paper is organized as follows. In section 2, we explain the dataset anddescribe the multilayer representation of the global trade system as well as the methods usedto analyse such a network. Section 3 summarizes the main results. We conclude in Section 4.Further information on the dataset and additional results are left in the Appendix.

2 Data, network representations and methods

2.1 Data

We analyse the mesoscale structure and the related topological properties of the global tradenetwork from the perspective of the multilayer architecture, using an updated version of theworld input-output database (WIOD, 2016 release).1 The updated version released in 2016 isthe second wave of WIOD data, providing yearly information for trade in input-output among56 different sectors in 43 countries and the aggregate of the rest-of-the-world over the periodfrom 2000 to 2014 (see Tables (4) and (5) in the Appendix, for a list of countries and sectors).Trade amounts between sectors are expressed in millions of dollars.

2.2 Multilayer representation and fundamental metrics of the world input-output network

Network representation in terms of tensor and supra matrices:

We shall now introduce the general mathematical representation of a multilayer trade net-work. Formally, a multilayer network that has a fixed set of N nodes in each of L layers canbe conveniently expressed in terms of a 4-order tensor Wtensor

N×N×L×L ∈ RN×N×L×L (e.g. see

1The database is described in detail by Timmer et al. (2016). It is publicly available at:http://www.wiod.org/database/wiots16.

4

De Domenico et al., 2013). We refer to each elementWtensori,j,α,β ofWtensor

N×N×L×L as the weight of the

link between node i in layer α and node j in layer β (for i, j = 1, 2, ..., N and α, β = 1, 2, ..., L).2

Notice that given two layers α and β, each matrix Wtensor:,:,α,β = {Wtensor

i,j,α,β } (for i, j = 1, 2, ..., N),which is 2-order tensor, captures all possible interlayer interactions between nodes in layer α andnodes in layer β. As the special case, Wtensor

:,:,α,α represents the intralayer interactions among N

nodes in layer α. The elements of the binary version AtensorN×N×L×L ∈ RN×N×L×L, which indicatethe existence of links, can be then defined as

Atensori,j,α,β =

{1 if Wtensor

i,j,α,β > 0,

0 if Wtensori,j,α,β = 0.

(1)

To deal with a multilayer network we can also adopt a slightly different approach, which willbe always used in the following. It consists in the so called unfolding procedure that allows toreduce the dimension of the 4-order tensor Wtensor into a supra matrix (equivalent to a 2-ordertensor). It is also referred to as the flattening or matricization method. In particular, we canre-express Wtensor as a block matrix, with L× L blocks, each one of order N , called the supraweighted adjacency matrix W supra with size Nsupra × Nsupra, where Nsupra = NL (Bianconi,2018):

W supra =

W [1,1] W [1,2] W [1,3] ... W [1,L]

W [2,1] W [2,2] W [2,3] ... W [2,L]

W [3,1] W [3,2] W [3,3] ... W [3,L]

... ... ... ... ...

W [L,1] W [L,2] W [L,3] ... W [L,L]

. (2)

Its binary version is characterized by a supra-adjacency matrix Asupra with the same size:

Asupra =

A[1,1] A[1,2] A[1,3] ... A[1,L]

A[2,1] A[2,2] A[2,3] ... A[2,L]

A[3,1] A[3,2] A[3,3] ... A[3,L]

... ... ... ... ...

A[L,1] A[L,2] A[L,3] ... A[L,L]

. (3)

Each W [α,α] = W:,:,α,α (or A[α,α] = A:,:,α,α) is a N -square block matrix representing theintralayer interactions among N nodes in layer α, while when α 6= β, each W [α,β] =W:,:,α,β (orA[α,β] = A:,:,α,β) with size NxN is a weighted (adjacency) block matrix capturing the interlayerinteractions between nodes in layer α and nodes in layer β.

The supra weighted matrix W supra can be further decomposed into two distinct partsW supra = W intra + W inter, in which the first part W intra consists of only intralayer linkagescaptured by different entries Wtensor

i,j,α,β (with α = β) and the second part W inter consists of only

2To avoid the confusion, throughout this study we use Latin letters (i, j, ...) to illustrate nodes and Greekletters (α, β, ...) to indicate layers.

5

external interlayer linkages represented by different elements Wtensori,j,α,β (with α 6= β):

W intra =

W [1,1] O O ... O

O W [2,2] O ... O

O O W [3,3] ... O

... ... ... ... ...

O O O ... W [L,L]

, (4)

and

W inter =

O W [1,2] W [1,3] ... W [1,L]

W [2,1] O W [2,3] ... W [2,L]

W [3,1] W [3,2] O ... W [3,L]

... ... ... ... ...

W [L,1] W [L,2] W [L,3] ... O

, (5)

where O is the square matrix (size NxN) whose elements are zero.Similarly, in the binary version, we can also define Aintra and Ainter as the intralayer and

interlayer adjacency matrices, respectively.Notice that if we only consider the intralayer links and exclude all information regarding the

interlayer connectivities, we will obtain a simple version of multiplex (or multislice) network thathas L layers showing different types of interactions among N nodes. Such a multiplex network iscaptured by the supra weighted matrix W intra and the supra adjacency matrix Aintra (e.g. seeBianconi, 2018, for a discussion on various types of multiplex economic and financial networks).3

Node strengths and degrees in a multilayer architecture:

Since each country or sector in the global trade network can be both the buyer and the sellerat the same time, it is necessary to distinguish between the incoming and outgoing linkages withtheir partners. Given the tensor Wtensor (or the supra weighted matrix W supra) showing theinteraction intensities and the tensor Atensor (or the supra adjacency matrix Asupra) representingthe existent interactions, one can compute the different types of degrees and strengths of nodesacross layers (e.g. see Bianconi, 2018).

The intralayer in-strength and in-degree for each node i in a layer α are given by

s[α←α]i,in =

∑j 6=iWj,i,α,α =

∑j 6=i

W[α,α]ji , (6)

andk[α←α]i,in =

∑j 6=iAj,i,α,α =

∑j 6=i

A[α,α]ji . (7)

In a similar vein, we can define the intralayer out-strength and out-degree of node i in layer αas

s[α→α]i,out =

∑j 6=iWi,j,α,α =

∑j 6=i

W[α,α]ij , (8)

3Note that, in a more general definition of a multiplex network, one could also take into account the linksamong replica nodes (identical nodes) in different layers Bianconi (2018).

6

k[α→α]i,out =

∑j 6=iAi,j,α,α =

∑j 6=i

A[α,α]ij . (9)

In addition, for every pair of different layers α and β, one can quantify the interlayer in-degree and in-strength of interactions from all nodes in a layer β to a specific node i in layer αas

s[α←β]i,in =

∑j 6=iWj,i,β,α =

∑j 6=i

W[α,β]ji , (10)

k[α←β]i,in =

∑j 6=iAj,i,β,α =

∑j 6=i

A[α,β]ji . (11)

Similarly, the interlayer out-strength and out-degree from a particular node i in the layer α toall nodes in a layer β are given by

s[α→β]i,out =

∑j 6=iWi,j,α,β =

∑j 6=i

W[α,β]ij , (12)

andk[α→β]i,out =

∑j 6=iAi,j,α,β =

∑j 6=i

A[α,β]ij . (13)

In the context of an input-output network, the aforementioned degrees and strengths rep-resent the distributions of the numbers of trading partners and of the magnitudes of trade ininput-output across nodes, respectively. A higher incoming (outgoing) degree of a node indicatesthat it has a more diversified portfolio of input sellers (output buyers) in another layer. Nodeswith higher incoming (outgoing) strengths trade their inputs (outputs) more intensively withthe rest of nodes in a certain layer.

The total in-strength and in-degree of a node i in a layer α from nodes in all L layers canbe straightforwardly extended as

s[α, total]i,in =

∑β

s[α←β]i,in , (14)

k[α, total]i,in =

∑β

k[α←β]i,in . (15)

Analogously, we can define the total out-strength and out-degree of a node i in a layer αacross all layers as

s[α, total]i,out =

∑β

s[α→β]i,out , (16)

k[α, total]i,out =

∑β

k[α→β]i,out . (17)

7

Notice that from the definitions of the total in-strength s[α, total]i,in and the intralayer in-

strength s[α←α]i,in , we can easily obtain the total interlayer in-strength, which is given by

s[α, inter]i,in = s

[α, total]i,in − s[α←α]i,in =

∑β 6=α

s[α←β]i,in . (18)

This captures the overall intensity of the dependency of node i in the layer α on the inputsprovided by nodes from the other layers. Similarly, one can derive the total interlayer in-degree,the total interlayer out-strength and the total interlayer out-degree:

k[α, inter]i,in = k

[α, total]i,in − k[α←α]i,in =

∑β 6=α

k[α←β]i,in , (19)

s[α, inter]i,out = s

[α, total]i,out − s[α→α]i,out =

∑β 6=α

s[α→β]i,out , (20)

and

k[α, inter]i,out = k

[α, total]i,out − k[α→α]i,out =

∑β 6=α

k[α→β]i,out . (21)

2.3 Interrelations between layers

Average connectivity and intensity between layers:

Recall that A[α,β] and W [α,β] capture the interdependencies between nodes in layer α withnodes in layer β in the binary and weighted versions, respectively. Hence, to examine how strongthe interaction between two layers α and β is, we measure the average connectivity and intensitybased on the elements of these two matrices. Let us define the average link and weight betweentwo layers α and β as

〈a[α,β]〉 =

∑i,j A

[α,β]ij

N2, (22)

and

〈w[α,β]〉 =

∑i,jW

[α,β]ij

N2. (23)

The average strength 〈w[α,β]〉 can be further normalized by w∗, the largest element of the supraweighted matrix W supra:

〈w[α,β]〉norm =

∑i,jW

[α,β]ij

N2w∗, (24)

which will lead to 0 ≤ 〈w[α,β]〉norm ≤ 1.

8

Overlaps and correlations between layers:

In following, we will briefly explain the methods used to measure the overall overlaps andcorrelations between layers. Following Gemmetto and Garlaschelli (2014); Gemmetto et al.(2016); Luu and Lux (2019), we define the overall (normalized) degree of overlaps between everypair of layers α and β in the binary version as

O[α,β]bin =

2∑

i,j min(A[α,α]ij , A

[β,β]ij )∑

i,j(A[α,α]ij +A

[β,β]ij )

. (25)

Analogously, for the weighted version, the overall (normalized) level of overlaps between twolayers α and β is given by

O[α,β]w =

2∑

i,j min(W[α,α]ij ,W

[β,β]ij )∑

i,j(W[α,α]ij +W

[β,β]ij )

. (26)

It can be easily shown that O[α,β]bin ranges in [0, 1], with O

[α,β]bin = 0 if and only if two layers α and

β have no overlaps at all, while O[α,β]bin = 1 if the two adjacency matrices are identical. Similar

interpretations apply to O[α,β]w used in the weighted version.

Alternatively, to analyse the overall degree of similarity between two layers α and β, one can

also measure the Pearson correlation coefficient (element by element) between A[α,α]ij and A

[β,β]ij

or between W[α,α]ij and W

[β,β]ij over all possible pairs of node indices (i, j):

R[α,β]bin =

〈A[α,α]ij A

[β,β]ij 〉 − 〈A

[α,α]ij 〉〈A[β,β]

ij 〉

σ[A[α,α]ij ]σ[A

[β,β]ij ]

, (27)

R[α,β]w =

〈W [α,α]ij W

[β,β]ij 〉 − 〈W [α,α]

ij 〉〈W [β,β]ij 〉

σ[W[α,α]ij ]σ[W

[β,β]ij ]

, (28)

where in general, 〈X〉 and σ[X] are the notations for the mean and standard deviation of X.Note that while the average connectivity and intensity are based on the interlayer linkages,

the level of overlaps and the degree similarity between two layers α and β depend on theintralayer linkages.

2.4 Multilayer communicability and its applications to community detection

2.4.1 Multilayer communicability

The communicability between every pair of nodes quantifies the number of all possible connec-tion paths between them (Estrada et al., 2012). It should be emphasized that the basic networkmetrics such as degrees or strengths can only capture the second order structural interdepen-dencies (from all other nodes to a node or from a node to all other nodes). Even the clusteringcoefficients, which are often used to analyse the clustering behaviours in complex networks (e.g.

9

see Luu et al. (2017) and the literature therein), can explain the third order of the structuralcorrelations among three nodes alone. In contrast, the communicability is able to catch richerinformation of the direct as well as indirect pathways associated with different orders of theinterconnectednesses between nodes.

In the context of monoplex networks, various measures based on communicability have beenproposed to analyse different properties of interactions among nodes, from microscopic to meso-scopic and macroscopic levels.4 The communicability among nodes has been shown as one ofthe main approaches to detect the community structure in complex networks (e.g. see Estradaand Hatano, 2008; Bartesaghi et al., 2020).

As a natural extension, the multilayer communicability indicates the number of paths throughboth possible intralayer and interlayer links that connect a given node in a particular layer tothe other nodes of the multilayer architecture. In the context of global input-output network,the communicability indicates the number of different upstream and downstream propagationchannels between sectors and between countries, via both direct and indirect pathways. In fact,it can be somewhat related to the concept of the average propagation length often used to mea-sure the economic distances between industries in the literature related to input-output analysis(e.g. Dietzenbacher et al., 2005; Miller and Blair, 2009).

Communicability matrix:

In the binary version, given a supra adjacency matrix Asupra, the communicability matrixof the multilayer network is

Gbin = I +(Asupra)1

1!+

(Asupra)2

2!+ ...+

(Asupra)k

k!+ ... = exp(Asupra). (29)

Note that Gbin can be further expressed in the form of a supra matrix as

Gbin = exp(Asupra) =

G[1,1] G[1,2] G[1,3] ... G[1,L]

G[2,1] G[2,2] G[2,3] ... G[2,L]

G[3,1] G[3,2] G[3,3] ... G[3,L]

... ... ... ... ...

G[L,1] G[L,2] G[L,3] ... G[L,L]

, (30)

where in general G[α,β] (size NxN) is the matrix containing the communicability betweenpairs of nodes belonging to layer α and layer β (Bianconi, 2018). As a special case, when α = β,G[α,β] represents communicability among nodes within layer α. However, it is worth to mentionthat G[α,α] may differ from exp(A[α,α]) if there exist couplings among replicas and/or interlayerconnections.

Communicability centralities:

In the directed version, for every node i in a layer α, it is necessary to distinguish theincoming paths from all nodes in a layer β to i and the outgoing paths from i to all nodes in

4See Estrada et al. (2012) for further details.

10

a layer β. Hence, we define the receive (communicability) centrality rc[α←β]i and the broadcast

(communicability) centrality bc[α→β]i in the binary version of the network as

rc[α←β]i =

N∑j=1

G[β,α]ji , (31)

and

bc[α→β]i =

N∑j=1

G[α,β]ij . (32)

The total communicability centralities of node i in layer α from/to all nodes in all L layersare then given by

rc[α, total]i =

L∑β=1

rc[α,β]i , (33)

and

bc[α, total]i =

L∑β=1

bc[α,β]i . (34)

Communicability in the weighted version:

It is straightforward to extend the communicability matrix and centralities to their weighedcounterparts. For the weighted version of the network, following Crofts and Higham (2009) and

Estrada (2011) we first normalize the supra matrix W supra by dividing each weight w[α,β]ij by the

product√sα,toti,in

√sβ,totj,out:

W supra = S− 1

2in W supraS

− 12

out , (35)

where Sin is the (supra) diagonal matrix whose diagonal elements are all in-strengths stotalin

and Sout is the (supra) diagonal matrix whose diagonal elements are all out-strengths stotalout

obtained from the supra weighted matrix W supra. Such normalization helps to avoid the exces-sive influence of links with higher weights in the network. In the next step, we can define thecommunicability matrix Gw for the weighted version as

Gw = I +W supra

1!+

(W supra)2

2!+ ...+

(W supra)k

k!+ ... = exp(W supra). (36)

Based on the weighted communicability matrixGw, one can also easily derive the weighted re-ceive (communicability) centralities and broadcast (communicability) centralities, both for layer-layer communicability centralities and for the total ones, similar to the binary counterpartsdefined in Eqs. (31) to (34).

11

2.4.2 Applications to community detection

In following, we will explain how to apply communicability centralities to identify communitiesin a multilayer network. We start by introducing on the network a suitable distance basedon the idea of communicability. As expressed by the definition, the communicability betweentwo nodes is a weighted sum of the number of all walks connecting the pair (see Estrada et al.(2012); Bartesaghi et al. (2020)). Indeed, the fact that two nodes can be connected by means ofall possible walks, and not only by shortest paths, is implicit in the very idea of communicability.

In general, the communicability-based distance ξ[α,β]ij between node i in layer α and node j

in layer β is defined as

ξ[α,β]ij = G

[α,α]ii − 2G

[α,β]ij +G

[β,β]jj , (37)

where G is computed as in Eq. 29 in the binary version and as in Eq. 36 in the weighted one.Let us also emphasize that we are assigning the same meaning to distances between nodes inthe same layer, distances between versions of the same node in different layers, and distancesbetween different nodes in different layers.

Following Chang et al. (2016); Bartesaghi et al. (2020) we can then compute the cohesion

function γ[α,β]ij as

γ[α,β]ij = ξ

[α,α]ii + ξ

[β,β]jj − ξ[α,β]ij − ξ, (38)

where ξ[α,α]ii is the average communicability distance of node i belonging to layer α from all the

other nodes in all layers and ξ is the average communicability distance over the whole network.

The cohesion function γ[α,β]ij 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 α and j in layer β in the same community of a given partition. Weassume then to maximize the global cohesion function Q:

Q =∑

[i,α],[j,β]

γ[α,β]ij x[i,α],[j,β], (39)

where x[i,α],[j,β] is the Kronecker delta function, which is equal to 1 if two nodes (i, j) are inthe same cluster and 0 otherwise. Hence, we obtain the best possible partition via maximizationof the function Q defined in (39).

To conclude this section we summarize the steps of the previous methodology to detectcommunities in the multilayer context:

1. let G be the original multilayer (in general, directed and weighted) network that has Llayers and N nodes per layer, and let W supra be the corresponding supra weighted matrix;

2. to apply the communicability-based method, we first build the undirected weighted net-work associated with the symmetric supra weighted matrix defined asW symm = 1

2(W supra+(W supra)T );

3. we then build the undirected weighted network associated with the normalised weightedsupra-adjacency matrix W symm = S−

12W symmS−

12 , where S is the diagonal matrix of the

strengths of each node in each level in the symmetrized network;

12

4. we compute the distances according to formula (37) and define the threshold interval[ξmin, ξmax], where ξmin and ξmax represent the minimum and the maximum communica-bility distances between couples of nodes, respectively and set ξh = ξmin with the initialindex h = 0;

5. we define a NL×NL matrix M suprah whose entries are 1 if ξ

[α,β]ij ≤ ξh and 0 otherwise and

build the undirected unweighted network from the binary supra-adjacency matrix M suprah ;

6. we select the partition Ph given by the components of the network associated to M suprah

and compute the partition quality function Q in equation (39)

7. we set the number of iterations r, compute the step increment k = ξmax−ξminr , set ξh =

ξh−1 + k and h = h + 1 and repeat steps 5-6 until ξh = ξmax; and select the optimalpartition P ?h as the partition Ph that provides the optimal Q.

We stress few key points of the presented methodology. We aim at clustering nodes andlayers (i, α) where i are countries and α are sectors on the basis of a specific communicabilitydistance. Varying the threshold we can disentangle the role of very tight relationships betweencouples of nodes. Of course, if we reduce the threshold distance, a great number of isolatednodes may appear. However, they are typically less significant pairs of country-sector whosetrade volume is very low and whose commercial links are few. Hence, they play a marginal rolein the global input-output network and they do not affect in a substantial way the structure ofthe network in terms of relevant communities. This is the reason why in the following we willfocus on the detection of the main and large communities.

3 Results on network properties and community detection

The first part of this section shall begin with an overview of the original input-output relationsamong sectors from different countries. It summarizes the basic statistics and properties ofindustries that indicate the trade diversification or concentration in the original input-outputtable, based on the information of the adjacency matrix Asupra and the weighted matrix W supra.In the next step, we restructure and consider the original table from the viewpoint of a multi-layer network in which each layer is associated with an industry while each node represents acountry member. We then analyse the interrelations between different layers to have an over-all assessment of interdependencies and similarities spanning across sector-based slices in themultilayer version of the global trade network in input-output. The rest of this section is thendevoted to the analysis of the community structure. In particular, we apply the extended versionof communicability approach described in section (2.4) to extract different communities in themultilayer architecture of the symmetrized network and analyse their properties.

Note that for the sake of illustration, for almost all of the contents in this section, we focus onthe analysis of the data in 2014, since it provides the latest available information on the globalnetwork that we can have from the WIOD database. However, in the Appendix 5.2, we alsosummarize the main community detection results for the other selected years over the periodfrom 2000 to 2014.

13

3.1 Basic network properties of the global input-output network

Trade diversity and concentration from the total degrees and strengths of nodes:

Figure 1 shows the distributions of the total degrees and strengths of the industries fromdifferent countries based on the original input-output table. It is interesting to observe that fromthe binary version, almost all of nodes have degrees larger than 1500 and a negative skewness, i.e.a longer tail on the left side of the distribution, is observed. This implies that industries indeedhave a relatively diversified list of trading partners from different countries. This is consistentwith the fact that the binary version of the network is very dense, for both intra-country andinter-country links.

In contrast, when the intensity of trading relationships is taken into account in the weightedversion, the strengths exhibit a right-skewed distribution, where only few nodes have the strengthsthat are much larger than those of the rest. To further investigate the overall heterogeneity levelof trading intensity across partners for each sector, we compute the Herfindahl–Hirschman in-dices (e.g. see Kvalseth (2018); Hirschman (1964)) associated with total in- and out-strengths,which are, respectively, defined as

h[α]i,in =

∑j,β

(w[α,β]ji /s

[α, total]i,in )2, (40)

andh[α]i,out =

∑j,β

(w[α,β]ij /s

[α, total]i,in )2. (41)

Generally speaking, the Herfindahl–Hirschman indices defined in (40) and (41) will range from1/Nsupra = 1/(NL) to 1, and a larger value indicates a higher level of the concentration of theinputs (purchased from fewer seller-sectors) or outputs (distributed across limited buyer-sectors).

In our present work, we find that while almost all industries in different countries havea moderate or negligible concentration of trading across partners, some other industries dointensively trade with few input suppliers or output customers in the global production network(see Figure (2)).

14

(a) (b)

(c) (d)

Figure 1: Distributions of total in- and out-degrees and distributions of total in- and out-strengths. Thered vertical line indicates the mean value.

15

(a) (b)

Figure 2: Distributions of the Herfindahl–Hirschman indices of total in- and out-strengths. The verticalgreen line correspond to the minimum 1/Nsupra. The vertical red line represents the critical value 0.25,which indicates a high level of concentration if h > 0.25.

We report the corresponding Pearson correlation coefficients in Figure (3) to illustrate theoverall correlations among different types of node degrees, strengths and the total communica-bility centralities. First, we can see that the correlation between in-degrees and out-degrees isrelatively high. This is indeed not an unexpected result since many nodes have a high level ofin- and out-degrees (see panels (a), (b) in Figure 1) and the binary trade links are very dense.However, the results also show that the correlation levels between degrees and strengths arerelatively weaker, which again confirms the distinguished characteristics of the two versions ofthe network. Furthermore, given the high degree of heterogeneity in the node strengths, it isinteresting to observe that interlayer in-strengths and interlayer out-strengths are synchronizedtogether, to a certain extent. This implies that, if a sector in a country exports more its outputs,it also purchases more inputs from sectors in other countries. For a more comprehensive assess-ment, we also add the total receive and broadcast communicability centralities in the weightedversion of the network and compute the correlations between them and the various degrees andstrengths. As expected, since these two additional measures are able to capture all possible trav-elling paths between nodes, they are less synchronized with the second order network metricslike node degrees and strengths.

16

(a)

Figure 3: Correlations among node degrees, strengths and the total (weighted) communicability central-

ities. Total in-degrees (total in-deg) and total in-strengths (total in-str): {k[α, total]i,in }, {s[α, total]i,in }. Total

out-degrees (total out-deg) and total out-strengths (total out-str): {k[α, total]i,out }, {s[α, total]i,out }. Interlayer

in-strengths (inter in-str) and interlayer out-strengths (inter out-str): {s[α, inter]i,in }, {s[α, inter]i,out }. The totalcommunicability centralities of a node from/to all nodes in all layers (in the weighted version of thenetwork): receive comm. and broadcast comm.

Interrelations between layers in a multilayer architecture:

We shall now reorganize the original input-output table into an equivalent network with amultilayer architecture. Nodes now represent countries and layers correspond to sectors. In sucha consideration, an intralayer link represents a trade link from one country to another countryin a particular sector. An interlayer link, in contrast, implies a trade link from a sector in onecountry to another sector in the same or another country.

To examine how strongly a layer interacts with another, we measure the average connectivity

17

and intensity between them. Figure (4) shows distinct behaviours in the two facets of themultilayer network: on the one hand, in the binary version, the density of the connectionsbetween many layers is very high; one the other hand, once the link weights (i.e. the tradedamounts) are considered, average intensities are highly heterogeneous. In particular, as we cansee in the panels (c) and (d) of Figure (4), few pairs of layers interact more intensively, suggestingthat these industry-based layers trade more between themselves than with the others.

(a) (b)

(c) (d)

Figure 4: Average connectivities and average intensities between layers. The measures for these averagesare explained in Eqs. (22) and (24). The vertical red line in each histogram shows the mean value. Notethat here layers represent 56 sectors, while nodes in each layer are 44 countries. The list of sector namesis shown in Table (4) in the Appendix.

18

Layer overlaps and correlations:

Comparing across layers, we want to examine whether layers exhibit similar internal struc-tures. In particular, if trade links are strong in a particular layer, do they also exist and tendto form intensive relations in the other layers? To answer this question, we measure the overalloverlap coefficient and Pearson correlation coefficient for every pair of layers (Gemmetto andGarlaschelli, 2014; Gemmetto et al., 2016; Luu and Lux, 2019). In general, when comparinglayer vs. layer, we find that some clusters of layers exhibit a relatively higher level of similarity(see Figures (5) and (6) as well as the dendrograms shown in Figures (13) and (14) in the Ap-pendix). Interestingly, this property for the network of trade in input-output is somewhat inlinewith what is observed in the multiplex trade networks in which each layer is associated with afinal good/service (see Gemmetto and Garlaschelli, 2014; Gemmetto et al., 2016).

19

(a) (b)

(c) (d)

Figure 5: Overlaps and correlations between layers in the binary version . The methods used to computethe overlap and correlation coefficients are explained in (2.3). Layers represent 56 sectors, while nodes ineach layer are 44 countries. The list of sector names is shown in Table (4) in the Appendix.

20

(a) (b)

(c) (d)

Figure 6: Overlaps and correlations between layers in the weighted version. The methods used to computethe overlap and correlation coefficients are explained in (2.3). Layers represent 56 sectors, and nodes ineach layer are 44 countries. The list of sector names is shown in Table (4) in the Appendix.

All in all, our analysis of the basic network properties suggests that the global trade networkin input-output, especially in its weighted version, has a heterogeneous structure of interactionsbetween sectors and countries. Furthermore, once various types of trade linkages are all con-sidered in a multilayer structure, we find that the interactions between layers vary across pairsof layers. On top of that, many layers have strongly dissimilar internal structures while fewothers are somewhat more overlapped or correlated. These observations naturally lead to the

21

following related question from a topological perspective at a broader scale of the global tradesystem in input-output: Are there clusters of countries whose members interact stronger amongthemselves simultaneously in different layers associated with different industries? To answer thisquestion, we shall now conduct an analysis of a multilayer community structure in the globalinput-output network.

3.2 Multilayer community structure in the global input-output network

3.2.1 Community structure based on multilayer communicability

In this section, we report main results obtained by applying the procedure based on multilayercommunicability defined in Section 2.4.1. The proposed approach has been applied directly to themultilayer network, which has been preliminarily transformed in an undirected one substitutingeach pair of bilateral directed links with one undirected link, with weight equal to the averageweight.Starting from a supradjacency matrix with 2464 nodes, given by 44 countries and 56 sectors,the methodology provides 117 communities (except some isolated nodes). Notice that clusteredgroups of nodes are located both intralayers and interlayers. This means that we can findcountries members of the same community in a specific sector, as well as countries sharing thesame community in more sectors. To provide a representation of the results, we report in Figure7 all members of the top twelve communities in terms of number of nodes. In particular, twolarge international communities are detected, with 443 and 318 nodes, respectively. The otherten communities have instead a smaller size, being between 30 and 50 nodes.

Looking at the clusters obtained, some noticeable elements can be identified. First of all, weobserve 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 inthe North American Free Trade Agreement (Canada, Mexico and USA) belong to the secondcommunity.Furthermore, it is interesting to explore the clustering behaviour of European countries thatare “in the middle” between these two relevant communities. Indeed, we find that almost all ofthem 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 report in Table 1for each country the number of sectors that belong to these two communities. We also computethe Gini heterogeneity index5 to give an indication of the dispersion of sectors for each countryalong the communities. A higher value of this index means that countries in that sector aremore spread out between different communities.As regard to European countries, it could be noticed how some of them (Bulgaria, Germany,Greece, Norway, Portugal) show a behaviour similar to the Australian-Asian countries, beingthese countries concentrated in the first community for most trade of sectors. However, lookingat the sectors, this participation occurs with a different degree of heterogeneity, as indicatedby the Gini index. For instance, Germany has 37 sectors in community 1 and only 6 sectors

5In particular, since we deal with nominal variables, we compute the index Hj to quantify the heterogeneityof a country j as Hj = 1 −

∑ci=1 p

2i,j where c is the total number of clusters detected by the procedure explained

in Section (2.4.2) and pi,j is the proportion of sectors of country j in the cluster i such that∑c

i=1 pi,j = 1. Thisformula is also known in the literature as the Gini-Simpson index.

22

Figure 7: Membership (in terms of countries and sectors) of larger communities with at least 30 nodes.The communities have been sorted according to the number of nodes. Empty cells represent nodes thatbelong to smaller communities (lower than 30 nodes) or isolated nodes. The lists of sector names andcountry names are shown in Tables (4) and (5) in the Appendix.

23

in community 2. In contrast, Czech Republic, France, Great Britain, Ireland, Poland, Russiaare instead more concentrated in community 2. The remaining European countries have onlya limited number of sectors belonging to the top two communities, while the other sectors areconcentrated in specific groups.Moreover, it is worth pointing out the presence of countries with a lower level of heterogeneityin the community location of their sectors, characterized by a concentration in the same com-munity for almost all sectors (except isolated ones). Indeed, we observe that besides the toplargest international clusters, the next large communities are formed by different sectors of thesame country. These are, for example, the cases of Spain (community 3), Italy (community 4),Brasil (community 5), Finland (community 6), Croatia (community 7), Turkey (community 8),Indonesia (community 9).

Now, if we look at each single sector-based layer, we can further explore how countries inthe same layer are classified in communities (see Table 2). Overall, we find that the analysis forsectors shows a higher level of heterogeneity (as shown by the Gini index). This implies that, inthe same layer, countries are on average split in several different communities. However, exceptfor sectors “T” and “U” where countries act almost always as isolated communities, the highernumber of countries in each sector belongs to one of the two top communities. It is also interest-ing to note that sectors with a lower heterogeneity (as “Mining (B)”and “Manufacture of refinedpetroleum products (C19)”) are characterized by a concentration of countries in Community 1.

To explore the similarity between sectors, we computed the Jaccard similarity index andwe reported only coefficients higher than 0.7 in Figure 8. In this way, we emphasize couple ofsectors characterized by a similar classification of countries in communities. We observe how thehighest similarity has been observed between the sectors “Crop and animal production, huntingand related activities (A01)”and “Manifacture of food products, beverages and tobacco products(C10-C12)”. We have indeed that all countries (except Ireland) have been classified in the sameway in these two sectors. This result can be easily explained by the fact that activities involvedin these two sectors are closely related. Another interesting case is represented by the sectors Fand L68 (Construction and Real Estate) that show 41 countries analogously classified (exceptMalta, Mexico and Ireland). The same behaviour is also observed for the other two couplesG47-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 versionof the global input-output network, with 44 nodes and 56 layers. In the next step, we willcompare previous results with the community structure detected from the mono-layer aggregateversion, where each node is a country and a link now considers the total trade between acouple of countries. For consistency, we apply the communicability-based approach proposedin Bartesaghi et al. (2020) to the mono-layer aggregate network. We obtain seven communities(except five isolated nodes) and report their composition in Table 3. It is noteworthy that two bigcommunities are detected also in this case but the composition is mainly driven by geographicalpatterns. This is in line with the results already detected in the literature (see Barigozzi et al.(2010), Bartesaghi et al. (2020)). It is interesting to note that the first community includesalmost all European Countries. Only some well-known couples are clustered alone (see forinstance 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.

24

Countries Number of Sectors Number of Sectors Number of Sectors Community Giniin Community 1 in Community 2 Isolated with highest heterogeneity

number of sectors Index

AUS 33 0 18 1 64.29%AUT 2 1 20 13 71.88%BEL 3 0 18 22 82.84%BGR 41 0 15 1 45.92%BRA 0 0 13 5 45.85%CAN 0 35 16 2 60.01%CHE 2 0 20 14 72.00%CHN 41 1 10 1 45.79%CYP 0 2 23 16 80.36%CZE 0 30 17 2 69.32%DEU 37 6 11 1 54.72%DNK 5 0 14 10 61.61%ESP 1 0 10 3 40.56%EST 0 0 20 18 76.85%FIN 2 0 16 6 53.32%FRA 1 36 10 2 57.65%GBR 8 33 12 2 62.56%GRC 38 0 18 1 53.38%HRV 0 0 16 7 53.32%HUN 0 1 27 24 86.86%IDN 0 0 15 9 57.78%IND 36 0 20 1 58.04%IRL 1 12 29 2 93.05%ITA 2 0 11 4 40.56%JPN 39 0 9 1 49.94%KOR 46 0 8 1 32.14%LTU 0 0 23 15 73.98%LUX 0 2 44 39 97.32%LVA 0 0 18 17 75.51%MEX 0 33 17 2 64.09%MLT 0 0 30 25 94.07%NLD 11 2 21 23 79.91%NOR 28 0 19 1 73.47%POL 0 45 11 2 35.08%PRT 30 0 14 1 69.77%ROU 0 0 16 19 77.74%RUS 0 28 26 2 74.11%SVK 0 4 19 20 79.59%SVN 0 0 17 12 66.26%SWE 3 0 10 11 63.58%TUR 0 0 18 8 53.38%TWN 2 0 14 21 78.83%USA 0 41 12 2 45.85%ROW 31 6 5 1 66.71%

Table 1: In this Table, we report for each country the number of sectors that belong to the two largestcommunities (1 and 2, respectively), the number of sector that does not belong to any community (i.e.isolated), the indication of the community with the highest number of sectors of that country and theGini heterogeneity index to measure the dispersion of sectors along the communities in the same country.

25

Sectors Number of Countries Number of Countries Number of Countries Community Giniin Community 1 in Community 2 Isolated with highest heterogeneity

number of countries IndexA01 10 5 0 1 92.05%A02 8 5 10 1 93.80%A03 4 1 32 1 97.00%B 21 6 7 1 74.48%C10-C12 10 5 1 1 92.05%C13-C15 4 0 35 1 97.11%C16 9 6 4 1 92.46%C17 6 3 19 1 95.87%C18 8 4 6 1 94.21%C19 17 6 8 1 82.02%C20 11 5 16 1 91.01%C21 9 4 18 1 93.39%C22 6 7 21 2 94.01%C23 11 7 4 1 89.88%C24 8 7 13 1 92.36%C25 9 8 6 1 91.12%C26 6 1 30 1 95.97%C27 5 2 32 1 96.59%C28 5 7 19 2 94.52%C29 6 11 14 2 90.08%C30 4 7 27 2 94.94%C31-C32 6 4 19 1 95.56%C33 2 2 27 1 97.52%D35 13 6 6 1 88.12%E36 4 1 24 1 97.11%E37-E39 2 3 22 2 97.21%F 11 8 0 1 89.15%G45 4 6 18 2 95.45%G46 12 8 1 1 88.02%G47 11 9 0 1 88.33%H49 11 8 2 1 89.15%H50 8 3 23 1 94.42%H51 7 7 11 1 93.39%H52 11 8 2 1 89.15%H53 6 6 21 1 94.63%I 11 6 2 1 90.50%J58 6 6 7 1 94.63%J59-J60 7 5 11 1 94.52%J61 10 7 9 1 90.91%J62-J63 9 8 7 1 91.12%K64 11 10 0 1 87.29%K65 9 10 7 2 89.36%K66 6 9 12 2 92.46%L68 11 8 1 1 89.15%M69-M70 10 7 10 1 90.91%M71 8 7 15 1 92.67%M72 2 4 30 2 97.00%M73 6 8 13 2 93.29%M74-M75 6 2 23 1 96.07%N 8 9 2 2 91.01%O84 10 8 2 1 90.19%P85 9 7 8 1 91.84%Q 8 4 5 1 94.21%R-S 10 7 4 1 90.91%T 1 0 40 1,7,10,16 97.73%U 0 0 44 97.73%

Table 2: For each sector, the Table reports the number of countries that belong to the two largestcommunities (1 and 2, respectively), the number of countries that does not belong to any community(i.e. isolated), the indication of the community with the highest number of countries of that sector andthe Gini heterogeneity index to measure the dispersion of countries along the communities in the samesector.

26

Figure 8: Matrix of Jaccard similarity coefficients between sectors. Only coefficients higher or equal than0.7 have been reported (it correspond to 90% quantile approximatively).

27

Community Country members1 AUT, BEL, BGR, CZE, DEU, DNK, FRA, GRC, HRV, HUN, ITA, LUX, NLD, NOR, POL,

ROU, RUS, SVK, SVN, SWE, TUR2 BRA, CAN, CHN, IND, JPN, MEX, USA, ROW3 AUS, IDN4 ESP, PRT5 EST, FIN6 GBR, IRL7 LTU, LTVIsolated CHE, CYP, KOR, MLT, TWN

Table 3: The composition of the communities computed on the aggregate mono-layer network in 2014.

These results allow to emphasize how the multilayer analysis, that takes into considerationdistinction among 56 industry-based layers, is able to catch more peculiar relationships betweencountries that cannot be detected at an aggregate level.

28

3.3 Further internal properties of different detected communities and theirevolution over years

The detection of different communities (shown in Figure (7) and Table (2)) leads to anotherimportant question, i.e. how do the internal topological properties in each community look like?To examine this, first, we select countries and sectors contributions to each identified communityand build a sub-network from the original one. More specifically, in such a sub-graph, only thelinks among its members are maintained and analysed, while those externally built with nodesin the other communities are discarded.

For the illustration purpose, we focus on the community 1 and the community 2, since thesetwo largest clusters consist of different sectors from various countries while almost all of thenext large ones are indeed mainly formed by those from the same country. Figure (9) showsthe ranking of the top sectors in each community based on the (total) node strengths definedin (2.2), using the data in 2014. The results again confirm the dominance of China in the firstlargest community and that of USA in the second community at the industry levels as many ofthe influential sectors in the communities 1 and 2 are actually from these two countries. Lookingat the industry codes of the most influential sectors, we also observe an interesting difference ineconomic structure between the two communities and hence between the two countries. Whileinfluential sectors from China are mostly manufacturing-related industries, those coming fromthe US are more dominated by real-estate, finance, technology or service-related sectors.

29

(a) Top industries in community 1, 2014 (b) Top industries in community 2, 2014

Figure 9: Ranking of top industries in two important international communities in 2014, based on the(total) node strengths.

Furthermore, to see how the structure of the largest international communities evolve overtime, in the next step, we select the three years 2000, 2004, 2009 in the subsample period toanalyse.6 Technically, we reapply the method based on the multilayer communicability distancedescribed in section (2.4) to extract different clusters for each of these selected years. Remark-ably, we find evidence that the clustering behaviours of trades in input-output among industriesand countries do restructure over the period from 2000 to 2014. In general, not only the commu-nity composition changes, but the centrality rankings of the members inside each community isalso reordered, with the diminishing of some industries and the emergence of new “key players”.

As shown in Figures (10) and (11), in 2000, the first largest community is composed bysectors from the US, Japan and interestingly several ones from China. In contrast, the secondlargest cluster mainly consists of industries from the former Eastern Bloc’s countries, led bythose from Russia and Poland. However, in 2004, while the composition of the first communityremains almost the same, the internal structure of the second community changes: sectors fromthe former Eastern Bloc’s countries play a less important role, but those from Belgium andFrance become more dominant. The internal structure of the largest international communitiesstill continues to evolve in the subsequent years. For example, Figure (12) demonstrates that,

6For the sake of illustration, we only select these three years before 2014 to report their results. However,results for other years are available upon request to the authors.

30

as for the data in 2009, a number of influential industries from China no longer stay in thefirst community with those from US but emerge as the “key players” in the second community.Subsequentially, they form the largest international community together with some sectors fromthe other Asian countries such as India, Japan, South Korea.

Altogether, the aforementioned observations may be the first signals that the global tradein input-output system has been moving from a former US-Russia bipolar system to anotherbipolar (or multipolar) one, with the rising role of sectors from China in the recent years.

(a) Top industries in community 1, 2000 (b) Top industries in community 2, 2000

Figure 10: Ranking of top industries in two important international communities in 2000, based on the(total) node strengths.

31

(a) Top industries in community 1, 2004 (b) Top industries in community 2, 2004

Figure 11: Ranking of top industries in two important international communities in 2004, based on the(total) node strengths.

(a) Top industries in community 1, 2009 (b) Top industries in community 2, 2009

Figure 12: Ranking of top industries in two important international communities in 2009, based on the(total) node strengths.

32

4 Conclusions

This work analyses the global trade network in input-output, using the recently released WIODdatabase. 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 industriesaround 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 andfind that although the list of trading partners is generally broad, some sectors trade moreintensively with a number of other sectors in different countries. We then view countries asnodes 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 strengthsare varied across pairs of layers and that few couples of layers tend to be more overlappedand/or more strongly interacting with each other. Such first insights motivate us to examinethe multilayer architecture at a broader scale rather the node-node or layer-layer interrelationsalone.

As investigated in our work, at the mesoscale level, there exist several large internationalcommunities in which some countries trade their inputs and outputs more intensively in somespecific industry-based layers. In such a network structure, the world somewhat seems to havebipolar or multipolar trade system with different clusters that are more internally cohesive.However, interestingly, we also observe that the clustering behaviours of trades in input-outputamong industries and countries restructure over the period from 2000 to 2014. In general, notonly the internal composition changes, but the centrality rankings of the members inside eachcommunity also reorder, with the diminishing role of industries from some countries and thegrowing 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 ofindustries mostly from the former Eastern Bloc’s countries (e.g. Russia and Poland). As in thedata 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, SouthKorea) and Australia are clustered together. On the other hand, those from countries involvedin the North American Free Trade Agreement (Canada, Mexico and USA) belongs to the nextlargest community.

Our present work opens several directions for future research. First, the findings on theemergence of several international communities and their structural changes over time need tobe 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 theglobal input-output network, in our future work, we plan to extend our analysis to study andquantify the other important network measures and properties such as the multilayer clusteringcoefficients and the multilayer centrality (e.g. see De Domenico et al., 2013). Last but not least,we believe that incorporating additional layers representing other economic relations such asfinancial links, trade in final goods and services on top of the input-output interrelations amongcountries would be able to give a more comprehensive and richer analysis of the multilayerarchitecture of the world-wide economic network.

33

References

Acemoglu, D., Akcigit, U., and Kerr, W. (2016). Networks and the macroeconomy: An empiricalexploration. NBER Macroeconomics Annual, 30(1):273–335.

Acemoglu, D., Carvalho, V. M., Ozdaglar, A., and Tahbaz-Salehi, A. (2012). The networkorigins of aggregate fluctuations. Econometrica, 80(5):1977–2016.

Acemoglu, D., Ozdaglar, A., and Tahbaz-Salehi, A. (2017). Microeconomic origins of macroe-conomic tail risks. American Economic Review, 107(1):54–108.

Alves, L., Mangioni, G., Cingolani, I., Rodrigues, F., Panzarasa, P., and Moreno, Y. (2018). Thenested structural organization of the worldwide trade multi-layer network. Scientific Reports.

Atalay, E. (2017). How important are sectoral shocks? American Economic Journal: Macroe-conomics, 9(4):254–80.

Bargigli, L., di Iasio, G., Infante, L., Lillo, F., and Pierobon, F. (2015). The multiplex structureof interbank networks. Quantitative Finance, 15(4):673–691.

Barigozzi, M., Fagiolo, G., and Garlaschelli, D. (2010). Multinetwork of international trade: Acommodity-specific analysis. Phys. Rev. E, 81:046104.

Bartesaghi, P., Clemente, G. P., and Grassi, R. (2020). Communicability in the world tradenetwork – a new perspective for community detection. Journal of Economic Interaction andCoordination.

Battiston, S., Caldarelli, G., and Garas, A. (2018). Multiplex and multilevel networks.

Bianconi, G. (2018). Multilayer networks: Structure and function. Oxford University Press.

Boehm, C. E., Flaaen, A., and Pandalai-Nayar, N. (2019). Input Linkages and the Transmissionof Shocks: Firm-Level Evidence from the 2011 Tohoku Earthquake. The Review of Economicsand Statistics, 101(1):60–75.

Bostandzic, D. and Weiß, G. N. (2018). Why do some banks contribute more to global systemicrisk? Journal of Financial Intermediation, 35:17 – 40.

Burstein, A., Kurz, C., and Tesar, L. (2008). Trade, production sharing, and the internationaltransmission of business cycles. Journal of Monetary Economics, 55(4):775–795.

Carvalho, V. M. (2014). From micro to macro via production networks. Journal of EconomicPerspectives, 28(4):23–48.

Carvalho, V. M., Nirei, M., Saito, Y., and Tahbaz-Salehi, A. (2016). Supply chain disruptions:Evidence from the great east japan earthquake. Working Paper No. 2017-01, Becker FriedmanInstitute for Research in Economics.

Cerina, F., Zhu, Z., Chessa, A., and Riccaboni, M. (2015). World input-output network. PLOSONE, 10(7):1–21.

34

Cetorelli, N. and Goldberg, L. (2011). Global banks and international shock transmission:Evidence from the crisis. IMF Economic Review, 59(1):41–76.

Chang, C., Liao, W., Chen, Y., and Liou, L. (2016). A mathematical theory for clustering inmetric spaces. IEEE Transactions on Network Science and Engineering, 3(01):2–16.

Crofts, J. and Higham, D. (2009). A weighted communicability measure applied to complexbrain networks. Journal of the Royal Society, Interface / the Royal Society, 6:411–4.

De Domenico, M., Sole-Ribalta, A., Cozzo, E., Kivela, M., Moreno, Y., Porter, M. A., Gomez,S., and Arenas, A. (2013). Mathematical formulation of multilayer networks. Phys. Rev. X,3:041022.

del Rıo-Chanona, R. M., Grujic, J., and Jeldtoft Jensen, H. (2017). Trends of the world inputand output network of global trade. PLOS ONE, 12(1):1–14.

Di Giovanni, J. and Levchenko, A. A. (2010). Putting the parts together: Trade, vertical linkages,and business cycle comovement. American Economic Journal: Macroeconomics, 2:95–124.

Di Giovanni, J., Levchenko, A. A., and Mejean, I. (2018). The micro origins of internationalbusiness-cycle comovement. American Economic Review, 108(1):82–108.

Dietzenbacher, E., Romero Luna, I., and Bosma, N. S. (2005). Using Average PropagationLengths to Identify Production Chains in the Andalusian Economy/Empleando LongitudesMedias de Propagacion para identificar Cadenas Productivas en la Economıa Andaluza. Es-tudios de Economia Aplicada, 23:405–422.

Eppinger, P., Felbermayr, G. J., Krebs, O., and Kukharskyy, B. (2020). Covid-19 ShockingGlobal Value Chains. CESifo Working Paper Series 8572, CESifo.

Estrada, E. (2011). The Structure of Complex Networks: Theory and Applications. OxfordUniversity Press.

Estrada, E. and Hatano, N. (2008). Communicability in complex networks. Phys. Rev. E,77:036111.

Estrada, E., Hatano, N., and Benzi, M. (2012). The physics of communicability in complexnetworks. Physics Reports, 514(3):89 – 119.

Fortunato, S. (2010). Community detection in graphs. Physics Reports, 486(3):75 – 174.

Gemmetto, V. and Garlaschelli, D. (2014). Multiplexity versus correlation: The role of localconstraints in real multiplexes. Scientific Reports, 5.

Gemmetto, V., Squartini, T., Picciolo, F., Ruzzenenti, F., and Garlaschelli, D. (2016). Multi-plexity and multireciprocity in directed multiplexes. Phys. Rev. E, 94:042316.

Hale, G., Kapan, T., and Minoiu, C. (2019). Shock Transmission Through Cross-Border BankLending: Credit and Real Effects. The Review of Financial Studies. hhz147.

35

Hirschman, A. O. (1964). The paternity of an index. The American Economic Review, 54(5):761–762.

Johnson, R. C. (2014). Trade in intermediate inputs and business cycle comovement. AmericanEconomic Journal: Macroeconomics, 6(4):39–83.

Kleimeier, S., Sander, H., and Heuchemer, S. (2013). Financial crises and cross-border banking:New evidence. Journal of International Money and Finance, 32(C):884–915.

Kok, C. and Montagna, M. (2013). Multi-layered Interbank Model for Assessing Systemic Risk.Kiel Working Papers 1873, Kiel Institute for the World Economy.

Kvalseth, T. O. (2018). Relationship between concentration ratio and herfindahl-hirschmanindex: A re-examination based on majorization theory. Heliyon, 4(10):e00846.

Lee, K.-M. and Goh, K.-I. (2016). Strength of weak layers in cascading failures on multiplexnetworks: Case of the international trade network. Scientific Reports, 6.

Leon, C., Berndsen, R., and Renneboog, L. (2014). Financial stability and interacting networksof financial institutions and market infrastructures. SSRN Electronic Journal.

Luu, D. T. and Lux, T. (2019). Multilayer overlaps and correlations in the bank-firm creditnetwork of spain. Quantitative Finance, 19(12):1953–1974.

Luu, D. T., Lux, T., and Yanovski, B. (2017). Structural correlations in the italian overnightmoney market: An analysis based on network configuration models. Entropy, 19(6).

Luu, D. T., Napoletano, M., Fagiolo, G., Roventini, A., and Sgrignoli, P. (2018a). Shock diffusionin the european production network:systemic importance and cascading failures. ISIGrowthWorking Paper 38/2018 July, ISIGrowth.

Luu, D. T., Napoletano, M., Fagiolo, G., Roventini, A., and Sgrignoli, P. (2018b). Uncovering thenetwork complexity in input-output linkages among sectors in european countries. ISIGrowthWorking Paper 17/2018 May, ISIGrowth.

Miller, R. E. and Blair, P. D. (2009). Input-Output Analysis: Foundations and Extensions.Cambridge University Press.

Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev.,45(2):167–256.

Park, C.-Y. and Shin, K. (2020). Contagion through national and regional exposures to foreignbanks during the global financial crisis. Journal of Financial Stability, 46:100721.

Poledna, S., Molina-Borboa, J. L., Martınez-Jaramillo, S., [van der Leij], M., and Thurner, S.(2015). The multi-layer network nature of systemic risk and its implications for the costs offinancial crises. Journal of Financial Stability, 20:70 – 81.

Shin, H. S. (2012). Global banking glut and loan risk premium. IMF Economic Review,60(2):155–192.

36

Timmer, M., Los, B., Stehrer, R., and de Vries, G. (2016). An anatomy of the global tradeslowdown based on the wiod 2016 release. GGDC Research Memorandum GD-162, GroningenGrowth and Development Centre, University of Groningen.

Timmer, M. P., Dietzenbacher, E., Los, B., Stehrer, R., and de Vries, G. J. (2015). An illustrateduser guide to the world input–output database: the case of global automotive production.Review of International Economics, 23(3):575–605.

37

5 Appendices

5.1 Lists of sectors and countries

In this part, we summarize the list of 56 sectors and the list of 44 countries and areas in theWIOD table (2016 release version).

no. sector name sector code

1 Crop and animal production, hunting and related service activities A012 Forestry and logging A023 Fishing and aquaculture A034 Mining and quarrying B5 Manufacture of food products, beverages and tobacco products C10-C126 Manufacture of textiles, wearing apparel and leather products C13-C157 Manufacture of wood and of products of wood and cork, except furniture; manufacture of articles of straw and plaiting materials C168 Manufacture of paper and paper products C179 Printing and reproduction of recorded media C1810 Manufacture of coke and refined petroleum products C1911 Manufacture of chemicals and chemical products C2012 Manufacture of basic pharmaceutical products and pharmaceutical preparations C2113 Manufacture of rubber and plastic products C2214 Manufacture of other non-metallic mineral products C2315 Manufacture of basic metals C2416 Manufacture of fabricated metal products, except machinery and equipment C2517 Manufacture of computer, electronic and optical products C2618 Manufacture of electrical equipment C2719 Manufacture of machinery and equipment n.e.c. C2820 Manufacture of motor vehicles, trailers and semi-trailers C2921 Manufacture of other transport equipment C3022 Manufacture of furniture; other manufacturing C31 C3223 Repair and installation of machinery and equipment C3324 Electricity, gas, steam and air conditioning supply D3525 Water collection, treatment and supply E3626 Sewerage; waste collection, treatment and disposal activities; materials recovery; remediation activities and other waste management services E37-E3927 Construction F28 Wholesale and retail trade and repair of motor vehicles and motorcycles G4529 Wholesale trade, except of motor vehicles and motorcycles G4630 Retail trade, except of motor vehicles and motorcycles G4731 Land transport and transport via pipelines H4932 Water transport H5033 Air transport H5134 Warehousing and support activities for transportation H5235 Postal and courier activities H5336 Accommodation and food service activities I37 Publishing activities J5838 Motion picture, video and television programme production, sound recording and music publishing activities; programming and broadcasting activities J59 J6039 Telecommunications J6140 Computer programming, consultancy and related activities; information service activities J62 J6341 Financial service activities, except insurance and pension funding K6442 Insurance, reinsurance and pension funding, except compulsory social security K6543 Activities auxiliary to financial services and insurance activities K6644 Real estate activities L6845 Legal and accounting activities; activities of head offices; management consultancy activities M69 M7046 Architectural and engineering activities; technical testing and analysis M7147 Scientific research and development M7248 Advertising and market research M7349 Other professional, scientific and technical activities; veterinary activities M74 M7550 Administrative and support service activities N51 Public administration and defence; compulsory social security O8452 Education P8553 Human health and social work activities Q54 Other service activities RS55 Activities of households as employers; undifferentiated goods- and services-producing activities of households for own use T56 Activities of extraterritorial organizations and bodies U

Table 4: List of 56 sectors in the WIOD table

38

no country name country code

1 Australia AUS2 Austria AUT3 Belgium BEL4 Bulgaria BGR5 Brazil BRA6 Canada CAN7 Switzerland CHE8 China CHN9 Cyprus CYP10 Czech Republic CZE11 Germany DEU12 Denmark DNK13 Spain ESP14 Estonia EST15 Finland FIN16 France FRA17 United Kingdom GBR18 Greece GRC19 Croatia HRV20 Hungary HUN21 Indonesia IDN22 India IND23 Ireland IRL24 Italy ITA25 Japan JPN26 Korea KOR27 Lithuania LTU28 Luxembourg LUX29 Latvia LVA30 Mexico MEX31 Malta MLT32 Netherlands NLD33 Norway NOR34 Poland POL35 Portugal PRT36 Romania ROU37 Russian Federation RUS38 Slovak Republic SVK39 Slovenia SVN40 Sweden SWE41 Turkey TUR42 Taiwan TWN43 United States USA44 Rest of the World ROW

Table 5: List of 44 countries and areas in the WIOD table.

39

5.2 Additional results for network properties and multilayer community struc-tures

Dendrograms obtained from the layer-layer correlations and the layer-layer overlaps:

As discussed in the main paper, the analysis of the layer correlation matrices ({R[α,β]bin },

{R[α,β]w }) and that of the layer overlap matrices ({O[α,β]

bin }, {O[α,β]w }) (see Figures (5), (6))) reveal

that some layers are more correlated or overlapped. To provide a more detailed refinementand visualization of clustering behaviours among them, we report in Figures (13) and (14) thedendrograms obtained from these four matrices. In these figures, the euclidean distance methodand average agglomeration method are used to plot the dendrograms.

40

(a) for binary version

(b) for weighted version

Figure 13: Dendrogram plot obtained from the layer-layer correlation matrices for the binary version(panel a) and the weighted version (panel (b)).

41

(a) for binary version

(b) for weighted version

Figure 14: Dendrogram plot obtained from the layer-layer overlap matrices for the binary version (panela) and the weighted version (panel (b)).

42

Detected communities in 2000, 2004 and 2009:

For illustration purpose, in our present work we mainly focus on the year of 2014 in whichthe latest information on the global trade network in input-output is available. However, inFigure (15), we also report the main results of the community structure in the three selectedprevious years 2000, 2004 and 2009.7

(a) Top communities in 2000

Figure 15: Detected communities in the global trade network in input-output in the different years 2000,2004, 2009. Cont.

7Results for other years over the period from 2000 to 2014 are available upon request to the authors.

43

(b) Top communities in 2004

Figure 15: Detected communities in the global trade network in input-output in the different years 2000,2004, 2009. Cont.

44

(c) Top communities in 2009

Figure 15: Detected communities in the global trade network in input-output in the different years 2000,2004, 2009.

45