Key Sectors in Input-Output Production Networks: An Application to Brexit

This paper presents the first detailed and holistic description of the European production network (EPN) and provides different rankings of the most 'systemically important' industries involved in Brexit. Employing techniques of complex networks analysis and Input-Output traditional tools, the study identifies those industries that are key in the complex structure of the UK-EU trade relationships. The method developed would help policy-makers to better understand which tariff would have a more distortive impact, which export sector should be pushed, which imports should be safeguarded. Such information may have foremost importance in the negotiations between the UK and EU. Our findings suggest that Brexit would be not just a problem for the UK, as it is often portrayed, but any form of Brexit could propagate affecting the global production system. Further, by inspecting industries centrality within the EPN, we find that the UK could be less exposed to trade barriers than EU countries.


Introduction
The structure of the global production system is nowadays characterised by a complex network of industries linked within and across different sectors and countries by means of input-output production ties (Amador and Cabral, 2017). The texture of the interdependencies between industries has relevant implications in the propagation throughout the economy of sectoral shocks and stimulus (Acemoglu et al., 2012). The primary role played by such interconnections in generating macro fluctuations was highlighted by the last economic crisis. Since the economic recession hit the USA and the world, there has been a large and growing body of research regarding the government bailout plans, in both the academic arena as well as in the popular press. Several criteria have emerged from the debate on the priority and choice of industries that the government should bailout in economic recessions (Luo, 2013). For example, focussing on the scale of the industry and its internal performance, some literature states that governments and institutions should come to the rescue of the 'too big to fail' firms and banks (White, 2014). Other studies highlighted the relevance of network effects, and suggest that should be prioritised 'too interconnected' (Battiston et al., 2012b;Markose et al., 2012) and 'too central to fail' (Battiston et al., 2012a) industries. The present paper aims to study the properties of the provides time-series of global input-output tables, covering, at the time of writing, 56 industries classified by the International Standard Industrial Classification revision 4 (ISIC Rev.4), in 43 countries in the world plus a region called 'Rest of the World', for the period 2000-2014, although we make use only of the 2014 data (see Timmer et al., 2014 and for sources and details). Figure 1 shows the schematic outline for a WIOT. Essentially, it includes a combination of national input-output tables in which the use of products is broken down according to countryindustry of origin.
Intermediate use (S columns per country) Final use (C columns per country) Total 1 … N 1 … N S Industries, country 1 … S Industries, country N The stylised WIOT depicted in Figure 1 illustrates a simplified WIOT with N countries and S sectors, which together constitute the world economy. The rows in the WIOT give the total dollar value of deliveries of output from a particular industry in a given country to another industry for intermediate use (block matrices labelled ), or to final user (block matrices labelled ), either within the same country or abroad. The fundamental accounting identity of any input-output table is that total use of output in a row equals total output of the same industry as indicated by the sum of inputs in the respective column in the left-hand part of the table. The columns indicate the amounts of intermediate inputs needed for production; hence, they are informative about the technology of production. What remains between total output and total intermediate inputs is value added ( ), i.e. the direct contribution of domestic factors to output.
Input-output tables, as one can guess, provide a natural source of information for representing the economy as a network. In particular, in order to build the EPN we consider the block matrices of the WIOT, for the 28 EU economies, as a weighted adjacency matrix of a network where the nodes are individual sectors in different countries, and edges are dollar goods flows within and across sectors. The direction of the flows goes from the supplier sector to the buyer sector. This data contain 1568 nodes (56 sectors in each of the 28 countries) and 2241747 directed weighted edges.

The Structure of the EPN
The aim of this section is to summarise the main topological properties of the EPN, from a Brexit perspective. Our primary interest is in illustrating the degree of industries connection, the density of sectoral interactions, the distance between country-sectors, and the presence of hub sectors or potential shock propagators in the network. These basic network statistics allow us to provide a descriptive analysis of the EPN and advance hypotheses on the propagation of a trade shock, as Brexit would be.
To study the extent to which industries are connected in the EPN we start analysing the degree and strength distributions. The degree of a node in a network is defined as the number of links incident upon a node, here, the number of input-output connections each sector has. When these connections are weighted, the strength of a node is measured, i.e. the sum of weights attached to the edges belonging to a node. Here, the dollar amount of input-output connections each sector has. Recall that the EPN is based on the weighted adjacency matrix that is suitable to study the strength distribution. On the other hand, to analyse the degree distribution of the EPN, as in Cerina et al. (2015), we need to define a regular binary adjacency matrix , where = = 1 if either > 0 or > 0, and = = 0 otherwise. Further, according to the direction of the connections, a sector has an in-degree ( ) and an in-strength ( ) 2 respectively defined as the sum of all elements in the column ℎ of the adjacency ( ) and weighted ( ) matrices: Conversely, a sector has an out-degree ( ) and an out-strength ( ) defined as the sum of all elements in the row ℎ of the adjacency (D) and weighted (Z) matrices, respectively: Summarising, the in(out)-degree of a node represents the number of supplier (buyer) sectors linked to sector . Similarly, the in(out)-strength of a node represents the dollar value of goods employed as inputs (delivered as outputs) by sector . The sum of in and out degree or in and out strength are respectively the total-degree and total-strength. As shown in Figure 2, the EPN is featured by highly left skewed degree distributions, showing that most of the sectors in the economy have many connections with other sectors. The average in-degree and out-degree is about 1478, i.e. every node is linked with almost every node. In particular, most of the values of the out-degrees are concentrated on the highest values. Therefore, there are sectors that act as general suppliers delivering inputs to many or all other sectors (Alatriste-Contreras, 2015 shows similar results). The high connectivity of the EPN is also highlighted by the density of the EPN that is 0.976, a high value which suggests that in the network under consideration sectors are highly dependent on almost all other sectors. Furthermore, the diameter, defined as the shortest distance between the two most distant nodes in the network, which is the largest number of steps that separate sector from sector for all possible pairs of sectors ( . ), is 3; and the average path length, i.e. the average of the number of steps it takes to get from sector to sector for all possible pairs of sectors ( , ), is 1.
Moving from the unweighted EPN to the weighted one, Figure 3 illustrates the empirical distributions of in-strength, out-strength and total-strength in the EPN. The x-axis is, respectively, the in, out and total strength for each country-sector presented on a log scale. The y-axis, also in log scale, represents the probability that the sector ℎ has a strength larger than or equal to x. Hence, the upper left-hand portion of all the three subgraph, shows that nearly 100 percent of country-sectors have an in, out and total strength greater than 0.01; moving down on the y-axis we see that only about one tenth 7 of all country-sectors have an in, out and total strength greater than 10000; and finally, the right-hand portion of all the distributions shows that only less than 1 percent of all country-sectors have an in, out and total strength greater than 100000. Therefore, on the contrary to the degree distributions observed, the in, out and total strength distributions for country-sectors in the EPN are all positively skewed. Our findings are coherent with Alatriste-Contreras and Fagiolo (2014), Alatriste-Contreras (2015) and Luu et al. (2017), which show that each European economy at sectoral level of aggregation is characterised by negatively skewed degree distributions and positively skewed strength distributions. The heavy tailed behaviour of the strength distributions means that there is a statistically significant probability that a node has a very large strength compared to the average, i.e. in the EPN many countrysectors have a low strength, whilst only a few have high strength values. The unequal distribution of in, out and total strength suggests the presence of hub-like countrysectors. In fact, as shown in Table 1, the EPN is dominated in terms of strength, i.e. dollar goods that flow through a sector, by a few industries placed in core countries, especially, Germany, the UK, and France. These key players could act as global propagators in the network. This implies that a shock affecting one of these hubs will spread quickly to most sectors, either domestically or abroad, thus affecting the performance of the aggregate economy (Carvalho, 2014). From a Brexit point of view, it is worth noting that the UK economy plays a primary role hosting more than 20 percent of top industries. Notably, according to the strength rankings, the UK and EU should take care of the trade relationships that involving the following UK's industries: construction (F), which is the largest sector in terms of total strength, health (Q), real estate (L68), electricity and gas (D35), food products (C10-12), administrative services (N), financial services (K64), retail trade (G47), legal and accounting (M69-70). country-sector in-strength country-sector out-strength country-sector tot-strength DEU_C29 272498,8 DEU_N 234098,36 GBR_F 424515,55 GBR_F 225088,8 FRA_N 213793,85 DEU_C29 402500,36 FRA_F 203144,5 GBR_F 199426,72 DEU_N 336599,33 DEU_F 188456,9 GBR_N 191547,49 DEU_F 309562,27 DEU_C10-12 170753 To sum up, the structure of the EPN in which sectors are both highly connected as shown by the degree distributions, and asymmetrically connected as reported by the strength distributions, combined with the remarks about the EPN density, diameter, average path length, and the presence of a small number of hubs, suggest the small-world nature of the EPN (on the definition of small-world networks see Caldarelli, 2007). In production networks characterised by these topological properties a local idiosyncratic shock, as it could be a trade shock due to Brexit, is able to propagate through the whole European economy and generate a sizeable global disturbance (Acemoglu et al., 2012;Carvalho, 2014;Cerina et al., 2015).

Central Nodes in the EPN
In the previous section, we have explored the EPN and identified the main sectors in terms of strength. However, this preliminary rough measure does not offer a complete view of the importance of a sector. For example, the strength of a node does not take into account the degree to which a specific sector is involved in global value chains (Bohn et al., 2018). Therefore, in this section we conduct a local analysis of the nodes and individualise the key sectors in the EPN employing the traditional methods of input-output literature and the PageRank centrality, a network-based measure also known as Google's algorithm (Brin and Page, 1998).
Consider an economy with industries and denote the interindustry flows by the × transaction matrix . Let be the vector of industry final demands and the vector of industry gross output. The accounting equations are given as = + , where is the summation vector, i.e. a vector of all ones. Define the direct input coefficients as the ratio of input supplied by and bought by over the gross output of sector as = / , which is the typical element of the economy's direct requirements matrix , also known as the technical coefficients matrix. Considering that, =̂− we can substitute = in the accounting equations to get = + . Solving for yields: where is the identity matrix and ≡ ( − ) − is the Leontief inverse or multiplier matrix, which makes clear the direct and indirect dependence of each of gross outputs on the values of each of the final demand. The ℎ column sum of the Leontief inverse describes the total output increase due to an increase of one unit in the final demand of sector . Thus, Rasmussen (1956) proposed to use the column sums of the matrix, ′ , to rank the industries and identify the key ones in the economy. One drawback of the Rasmussen method of backward linkages is that it assumes homogeneous sectors, assigning the same weight to all the industries, which is far from the reality. In particular, the industries composing the EPN are very heterogeneous as are the 28 economies that host them. Therefore, as in Cerina et al. (2015) we use the final-demand-weighted version of the Rasmussen method, i.e. the Laumas (1976) key sector measure: where ∘ is the element-wise multiplication operator. However, in the Laumas method the weighting scheme is arbitrary. Furthermore, this measure, although weights the industries according to their final demand, does not take into account the heterogeneity of intersectoral relationships, i.e. it assumes that all the neighbouring industries have the same importance. To solve this issue, Dietzenbacher (1992) proposed the eigenvector method of backward linkages, which is based on the reasoning that the inputs from a sector with stronger pulling power should be weighted more than the inputs from a sector with weaker power (Luo, 2014). In other words, not all the connected industries are equal but the one with more strength should be weighted more. Dietzenbacher (1992) proved that sectors can be ranked by importance computing a sector power indicator, which we denote as , that coincides to the left-hand eigenvector corresponding to the dominant eigenvalue of the technical coefficients matrix . In the input-output literature, Dietzenbacher method is de facto in line with the eigenvector centrality one of the best known 'influence measures' employed in network theory and social network analysis, according to which nodes are considered to be central in the network if their connections in the network are themselves well-connected nodes. One drawback of this indicator is that it does not penalise the distant connections, this means that it can overestimates the importance of some peripheral industries if they have even only an insignificant indirect connection with a hub industry (Cerina et al., 2015). Therefore, other 'influence measures' of network centrality such as Katz-Bonachic centrality (Katz, 1953;Bonachic, 1987) and PageRank centrality (Brin and Page, 1998) have been preferred in recent studies on input-output networks (Acemoglu et al., 2012;Carvalho 2014;Cerina et al. 2015). Here, we refer mainly to the weighted version of PageRank centrality used in Cerina et al., 2015. The PageRank ( ) also relates the importance of a sector with the quality of its connection but it contains a damping factor that penalises distant connections. It is computed iteratively for each node as follows: where is the total number of nodes (sectors), is the damping factor set to its default value, 0.85, ( ) are the in-neighbours of (input supplier for the sector), is the weight of the link connecting the nodes and , ( ) is the sum of the weights of the outgoing edges from (the sum of the output delivered by sector ). Note that the algorithm starts at time step = 0, assuming a probability distribution such that ( ; 0) = 1⁄ . As in the strength distributions, Figure 4 shows that the network centrality of different nodes is distributed as a power-law. Far out in the right tails, we find the central production nodes in the network, which we rank for each centrality measures in Table 2. Again, as in Table 1 we find that key sectors in the EPN are placed in core countries. In particular, the Laumas indicator (w), which emphasises the role of final demand, indicates the construction (F) sector in France as the key EPN sector, followed by two UK sectors, respectively real estate (L68) and health (Q). Differently, the Dietzenbacher eigenvector indicator ( ) shows the relevance of German sectors. Especially according to this measure, almost fifty percent of the top 30 sectors in the EPN are from Germany that hosts even the first four key sectors. However, the presence of many German sectors in the ranking reveals another drawback of this measure already noted in Cerina et al. (2015). Indeed, in the presence of clusters in the network, such as in the EPN where sectors usually cluster domestically, the eigenvector centrality measure tends to overestimate the importance of some nodes. For example, if some industries in Germany have strong linkages, the eigenvector method imputes a high strength to almost all other industries in Germany due to the national connections and this process will reinforce itself. In addition, to penalising ties with distant nodes, the other 'influence measure', namely PageRank centrality ( ), addresses also this problem. According to , Germany still plays a central role in the EPN, hosting the first two sectors, which are motor vehicles (C29), and machinery and equipment (C28), respectively. However, what is noteworthy from a Brexit point of view is that with eleven industries, the UK is the most represented country in the top 30 sectors ranking. In other words, more than 35 percent of key sectors in the EPN are hosted by the UK. Recalling the definition of the , this means that UK sectors are among the most influential sectors, i.e. they are very important sectors and are well connected with other EPN key sectors. FRA_D35  ITA_G46  ITA_Q  DEU_C25  GBR_G46  FRA_G47  ITA_G46  DEU_Q  ITA_L68  DEU_H49  ITA_C28  DEU_G47  GBR_K64  ITA_C10-12  FRA_C10-12  GBR_M69_70  GBR_K65  DEU_P85  GBR_B  ESP_C10-12  ESP_F  ITA_D35  GBR_K64  GBR_P85  DEU_K64  ESP_C29  ESP_I  DEU_D35  DEU_L68  ITA_O84  ITA_C24  GBR_N  GBR_I  NLD_M69_70  FRA_O84  ITA_I  FRA_C20  GBR_I  ITA_G47 DEU_J62-63 GBR_C10-12 DEU_R-S FRA_K64 ITA_I Our findings on the structure of the EPN help the understanding of the UK relevance within the EPN and suggest that a shock hitting key sectors placed in the UK could propagate through other key sectors and generate macro disturbances in other European economies. However, they merely give us a descriptive and qualitative view, whereas it does not provide any effective quantitative measure of the possible economic implications of Brexit. This will be the object of the next section.

The Hypothetically Extraction method to unveil Key Industries from a Brexit perspective
The Brexit debate has been enriched by numerous studies of academics and governing bodies that attempt to quantify the economic impacts of Brexit on the UK, the EU and the rest of the world (see Gasiorek et al., 2019;Hantzsche et al., 2019;Minford, 2019  However, as the outcome of the negotiations between the UK and Europe is not known yet, most of these studies are based on assumptions about possible future scenarios. Furthermore, these analyses require also assumptions on the strength of international substitution patterns. One exception is W. , which opt for a different approach to study the degree to which EU regions and countries are exposed to negative trade-related consequences of Brexit. In particular, using an extended version of the general formula proposed by Los et al. (2016), they get estimates of domestic value added (DVA) in exports of EU regions to the UK and DVA exports of UK regions to the EU. Dividing these estimates by regional GDP, they compute an index of the share of GDP exposed to Brexit, for EU regions and countries, which takes into account all the effects due to the fragmented production processes within the UK, the EU and beyond. This accounting exercise, which not allows for an actual quantification of changes in regional GDP due to Brexit, helps in answering the question: what if the UK and EU regions would stop trading? In other words, W.  are able to rank EU regions and countries by the risk they face due to Brexit. The method employed by Los et al. (2016) and W.  is called "hypothetical extraction" and it is used in the input-output literature to identify key sectors (for a complete review and insights see Miller andBlair, 2009). The aim of this technique is to quantify how much the output of an n-sectors economy would decrease if a particular industry were not present. Extracting industry requires that the ℎ row and column of the matrix are set equal to zero. We define this matrix by * . Equally, the final demand for goods and services provided by industry is set to zero, i.e. = 0, which gives the new final demand vector * . Thus, the estimated new vector of sector gross outputs will be: * = ( − * ) −1 * The change before and after extraction is equal to the difference ′ = ( − * ). This method can be easily extended to an inter-country input-output framework with N countries and n production sectors in each country to quantify the effect on the output of the rest of the economy, as induced by hypothetically extracting a country (see Ditezenbacher et al., 1993;Dietzenbacher and van der Linden, 1997). As shown by W. , this approach is suitable in the case of Brexit to quantify how much the GDP of UK and EU would change if these two macro regions stop trading.  Electronic copy available at: https://ssrn.com/abstract=3347545 Figure 5 shows a simplified version of the global WIOT presented in Figure 1, with one sector and three countries, namely the UK, an EU country (EU), and the rest of the world (ROW Again, the estimated new vector of sector gross outputs is given by equation (8), and the change before and after extraction will be equal to the difference ′ = ( − * ). To express this change in GDP terms we pre-multiply equation (8) by the value added coefficients matrix ̂, i.e. a diagonal matrix, of which the typical element on the main diagonal, ⁄ , is the value added coefficient of industry j in country s. This leads to: * =̂( − * ) −1 * Finally, the change in value added is derived by the difference ′ = ( − * ). Briefly, this is the technique employed by W. . Here, we build on this approach and develop a more granular monetary indicator able to quantify the impact of sectoral hypothetical extraction on the GDP of the UK and EU countries. One can consider such a measure as the exposure of the UK and EU countries to sectoral tariff and non-tariff barriers (on the determinants and relevance of trade barriers see Ennew et al, 1990;Greenaway andMilner, 1994 andGreenaway andMilner, 2003). Indeed, if trade barriers, in general, can reduce bilateral trade between two countries, applying country-sector hypothetical extractions allows us to identify those sectors for which a reduction in trade flows implies a higher loss for the economies involved (on the impact of trade barriers and border effects see Capello et al., 2017 andCapello et al., 2018).
Our exposure measure is closely related to the concept of industries vulnerability to Brexit developed by Gasiorek et al. (2019). Employing a multisector and partial equilibrium framework, Gasiorek et al. (2019) analyse effects on 122 UK manufacturing sectors, using 2016 trade data. The authors provide one of the most detailed and granular analysis on the possible impacts of Brexit on prices, output and trade on specific manufacturing industries. Gasiorek et al. (2019) model five different Brexit scenarios, and achieve results that point to considerable variation across manufacturing sectors, and across skill categories of labour. Here, adopting a different modelling strategy, we extend Gasiorek et al. (2019) by including raw material and services industries and exploring the sectoral vulnerability to Brexit in both the UK and EU27 countries. Furthermore, the indicator that we propose, in addition to being a measure of risk, provides answers to questions like: to what extent the UK (EU) GDP depends on the export of sector to EU (UK), or conversely, to what extent the UK (EU) GDP depends on the import from the ℎ EU (UK) sector? In this sense, the measure we develop could be seen as a kind of sector external centrality measure. In other words, our measure identifies also key import sectors and key export sectors.

Methodology
As in section 3, in our accounting exercise we use the last available WIOT released by the WIOD (2014), but we consider all the 44 economies in order to quantify the impact the extraction of sectoral trade flows between the UK and EU will have on these directly involved countries and the rest of the world.
Using partitioned matrices, the coefficients matrix and the final demand matrix of the WIOT are presented in summary form as: Finally, as in W. , we estimate the new vector of sector value added using equation (11), and the hypothetical loss in value added (LiVA) derived from trade flows extraction, as the difference ′ = ( * − ).
Clearly, one can consider also the opposite case, in which the ℎ row of the sub matrices UE 1 to UE 27 are set equal to zero, or both cases simultaneously, i.e. the UK and EU countries will stop importing each other product delivered by sector .
In the next results section, we contemplate all these three scenarios extracting one at a time all the 56 UK and EU sectors included in the WIOT.

The exposure to sectoral hypothetical extractions due to Brexit
In this sub-section, we discuss the results about the hypothetical extractions of sectoral bilateral trade flows between the UK and EU countries. The results are presented in Tables 3, 4 and 5, more detailed information can be found in tables from A.3 to A.8 in the Appendix. Table 3 summarises the results in tables A.3 to A.6, and shows the top 30 sectors ranked by LiVA aggregates at the country level, i.e. those sectors delivering products that if excluded in bilateral trade between the UK and Europe would generate a greater loss in terms of aggregate domestic value added. The UK would be most affected by the exclusion of wholesale trade products (G46), administrative and support activities (N) and auxiliary financial services (K66). On the other side of the channel, EU countries appear to be very sensitive to the dynamics affecting motor vehicles industries (C29), food products (C10-12) and wholesale trade (G46). Furthermore, the paths designed by motor vehicle (C29) and food (C10-12) sectors, together with other manufacturing industries, such as petroleum products (C19), chemicals (C20), electronics and computers (C26) etc., are also significant for extra-EU countries. This evidence suggests that EU manufacturing industries are highly integrated in global value chains, thus the economic impact of Brexit would propagate worldwide. The automotive industry (C29) is the sector most exposed to Brexit. Consistently with the PageRank ranking in Table 2, Table  A.3 suggests that this finding depends largely on the relevance of the German motor vehicles industry, which is a driving sector in Europe and has many input-output connections with other key sectors both in Europe and in the UK.
As revealed by Tables A.3 and A.4, unsurprisingly, the UK is the most exposed country in the world. In particular, the most vulnerable goods sector is food products (C10-12) and the most exposed services sector is the wholesale trade industry (G46). The fact that Brexit is risky and costly especially for the UK is in line with W.  and the main Brexit literature. Here this finding is obtained by applying our technique by extracting all sectoral trade inflows and outflows between the UK and the EU, thus in a context of 1 against 27 countries. On the other hand, it is noteworthy that some EU countries such as Germany, the most exposed EU country in absolute LiVA terms, Ireland, France, Italy, the Netherlands, and Belgium appear significantly vulnerable as well. Outside Europe, Tables A.5 and A.6 show that the USA is the most exposed country along with the region labelled in the WIOD as rest of the world (ROW).
The exposure to aggregate LiVA, as a result of sectoral bilateral trade flows extractions, could also be seen as a measure of economic exposure to sectoral trade barriers. Generally, trade barriers include tariff and non-tariff barriers (see Greenaway, 1983;Greenaway, and Milner, 2003 for more insight on this). The goods sector could face both, whereas only non-tariff barriers can be applied to the service sectors. Table 3 shows that the UK most exposed sectors are services, whereas the most vulnerable sectors in EU countries are goods industries. Therefore, the UK main trade flows are exposed to nontariff barrier, whilst EU countries are exposed to both tariff and non-tariff barriers. Hence, we can conclude that the UK is less exposed to the economic impact of trade barriers than Europe. This last remark, clearly, holds if EU does not impose huge non-tariff barriers. For example, the picture could change in the extreme case in which EU forbids the UK from selling financial products to EU countries.

Brexit strategic sectors
In this sub-section, we discuss the results about the hypothetical extractions of sectoral trade inflows and outflows between the UK and EU countries. We first extract UK sectoral exports to EU, and then we extract UK sectoral imports from EU. The results are presented in tables 6 and 7, more detailed information can be found in tables from A.9 to A.20 in the Appendix. Tables 6 and 7 present the top 30 sectors ranked by the expected aggregate LiVA as a result of respectively the extraction of sectoral UK export flows to EU countries, and the extraction of sectoral UK import flows from EU countries. The results shown in these two tables can be interpreted as measures of sectors external centrality. In other words, tables 6 and 7 indicate the Brexit strategic sectors, i.e. those sectors that play a key role in the import-export relations between the UK and EU countries. In particular, Table 6 provides a ranking of key export sectors for the UK and reveals that the most important products exported to EU countries are delivered respectively by the wholesale trade industries (G46) administrative and support activities sector (N) and auxiliary financial services (K66). On the other side of the channel, EU countries in order to safeguard their domestic value-added should import from the UK automotive (C29), chemicals (C20) and wholesale trade (G46) industries. Conversely, Table 7 indicates the key import sectors for the UK and shows the relevance of food products (C10-12), motor vehicles industries (C29) and financial services (K64). Losing UK imports could have significant repercussions for EU countries, especially if the UK would stop importing from the automotive (C29), food products (C10-12) and wholesale trade (G46) industries. Again, the UK industries most involved in direct and indirect trade relationships with EU countries are mainly services sectors, whilst the most important EU industries are goods sectors. Thus, as aforementioned, the UK is, in general, less exposed to sectoral trade barriers than EU. This last remark could strengthen the position of the UK in the negotiation of a Brexit deal with EU.   The results shown in this and the previous sub-section do not provide any prediction about the economic impact of Brexit. In fact, the aim of the present study is different. Our findings would allow indicating those sectors that are key in the complex structure of the UK-EU trade relationships. In particular, our sectoral hypothetical extraction technique would help policy-maker to better understand which tariff would have a more distortive impact, which export sector should be pushed, which imports should be safeguarded. Such information may have foremost importance in the negotiations between the UK and EU.

Conclusion
This paper aimed to provide a detailed and holistic description of the EPN and to identify those sectors that are key in the complex structure of the UK-EU trade relationships. Studying the structure of the EPN is crucial in establishing whether and how a potential shock due to Brexit can propagate throughout the economy and lead to significant aggregate fluctuations. Furthermore, the analysis of this production network and the identification of 'systemically important' sectors, is of a foremost importance to design predictive tools, rather than bailout post-recession arguments, and better inform regulators on how to dampen aggregate variability and reduce the likelihood of systemic risk.
Our results can be summarised in three major points. First, the sectors in Europe are both highly connected and asymmetrically connected, i.e. most of the sectors have many connections with other sectors, whilst most of the goods and services flow through just a few sectors. Therefore, a few industries placed in core countries, especially, Germany, the UK, and France dominate the EPN. In particular, the UK hosts the most important sectors both in standard input-output key sectors measure and in terms of network centrality. This means that a shock affecting one of these UK hubs will spread quickly to most sectors and countries, thus affecting the performance of the aggregate economy. Therefore, both macro-regions, the UK and the EU27, should safeguard UK key sectors from the potential negative impact of Brexit.
Second, the measure of country and sectors exposure to tariff and non-tariff barriers, that we developed inspired by the 'hypothetical extraction' method used in W. , shows that the UK would be less exposed than EU countries to trade barriers. Indeed, although in our simulation as well as in the main literature, the UK is the country most exposed to the economic risk deriving from Brexit, we find that the most vulnerable UK sectors are services industries whose products can only be subject to non-tariff barriers, whereas the most exposed EU industries are goods sectors, mainly manufacturing, which can be subject to both tariff and non-tariff barriers.
Third, our measure identifies Brexit key import and export sectors for the UK, EU27, i.e. those sectors that play a key role in the import-export relations between the UK and EU countries. Results show that the UK industries most involved in direct and indirect trade relationships with EU countries are mainly services sectors, whilst the most important EU industries are goods sectors.
Therefore, the main implication of our results is that Brexit could be risky and costly not just for the UK, as it is often portrayed, but any form of Brexit could propagate within the EPN and affect businesses and governments in the EU and globally. Further, our findings of the exposure to trade barriers could strengthen the position of the UK in the negotiation of a Brexit deal with EU.