The roles that countries occupy within the international GVNs can be unmasked through the application of suitable centrality measures to the international trade network of intermediate and finished products. To this end, here we propose the following three measures of centrality for capturing the degree to which a given country plays a prevailing role in the upstream, midstream, and downstream stages of the production in a given industry’s GVN:

(a)
A country’s upstreamness centrality in an industry captures the tendency of the country to preferentially export intermediate goods to other countries that, in turn, have a tendency to preferentially export finished products and import intermediate inputs.

(b)
A country’s downstreamness centrality in an industry captures the tendency of the country to preferentially import final products from countries that, in turn, tend to preferentially export finished products and import intermediate inputs.

(c)
A country’s midstreamness centrality in an industry is a function of both the upstreamness of the countries from which the focal country imports the intermediate products and the downstreamness of the countries to which the focal country exports the finished products. More specifically, a country’s midstreamness centrality in an industry captures the tendency of the country to import intermediate goods preferentially from countries with high upstreamness centrality and to export final products preferentially to countries with high downstreamness centrality.
Given an industry, these three measures will be applied to the directed and weighted international global value tripartite network defined above, in which nodes are suitably labelled according to the roles they play as exporters or importers of intermediate or finished products.
We now proceed to formalise the three recursive measures of centrality. We begin by formalising the firstorder midstreamness centrality, and we then formalise the firstorder upstreamness and firstorder downstreamness centralities as a function of firstorder midstreamness centrality. In a similar way, we show how a horder midstreamness centrality can be defined in terms of the h−1order upstreamness and h−1order downstreamness centralities. The three centralities then converge to higherorder values that are the solutions of a linear system of equations.
First, we define the firstorder midstreamness centrality of a country as the sum of the intermediate inputs imported and the finished products exported by the country. This measure thus broadly captures the country’s centrality as an intermediary actor in the production of products in a certain industry. Formally, a country m in set M can be characterised by the sum of the weights of all its incoming and outgoing links, i.e., the total strength
s
_{
m
}, as follows
$$ \mu_{m}^{1}=s_{m}=\sum_{u\in U}w_{um}^{I}+\sum_{d\in D}w_{md}^{F}. $$
(4)
To normalise the firstorder midstreamness centrality of each country, we divide Eq. 4 by the maximum midstreamness value across all countries
$$ \frac{\mu_{m}^{1}} {\mu^{*}} \times 100, $$
(5)
where \(\mu ^{*}=\underset {m}{max}\,(\mu _{m}^{1})\).
Next, the firstorder upstreamness centrality of a country u included in set U is defined as the weighed sum of the values of its outgoing links representing the exports of intermediate inputs toward countries in set M, where the weight of the value of each link in the sum is the firstorder midstreamness centrality \(\mu _{m}^{1}\) of the country m in set M to which the link is directed. Formally, we define the firstorder upstreamness centrality of a country u in set U as follows
$$ \upsilon_{u}^{1}=\sum_{m\in M}\mu_{m}^{1}w_{um}^{I}. $$
(6)
To normalise the firstorder upstreamness centrality of each country, we then divide Eq. 6 by the maximum upstreamness value across all countries
$$ \frac{\upsilon_{u}^{1}} {\upsilon^{*}} \times 100, $$
(7)
where \(\upsilon ^{*}=\underset {u}{max}\,(\upsilon _{u}^{1})\).
The firstorder downstreamness centrality of a country d included in set D is instead defined as the weighed sum of the values of the country’s incoming links representing the imports of finished products from countries in set M, where the weight of the value of each link in the sum is the firstorder midstreamness centrality \(\mu _{m}^{1}\) of the country m in set M from which the link originates. Formally, we define the firstorder downstreamness centrality of a country d in set D as follows
$$ \delta_{d}^{1}=\sum_{m\in M}\mu_{m}^{1}w_{md}^{F}. $$
(8)
To normalise the firstorder downstreamness centrality of each country, we then divide Eq. 8 by the maximum downstreamness value across all countries
$$ \frac{\delta_{d}^{1}} {\delta^{*}} \times 100, $$
(9)
where \(\delta ^{*}=\underset {d}{max}\,(\delta _{d}^{1})\).
Next, we define the secondorder midstreamness centrality of a country m in set M as the weighted sum of the values of the links that m receives from countries in set U and directs to countries in set D, where the weights in the sum of the values of the links from countries in set U and to countries in set D are, respectively, the firstorder upstreamness centralities of the countries in U and the firstorder downstreamness centralities of the countries in D. Formally, we define the secondorder midstreamness centrality of a country m in set M as
$$ \mu_{m}^{2}=\sum_{u\in U}\upsilon_{u}^{1}w_{um}^{I}+\sum_{d\in D}\delta_{d}^{1}w_{md}^{F}. $$
(10)
To normalise the secondorder midstreamness centrality of each country, we then divide Eq. 10 by the maximum value of midstreamness across all countries
$$ \frac{\mu_{m}^{2}} {\mu^{*}} \times 100, $$
(11)
where \(\mu ^{*}=\underset {m}{max}\,(\mu _{m}^{2})\).
Finally, the higherorder upstreamness, downstreamness, and midstreamness centrality measures are the solutions of the following linear system
$$ \left\{ \begin{array}{l} \upsilon_{u}^{h}=\sum_{m\in M}\mu_{m}^{h}w_{um}^{I}\\ \delta_{d}^{h}=\sum_{m\in M}\mu_{m}^{h}w_{md}^{F}\\ \mu_{m}^{h}=\sum_{u\in U}\upsilon_{u}^{h1}w_{um}^{I}+\sum_{d\in D}\delta_{d}^{h1}w_{md}^{F} \end{array}\right. $$
(12)
All countries that are involved in at least one transaction (exports or imports) of (intermediate or finished) products will be assigned at least one value of the centralities defined above. In the previous section we mentioned that the international GVNs capture the nuances of international trade, such as chains of multiple transactions of intermediate or finished products. In particular, two cases can be identified. First, a country may well control the whole production of a given product, and contribute to the exports without importing any input. Second, a country may play a prominent role in many stages of the transformation process without necessarily exporting the final product. In the former case, the country that exports finished products but does not import intermediate ones would still occupy the midstream stage of the GVN and thus be classified as part of set M. However, since there would be no incoming links pointing to the country, its midstreamness centrality would not be defined in terms of the upstreamness of other countries. Similarly, in the latter case, the country that exports and imports only intermediate products would occupy both an upstream and a midstream stage in the GVN, and would thus be classified as a member of both set U and set M. However, since there are no outgoing flows of finished products from that country toward other countries, its midstreamness centrality would not be defined in terms of the downsteamness centralities of the other countries.