  Nestedness has traditionally been used to detect assembly patterns in
meta-communities and networks of interacting species. Attempts have also been
made to uncover nested structures in international trade, typically represented
as bipartite networks in which connections can be established between countries
(exporters or importers) and industries. A bipartite representation of trade,
however, inevitably neglects transactions between industries. To fully capture
the organization of the global value chain, we draw on the World Input-Output
Database and construct a multi-layer network in which the nodes are the
countries, the layers are the industries, and links can be established from
sellers to buyers within and across industries. We define the buyers' and
sellers' participation matrices in which the rows are the countries and the
columns are all possible pairs of industries, and then compute nestedness based
on buyers' and sellers' involvement in transactions between and within
industries. Drawing on appropriate null models that preserve the countries' or
layers' degree distributions in the original multi-layer network, we uncover
variations of country- and transaction-based nestedness over time, and identify
the countries and industries that most contributed to nestedness. We discuss
the implications of our findings for the study of the international production
network and other real-world systems.
