Trade networks, across which countries distribute their products, are crucial components of the globalized world economy. Their structure affects the mechanism of propagation of shocks from country to country, as observed in a very sharp way in the past decade, characterized by economic uncertainty in many parts of the world. Such trade structures are strongly heterogeneous across products, given the different features of the countries which buy and sell goods. By using a diversified pool of indicators from network science and product complexity theory, we quantitatively demonstrate that, overall, products with higher complexity-i.e., with larger technological content and/or number of components-are traded through more centralized networks-i.e., with a smaller number of countries concentrating most of the export flow. Since centralized networks are known to be more vulnerable, we argue that the current composition of production and trading is associated to high fragility at the level of the most complex-thus strategic-products.