  We present a new metric estimating fitness of countries and complexity of
products by exploiting a non-linear non-homogeneous map applied to the publicly
available information on the goods exported by a country. The non homogeneous
terms guarantee both convergence and stability. After a suitable rescaling of
the relevant quantities, the non homogeneous terms are eventually set to zero
so that this new metric is parameter free. This new map almost reproduces the
results of the original homogeneous metrics already defined in literature and
allows for an approximate analytic solution in case of actual binarized
matrices based on the Revealed Comparative Advantage (RCA) indicator. This
solution is connected with a new quantity describing the neighborhood of nodes
in bipartite graphs, representing in this work the relations between countries
and exported products. Moreover, we define the new indicator of country
net-efficiency quantifying how a country efficiently invests in capabilities
able to generate innovative complex high quality products. Eventually, we
demonstrate analytically the local convergence of the algorithm involved.
