  We investigate the convergence properties of an algorithm which has been
recently proposed to measure the competitiveness of countries and the quality
of their exported products. These quantities are called respectively Fitness F
and Complexity Q. The algorithm was originally based on the adjacency matrix M
of the bipartite network connecting countries with the products they export,
but can be applied to any bipartite network. The structure of the adjacency
matrix turns to be essential to determine which countries and products converge
to non zero values of F and Q. Also the speed of convergence to zero depends on
the matrix structure. A major role is played by the shape of the ordered matrix
and, in particular, only those matrices whose diagonal does not cross the empty
part are guaranteed to have non zero values as outputs when the algorithm
reaches the fixed point. We prove this result analytically for simplified
structures of the matrix, and numerically for real cases. Finally, we propose
some practical indications to take into account our results when the algorithm
is applied.
