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.

For models whose evolution takes place on a network it is often necessary to augment the mean-field approach by considering explicitly the degree dependence of average quantities (heterogeneous mean-field). Here we introduce the degree dependence in the pair approximation (heterogeneous pair approximation) for analyzing voter models on uncorrelated networks. This approach gives an essentially exact description of the dynamics, correcting some inaccurate results of previous approaches. The heterogeneous pair approximation introduced here can be applied in full generality to many other processes on complex networks.

We show that the space in which scientific, technological and economic activities interplay with each other can be mathematically shaped using techniques from statistical physics of networks. We build a holistic view of the innovation system as the tri-layered network of interactions among these many activities (scientific publication, patenting, and industrial production in different sectors), also taking into account the possible time delays. Within this construction we can identify which capabilities and prerequisites are needed to be competitive in a given activity, and even measure how much time is needed to transform, for instance, the technological know-how into economic wealth and scientific innovation, being able to make predictions with a very long time horizon. We find empirical evidence that, at the aggregate scale, technology is the best predictor for industrial and scientific production over the upcoming decades.