Cost Functions in Economic Complexity

Abstract: Economic complexity algorithms, including the Economic Complexity Index (ECI) and Non-Homogeneous Economic Fitness and Complexity (NHEFC), have proven effective at capturing the intricate dynamics of economic systems. Here, we present a fundamental reinterpretation of these algorithms by reformulating them as optimization problems that minimize specific cost functions. We show that ECI computation is equivalent to finding eigenvectors of the network's transition matrix by minimizing the quadratic form associated with the network's Laplacian. For NHEFC, we derive a novel cost function that exploits the algorithm's intrinsic logarithmic structure and clarifies the role of its regularization parameter. Additionally, we establish the uniqueness of the NHEFC solution, providing theoretical foundations for its application. This optimization-based reformulation bridges economic complexity and established frameworks in spectral theory, network science, and optimization. The theoretical insights translate into practical computational advantages: we introduce a conservative, gradient-based update rule that substantially accelerates algorithmic convergence. Beyond advancing our theoretical understanding of economic complexity indicators, this work opens new pathways for algorithmic improvements and extends applicability to general network structures beyond traditional bipartite economic networks.