Cost functions in economic complexity

Economic complexity algorithms aim to uncover the hidden capabilities that drive economic systems. Here we present a fundamental reinterpretation of two of these algorithms, the economic complexity index (ECI) and the economic fitness and complexity (EFC), 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 EFC, we derive a novel cost function that exploits the algorithm's intrinsic logarithmic structure and clarifies the role of the regularization parameter in its nonhomogeneous version. Additionally, we establish the existence and uniqueness of its 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, with potential implications for a broader class of algorithms, including the Sinkhorn-Knopp method. Finally, we apply the energetic framework to a real-world trade network, demonstrating how linkwise energy provides a direct way to identify structurally relevant and vulnerable regions of the export matrix, thus complementing and enriching standard economic complexity analyses. 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.