Reconciling contrasting views on economic complexity

The study addresses how to consistently measure the economic complexity (EC) of countries using export data. Existing EC metrics—most notably the Method of Reflections (MR) and the Fitness-Complexity (FC) algorithm—seek to infer countries’ productive knowledge from the diversification and sophistication of their export baskets. MR computes country complexity as an average over product complexities (and vice versa), but averaging tends to lose information about diversification and product ubiquity. FC rejects a linear relationship and introduces a non-linear coupling where product quality depends on the least-fit exporters, preserving diversification information in country fitness. Despite a common motivation, MR and FC yield markedly different country rankings and scatter substantially, limiting their policy usefulness. The purpose of this paper is to reconcile MR and FC within a single mathematically sound framework using linear algebra on bipartite networks, preserving the strengths of both and enabling robust, interpretable rankings and temporal analyses of growth and innovation potential.

The paper situates economic complexity as a data-driven alternative or complement to traditional growth theories (e.g., Schumpeterian, endogenous growth), which can oversimplify development dynamics. Initial efforts linked productive knowledge to export diversity; the MR approach formalized ECI/PCI via iterative averages mapped to an eigenvector solution, but at the cost of losing diversification and ubiquity details. Critiques highlighted information loss in MR and ambiguity from asymmetric mappings. The FC algorithm proposed non-linear metrics (Fitness for countries, Quality for products) that better capture nestedness and preserve diversification, though the iterative method can suffer convergence issues. Subsequent work (e.g., ECI+) aimed to improve MR and has been linked to FC, suggesting conceptual proximity. EC metrics have seen applications in forecasting growth, strategy, and macro-finance, yet discrepancies across methods have hindered broader adoption. Network science provides tools (eigenvector centrality, multidimensional centrality) relevant to reconcile these approaches within a symmetric proximity matrix framework.

Data: International trade data (BACI-CEPII) for 1995–2017 at HS-1992 6-digit, downscaled to 4-digit, including countries with export share ≥ 10^-5 of world exports. The incidence matrix M is constructed via Revealed Comparative Advantage (RCA): M_cp = 1 if RCA_cp ≥ 1, 0 otherwise; analyses also consider using RCA values directly as weights.

General framework: Countries and products form a bipartite network. The goal is to determine country properties X_c and product properties Y_p via coupled linear relations using a single transformation matrix W derived from M:

• X_c = Σ_p W_cp Y_p
• Y_p = Σ_c W_cp X_c This yields equivalent eigen-problems for symmetric proximity matrices N = W W^T (countries) and G = W^T W (products):
• X = λ N X, Y = (1/λ) G Y N_cc' and G_pp' measure similarity of export baskets and product co-export patterns, respectively. Symmetry allows interpreting X and Y as eigenvector centralities.

Recasting MR and FC:

• MR mapping (example A): choose W^A = M_cp / √(k_c k_p), with k_c = Σ_p M_cp (diversity) and k_p = Σ_c M_cp (ubiquity). Transformations X_c^A = ECI_c √k_c and Y_p^A = PCI_p √k_p recover MR. The first eigenvectors correspond to √k_c and √k_p (uninformative unitary ECI/PCI in MR), so MR uses second eigenvectors to define ECI/PCI.
• FC mapping (example B): linearize the non-linear FC coupling around F ≈ k (diversity) via Taylor expansion, obtaining a linear eigen-system: F_c = Σ_p M_cp Q_p, Q_p = Σ_c M_cp F_c / K_p with K_p = Σ_c M_cp / k_c. In the framework set X_c = F_c / √k_c (or F_c / k_c per the final linear mapping), Y_p = Q_p / √k_p (or Q_p / k_p), and W^B = M_cp / √(k_c k_p) (or M_cp / (k_c k_p) for the linearized FC), yielding symmetric proximity matrices:
  • N_cc' = Σ_p M_cp M_c'p / (k_c k_c')
  • G_pp' = Σ_c M_cp M_cp' / (k_p k_p') This linearization removes iterative convergence issues and closely preserves the information of the original non-linear FC outputs.

Integrated multidimensional index (GENEPY): Define GENEPY for countries by combining the first two eigenvectors X_1 and X_2 of the country proximity matrix N (with diagonal set to a constant, here zero, to remove self-proximity) using a least-squares, multidimensional centrality framework:

• GENEPY_c = (λ_1 X_{c1}^2) + 2 (λ_2 X_{c2}^2) X_1 captures overall centrality tied to diversification/fitness-like information; X_2 captures spectral clustering akin to ECI. The weighting reflects unique contributions from commonality analysis in a least-squares reconstruction of N using eigenvector outer products.

Empirical analyses: Compute X_1, X_2 for each year 1995–2017; analyze country positions in the (X_1, X_2) plane; derive country trajectories over time; characterize three growth regimes (Impasse, Bounce, Arena). Compute the world’s GENEPY-weighted barycentre annually (weighting GENEPY by population to make an intensive metric comparable to GDP levels) and compare with GDP PPP-weighted and population-weighted barycentres. Robustness checks include using RCA as weighted inputs rather than binarized M and comparing linearized FC against the original non-linear algorithm.

• Reconciliation of MR and FC: Both are recast into a single eigenvector-based framework on a symmetric proximity matrix, recovering their key outputs via mappings. Divergences in prior rankings largely stem from using different-order eigenvectors (MR’s second vs FC’s first), not fundamentally different information.
• Linearization validity: The linearized FC closely matches the original non-linear FC, with an average Pearson correlation of about 99.5% over time between linear and non-linear fitness/quality values (Supplementary Fig. 2), and without convergence issues.
• Component correlations (2017):
  • First eigenvector X1 correlates very strongly with rescaled Fitness (F/k): ρ ≈ 0.982 (Fig. 1b).
  • Second eigenvector X2 correlates well with rescaled ECI (ECI·k^{-1}): ρ ≈ 0.885 (Fig. 1c).
• GENEPY integrates ECI-like and Fitness-like information, ranking advanced, innovation-driven economies (e.g., EU-28, CHE, CHN, JPN, SGP, USA) far from the origin in the (X1, X2) plane, while less stable economies populate the bottom-left.
• Growth regimes and trajectories: Countries’ movements in the (X1, X2) plane exhibit three regimes—Impasse (low X1, negative X2), Bounce (crossing to X2 > 0 with rising quantity/quality of exports), and Arena (competition among advanced economies with diverging GENEPY trends). Examples: CHN, IND, SGP moved into the advanced cluster; NGA and VEN remained in Impasse; JPN, USA, DEU, CHE exemplify Arena dynamics. Effects of shocks are visible (e.g., 2008 crisis; CHN’s 2015–2017 adjustments).
• Global barycentre shifts (1995–2017): The GENEPY- and GDP PPP-weighted barycentres both shift eastward, unlike the population barycentre which trends westward due to African population growth. The gap between GDP and GENEPY centres suggests further growth potential in Asia.
• Robustness: GENEPY results are coherent when using RCA weights instead of binarized M, unlike MR/FC which are more sensitive to binarization choices.
• Conceptual linkage: The framework reveals a formal similarity between GENEPY equations and the EXPY/PRODY construction (with EXPY embedding exogenous GDP per capita), hinting at microeconomic underpinnings for EC.

The findings address the central question of reconciling disparate EC measures by demonstrating that MR and FC can be unified within a symmetric, linear-algebraic framework on bipartite networks. Using a single transformation matrix and symmetric proximity matrices (N, G) clarifies the eigenvector centrality basis of EC measures and resolves ambiguities from asymmetric formulations (e.g., left vs right eigenvectors). The integrated GENEPY index leverages complementary information: X1 captures diversification/fitness-like centrality, while X2 captures clustering/ECI-like structure. This multidimensionality allows coherent rankings and interpretable growth trajectories without exogenous variables like GDP per capita. The framework’s robustness to input weighting (RCA vs binarized incidence) strengthens its applicability. While the linearization of FC works very well in trade networks—likely due to flexible human-driven decision-making and less rigid nestedness—the authors caution that genuinely non-linear dynamics may be essential in other bipartite systems (e.g., ecological networks). The symmetric proximity approach also aids interpretation of bilateral similarities and capability sharing across countries and products, potentially informing policy on diversification paths and industrial upgrading.

This work introduces GENEPY, a multidimensional economic complexity index that reconciles MR and FC within a unified, symmetric eigenvector framework. By mapping both methods to a common linear-algebraic formulation and linearly approximating FC, the approach preserves key information (diversification and nestedness effects), avoids convergence issues, and yields robust, interpretable rankings and temporal trajectories of economic growth. Empirically, GENEPY integrates ECI-like clustering and Fitness-like centrality, identifies three characteristic growth regimes, and reveals an eastward shift in the global centre of economic complexity. The framework’s formal connections to EXPY suggest avenues toward micro-foundations of EC. Future work could deepen microeconomic grounding, explore non-linear extensions where warranted (e.g., highly nested ecological systems), enhance product-level analyses, and assess policy interventions for capability accumulation and export upgrading using the proximity structure.

• Generalizability of linearization: The high agreement between linearized and non-linear FC holds in international trade networks but may not extend to systems with strong non-linear nested dynamics (e.g., pollinator–plant networks).
• Data construction choices: Results depend on RCA thresholds, product aggregation (HS downscaling), and inclusion criteria (export share ≥ 10^-5). Although robustness to using RCA as weights is shown for GENEPY, MR/FC are more sensitive.
• Proximity matrix adjustment: Setting diagonal elements of N and G to zero (to remove self-proximity) improves interpretability but slightly alters exact correspondences (minor deviations in Fig. 1b).
• Scope: The analysis focuses on merchandise trade; services, non-tradables, and domestic capability measures are not included. Exogenous factors (institutions, policy, geography) are not modeled explicitly.
• Interpretation: While symmetric proximity clarifies centrality, causality between complexity changes and growth outcomes remains inferred rather than identified within a structural model.