Source-linked AI summary
On the convergence of the Fitness-Complexity Algorithm
Emanuele Pugliese, Andrea Zaccaria, Luciano Pietronero
TL;DR
The paper asks how the Fitness–Complexity algorithm converges and which countries and products retain nonzero values. It analyzes structured matrices analytically, tests the resulting conditions numerically on real cases, and derives practical guidance for applying the algorithm. The ordered matrix’s shape is decisive: diagonal crossings of its empty region are associated with zero-converging fitness and complexity, while real datasets show substantially different convergence patterns.
Problem
The paper investigates how the adjacency matrix’s structure determines which Fitness and Complexity values converge to nonzero limits and how quickly convergence to zero occurs.
Method
The paper analyzes a structured matrix class analytically, formulates an ansatz, and tests the resulting convergence insight numerically on random and economic-complexity matrices.
Results
The ordered matrix’s shape is decisive: matrices whose diagonal crosses the empty part are not guaranteed to yield nonzero fixed-point values, and real datasets display widely varying numbers of zero-converging countries.
Takeaways & Limitations
Applying the algorithm requires checking matrix shape and convergence behavior because different datasets can produce finite fitness for most countries, many zero values, or nearly universal zero values.
Takeaways & Limitations
Very slow decay can make some values fall below machine precision while others are still changing, and zero fitness makes the algorithm ill defined.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper studies convergence of the Fitness–Complexity algorithm, whose outputs are determined by a country-product matrix but can extend to any bipartite network. It identifies matrix characteristics needed for strictly positive fixed-point values and develops analytical, numerical, and practical analyses.
- Economic Complexity: Fitness F measures countries’ growth potential through the quality, or Complexity Q, of their exported products.The algorithm was proposed as a revision of earlier approaches using the country-product matrix M.
- Algorithm scope: The algorithm is determined by the country-product matrix M but can be applied to any bipartite network.Countries and products are used as names for row and column elements because economic complexity is the algorithm’s main application.
- Research aim: The study analyzes matrix characteristics required for every country’s Fitness and product’s Complexity to converge to strictly positive values.It develops an analytical treatment for a class of matrices, tests the resulting insight numerically, and examines economic-complexity matrices.
- Paper scope: The paper extends its convergence analysis with an appendix addressing a wider class of similar algorithms.This generalization is presented as an additional appendix beyond the main sections.
2 A theoretical example
The theoretical example reduces the algorithm to a symmetric four-block matrix and shows that convergence depends primarily on its geometry, especially the relative areas of the internal and external blocks. The resulting cases motivate broader ansätze linking the ordered matrix’s diagonal and belly shape to whether fitnesses and complexities remain nonzero.
- 2.3 Convergence: If A2 > 1, F2 converges exponentially to zero; if A2 = 1, it converges as n^-1; and if A2 < 1, it converges exponentially to (1 − A2)/A1.Here A2 is the ratio of the external block’s area to the internal block’s area.
- 2.3 Convergence: The convergence threshold A2 = 1 corresponds geometrically to the diagonal separating the internal and external areas, while an inward belly makes the external area larger and drives F2 to zero.The same criterion is expressed as the external area being larger than the internal area.
- 2.3 Convergence: When A2 is close to 1, convergence can be extremely slow, with characteristic time n* = 1/(A2 − 1) for A2 > 1.For A2 near 1, the exponential decay is approximately proportional to e^−(A2−1)n.
- 2.4 Heterogeneous Density: Changing the densities of the nonempty blocks does not alter the convergence criterion: shape determines whether the limit is nonzero, while frontier density determines its specific value.The internal block’s positive density is irrelevant to the convergence criterion, whereas densities in frontier blocks affect the convergence point.
- 2.4.1 Zeros outside the frontier: If the internal block has zero density, the two country groups are disconnected; unequal block areas make one group’s fitness vanish, whereas equal areas leave the starting fitnesses stationary.In this case, inside and outside cannot be uniquely defined because rows and columns can be rearranged to exchange them.
- 2.5 Ansatz: The proposed ansätze state that an outward belly yields nonzero fitnesses and complexities, whereas an inward belly or a diagonal crossing the external area causes some values to converge to zero.After removing zero-converging countries and their exported products, the remaining ordered matrix is expected to have finite nonzero values when its diagonal avoids the external area.
3 A numerical investigation
Numerical simulations show that matrix structure controls whether Fitness values remain positive, decay by power laws, or decay exponentially. Inward-bellied matrices and external connections determine the surviving countries and the decay rates.
- 3 A numerical investigation: After sufficiently many iterations, fitness values may be numerically positive even when their limits are zero, while rankings stabilize once country-specific convergence speeds become constant.This distinction separates finite-iteration rankings from fixed-point convergence.
- 3.2 The importance of oligopolies: Common products can make one diversified country the only country with nonzero fitness, while other countries differ mainly in their power-law decay exponents.Once decay rates differ, the fitness ranking becomes constant after a sufficiently large iteration number.
- 3.2 The importance of oligopolies: In N-polistic competitions, external products break symmetry and produce power-law decay rates determined by the relative sizes of connected blocks.A representative block-size ratio gives exponent 3/(3 + 2) = 0.6, while an alternative configuration yields 3/4.
- 3.3 Exponential decays: A matrix whose diagonal crosses the external empty area causes all non-crossing countries to converge to zero exponentially.The crossing country is the only exception in the illustrated inward-bellied example.
- 3.4 Numerical verification of the ansatz: Inward-bellied matrices leave only the crossing country and product with nonzero limiting values, while the others decay exponentially.For matrix B, removing countries until an outward-bellied submatrix is found leaves only the trivial 1x1 crossing submatrix; outward-bellied matrix C instead supports nonzero fitness for all countries.
- 3.4 Numerical verification of the ansatz: Removing a single product can change a rectangular matrix from full convergence to zero convergence for two countries, and further removal can make the decay exponential.The example also shows that a country's fitness can change even when its own number and category of products remain unchanged, because matrix values are evaluated relationally.
4 Real cases
Applications to trade, patent, and other real matrices support the convergence ansatz: ordered-matrix geometry predicts which countries and products retain nonzero values and how many must be removed.
- 4.1 UN COMTRADE dataset, 1995–2010: Across the BACI years 1995–2010, matrices are outward bellied, so most countries converge to nonzero fitness, with exceptions among the least fit.The dataset contains 1,131 products and roughly 146–148 countries, with entries defined using an RCA threshold.
- 4 Real cases: Fewer countries and products need removal in later years to obtain a matrix converging to nonzero values, a pattern reported across all checked datasets.
- 4.2 UN COMTRADE dataset, 1963–2000: In the 1963–2000 dataset, large matrix regions lie above the diagonal, making the convergence ansatz especially relevant.Figure 2 illustrates removal of zero-converging countries and products until the remaining matrix diagonal no longer crosses the external area.
- 4.2 UN COMTRADE dataset, 1963–2000: The boundary between zero-converging and positive-converging regions is smooth in the former and rougher in the latter because exponential decay separates finite-iteration values below the crossing country.The same pattern appears in the BACI example's last four countries.
- 4 Real cases: Sparse patent matrices make convergence to zero especially prominent, and a country diversified across sectors ignored by others can drive the other countries' fitnesses toward zero.
5 Convergence in ranking: a practical note
Ranking convergence can remain unresolved for thousands of iterations because fitness decays slowly and rank crossings continue long after apparent stabilization. The paper proposes Minimum Crossing Iteration as a lower-bound stopping criterion, while warning that standard vector-difference convergence tests may be misleading.
- The usual condition |F(n) − F(n−1)| < ϵ does not ensure that individual fitness components stop decreasing, so computed values may depend on ϵ.This matters especially for nonlinear quantities such as log fitness.
- Small decay exponents can delay rank changes dramatically: αc = 0.1 requires 10^10 iterations for a tenfold fitness reduction.Similar decay rates can therefore keep rankings unstable for very long periods.
- Minimum Crossing Iteration estimates a lower bound on when the next rank change can occur, using fitted country growth rates and valid crossing estimates.The estimate is meaningful for a pair only when the predicted crossing lies after the current iteration; otherwise the fitnesses diverge.
- Fitness rankings can still switch after around 500 iterations, even when earlier trajectories appear to converge exponentially toward zero.The change in other countries’ rankings alters the apparent convergence behavior of the two sampled countries.
- The lower-bound estimate closely tracks observed crossings, with plateaux persisting until the next crossing and jumps occurring after each rank change.For ranking-only applications, the algorithm can be stopped once the predicted next change exceeds the permitted iteration budget.
- A practical limitation is that achieving a satisfying Minimum Crossing Iteration may require so many iterations that some values underflow to zero, making the algorithm ill defined.Numerical safeguards are needed when different countries or products decay at very different rates.
6 Conclusions and discussion
The paper links Fitness–Complexity convergence to the structure and ordering of the underlying bipartite matrix. It finds that empty regions and diagonal crossings can drive countries and products to zero, while ranking stabilization can be estimated practically despite very slow convergence.
- The ordered matrix’s shape determines whether countries and products can retain nonzero fixed-point values, with diagonal crossings through empty regions producing zero convergence.The condition is supported analytically for simplified matrices and numerically for random and real economic-complexity datasets.
- The analysis finds sharply different outcomes across datasets: almost all countries have finite fitness in UN COMTRADE 1995–2010, whereas over half converge to zero in many 1963–2000 years.In some patent datasets, all but one country’s fitness tends to zero.
- After the last observed crossing at iteration 1892, Figure 7 predicts the next ranking change only after 3.7·10^6 iterations, supporting an earlier stopping point for ranking-only use.Minimum Crossing Iteration behaves as an increasing step-like lower bound that jumps after crossings.
A A generalization to a wider class of algorithms
The generalized algorithm varies the harmonic-mean parameter γ, but avoiding zero fitness values imposes a trade-off between high- and low-fitness countries. Changing γ cannot overcome a frontier that rises above the diagonal in the middle.
- For γ = −1, the generalized equation recovers the original algorithm, whereas γ = 1 turns it into a symmetric sum.The γ = 1 case is excluded because it would assign greater complexity to products made by more countries.
- The proposed variations change the parameter conditions defining the boundary between convergence to nonzero values and convergence to zero.For the alternative formulation, A2 = 1 identifies a curve rather than the diagonal line.
- Changing γ trades a looser condition for high-fitness countries against a stricter condition for low-fitness countries, and conversely.
- When the frontier lies above the diagonal in the middle, changing γ does not prevent convergence of some fitness values to zero.