# Luciano Pietronero's research while affiliated with Centro FERMI and other places

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## Publications (69)

A central problem in economics and statistics is the assessment of income or wealth inequality starting from empirical data. Here we focus on the behavior of Gini index, one of the most used inequality measures, in presence of Zipf’s law, a situation which occurs in many complex ﬁnancial and economical systems. First, we show that the application o...

We reply to the criticisms moved in [1] against our results presented in [2]. In particular, we show that our fractal model has none of the problems claimed in [1]. The latters can be addressed to the overlooked nonlinear behaviour of the Einstein's equations.

Economic complexity methods, and in particular relatedness measures, lack a systematic evaluation and comparison framework. We argue that out-of-sample forecast exercises should play this role, and we compare various machine learning models to set the prediction benchmark. We find that the key object to forecast is the activation of new products, a...

We present an integrated database suitable for the investigation of the economic development of countries by using the Economic Fitness and Complexity framework. Firstly, we implement machine learning techniques to reconstruct the export flow of services and we combine them to the export flow of the physical goods, generating a complete view of the...

Many natural and socioeconomic systems are characterized by power-law distributions that make the occurrence of extreme events not negligible. Such events are sometimes referred to as Black Swans, but a quantitative definition of a Black Swan is still lacking. Here, by leveraging on the properties of Zipf-Mandelbrot law, we investigate the relation...

We reconstruct the innovation dynamics of about two hundred thousand companies by following their patenting activity for about ten years. We deﬁne the technological portfolios of these companies as the set of the technological sectors present in the patents they submit. By assuming that companies move more frequently towards related sectors, we lev...

When analyzing countries’ medium- and long-term economic performance, it is important to study jointly the dynamic of growth and the industrial evolution that determines how the productive structure changes over time. In this paper, we use the Economic Fitness metric to describe the competitiveness of the countries' industrial structure, and classi...

A key element to understand complex systems is the relationship between the spatial scale of investigation and the structure of the interrelation among its elements. When it comes to economic systems, it is now well-known that the country-product bipartite network exhibits a nested structure, which is the foundation of different algorithms that hav...

Many natural and socio-economic systems are characterized by power-law distributions that make the occurrence of extreme events not negligible. Such events are sometimes referred to as Black Swans, but a quantitative definition of a Black Swan is still lacking. Here, by leveraging on the properties of Zipf-Mandelbrot law, we investigate the relatio...

We reconstruct the innovation dynamics of about two hundred thousand companies by following their patenting activity for about ten years. We define the technological portfolios of these companies as the set of the technological sectors present in the patents they submit. By assuming that companies move more frequently towards related sectors, we le...

We present an integrated database suitable for the investigations of the Economic development of countries by using the Economic Fitness and Complexity framework. Firstly, we implement machine learning techniques to reconstruct the database of Trade of Services and we integrate it with the database of the Trade of the physical Goods, generating a c...

Economic complexity methods, and in particular relatedness measures, lack a systematic evaluation and comparison framework. We argue that out-of-sample forecast exercises should play this role, and we compare various machine learning models to set the prediction benchmark. We find that the key object to forecast is the activation of new products, a...

The statistical characterization of the distribution of visible matter in the universe is a central problem in modern cosmology. In this respect, a crucial question still lacking a definitive answer concerns how large the greatest structures in the universe are. This point is closely related to whether or not such a distribution can be approximated...

The statistical characterization of the distribution of visible matter in the universe is a central problem in modern cosmology. In this respect, a crucial question still lacking a definitive answer concerns how large are the greatest structures in the universe. This point is closely related to whether or not such a distribution can be approximated...

Relatedness is a quantification of how much two human activities are similar in terms of the inputs and contexts needed for their development. Under the idea that it is easier to move between related activities than towards unrelated ones, empirical approaches to quantify relatedness are currently used as predictive tools to inform policies and dev...

The rank-size plots of a large number of different physical and socio-economic systems are usually said to follow Zipf's law, but a unique framework for the comprehension of this ubiquitous scaling law is still lacking. Here we show that a dynamical approach is crucial: during their evolution, some systems are attracted towards Zipf's law, while ot...

This paper applies the Economic Fitness and Complexity approach to analyze the underlying factors behind the wide and persistent economic disparities across the Italian regional units. Measures of regional fitness are obtained from their revealed comparative advantage and their patent performance. Southern regions tend to be characterised by a lowe...

Predicting innovation is a peculiar problem in data science. Following its definition, an innovation is always a never-seen-before event, leaving no room for traditional supervised learning approaches. Here we propose a strategy to address the problem in the context of innovative patents, by defining innovations as never-seen-before associations of...

A recent paper by Hausmann and collaborators (1) reaches the important conclusion that Complexity-weighted diversification is the essential element to predict country growth. We like this result because Complexity-weighted diversification is precisely the first equation of the Fitness algorithm that we introduced in 2012 (2,3). However, contrary to...

Rank size plots of very different systems are usually fitted with Zipf's law, however, one often observes strong deviations at large sizes. We show that these deviations contain essential and general information on the evolution and the intrinsic cutoffs of the system. In particular, if the first ranks show deviations from Zipf's law, the empirical...

We show that the space in which scientific, technological and economic developments interplay with each other can be mathematically shaped using pioneering multilayer network and complexity techniques. We build the tri-layered network of human activities (scientific production, patenting, and industrial production) and study the interactions among...

[This corrects the article DOI: 10.1371/journal.pone.0197616.].

In this paper we attempt to answer the question: can cosmic acceleration of the universe have a fractal solution? We give an exact solution of a Lemaitre-Tolman-Bondi (LTB) universe based on the assumption that such a smooth metric is able to describe, on average, a fractal distribution of matter. While the LTB model has a center, we speculate that...

The advent of social networks revolutionized the way people access to information sources. Understanding the complex relationship between these sources and users is crucial. We introduce an algorithm, that we call PopRank, to assess both the Impact of Facebook pages as well as users’ Engagement on the basis of their mutual interactions. The ideas b...

Two-dimensional transition-metal dichalcogenides MX2(MoS2,WS2,MoSe2,...) are among the most promising materials for band-gap engineering. Widely studied in these compounds, by means of ab initio techniques, is the possibility of tuning the direct-indirect gap character by means of in-plane strain. In such kind of calculations however the lattice de...

This work contributes to the literature in the field of innovation by proposing a quantitative approach for the prediction of the timing and location of patenting activity. In a recent work, it was shown that focusing on couples of technological codes allows for the formation of testable predictions of innovation events, defined as the first time t...

In this paper we attempt to answer to the question: can cosmic acceleration of the Universe have a fractal solution? We give an exact solution of a Lema\^itre-Tolman-Bondi (LTB) Universe based on the assumption that such a smooth metric is able to describe, on average, a fractal distribution of matter. While the LTB model has a center, we speculate...

We present a new metric estimating fitness of countries and complexity of products by exploiting a non-linear non-homogeneous map applied to the publicly available information on the goods exported by a country. The non homogeneous terms guarantee both convergence and stability. After a suitable rescaling of the relevant quantities, the non homogen...

We present a new method of estimating fitness of countries and complexity of products by exploiting a non-linear non-homogeneous map applied to the publicly available information on the goods exported by a country. The non homogeneous terms guarantee both convergence and stability. After a suitable rescaling of the relevant quantities, the non homo...

Users online tend to acquire information adhering to their system of beliefs and to ignore dissenting information. Such dynamics might affect page popularity. In this paper we introduce an algorithm, that we call PopRank, to assess both the Impact of Facebook pages as well as users' Engagement on the basis of their mutual interactions. The ideas be...

We present a non-linear non-homogeneous fitness-complexity algorithm where the presence of non homogeneous terms guarantees both convergence and stability. After a suitable rescaling of the relevant quantities, the non homogeneous terms are eventually set to zero so that this new method is parameter free. This new algorithm reproduces the findings...

In this paper we introduce a novel algorithm, called Exogenous Fitness, to calculate the Fitness of subnational entities and we apply it to the states of Brazil. In the last decade, several indices were introduced to measure the competitiveness of countries by looking at the complexity of their export basket. Tacchella et al (2012) developed a non-...

[This corrects the article DOI: 10.1371/journal.pone.0177360.].

Additional supporting information.
Here we report and discuss a number of auxiliary results supporting the main findings of the principal paper and assess the robustness of our calculations.
(PDF)

We study the hopping-resolved contribution of the tight-binding parameters to the Van Hove singularities of H\(_3\)S in the \(Im-3m\) cubic structure at 180 GPa close to the Fermi level. By means of a \(7\times 7\) tight-binding model, we study the sensitivity of the DOS close to the Fermi level to a change in the tight-binding parameters. We show...

Robustness check: A Gini Coefficient of wage inequality between sectors.
In this appendix we test the robustness of our results and show that they are unaltered when measuring wage inequality among sectors within countries/counties with a Gini coefficient instead of a Theil index.
(PDF)

Shapley values.
In this appendix we describe the procedure used to compute the Shapley values that quantify the sectoral contributions to overall wage inequality that are shown in Fig 11.
(PDF)

Colorblind friendly figures.
In this appendix we reproduce all the figures of the paper in a palette more accessible to red-green blind people.
(PDF)

This note is a contribution to the debate about the optimal algorithm for Economic Complexity that recently appeared on ArXiv [1, 2] . The authors of [2] eventually agree that the ECI+ algorithm [1] consists just in a renaming of the Fitness algorithm we introduced in 2012, as we explicitly showed in [3]. However, they omit any comment on the fact...

The observation of a superconducting critical temperature \(T_\mathrm{c}\) exceeding 200K in ultra-dense hydrogen sulfide has demonstrated that high-\(T_\mathrm{c}\) superconductivity can be achieved also in compounds where the superconducting pairing is mediated by phonons. This poses interesting challenges and opportunities. In particular, in thi...

Recently a measure for Economic Complexity named ECI+ has been proposed by Albeaik et al. We like the ECI+ algorithm because it is mathematically identical to the Fitness algorithm, the measure for Economic Complexity we introduced in 2012. We demonstrate that the mathematical structure of ECI+ is strictly equivalent to that of Fitness (up to norma...

We study the relationship between firms' performance and their technological portfolios using tools borrowed from the complexity science. In particular, we ask whether the accumulation of knowledge and capabilities related to a coherent set of technologies leads firms to experience advantages in terms of productive efficiency. To this end, we analy...

Comparison between the 1-year average velocity field, v→, and the field obtained from the gradient of H defined in the paper.
Calculations done on the BACI dataset. Top: comparison between the horizontal and vertical components of the field, the latter regressed with RANSAC. Bottom: Comparison between orientations and moduli of the field.
(PDF)

Bootstrapping the regression of v→y versus -∇→kH at 1 year displacement time and 10×10 grid.
Each iteration of the bootstrap randomly removed 10% of the samples, for 50k iterations. Two peaks are clearly visible: one at about R2 = 4, and one at about .7. The bimodal distribution confirms our hypothesis that the points follow a very clear linear tre...

Comparison between the 1-year average velocity field, v→, and the field obtained from the gradient of H.
Calculations done on the Feenstra dataset. Top: comparison between the horizontal and vertical components of the field. In the left panel, showing the regression for velocities along the logPRODY axis, the points removed by the RANSAC regression...

The field obtained by calculating the gradient of the HR field.
(PDF)

A plot of the R2 values obtained by regressing v→y versus -∇→kH at various resolution and displacement time values, Feenstra dataset.
On the horizontal axis there is the resolution of the grid, with the number of boxes per side. In blue to red color, we show the time interval used to calculate the products’ displacements, which are then averaged in...

In green, the field obtained by regressing the average value of U (Ubiquity) per box in the RCLP plane.
It is different from the field obtained by regressing H, because the average Ubiquity of products decreases with increasing Complexity; therefore the left part of the RCLP plane has higher values than the right. The plot shows the regression for...

Comparison between the 1-year average velocity field, v→, and the field obtained from the gradient of S.
Calculations done on the Feenstra dataset. Top: comparison between the horizontal and vertical components of the field Bottom: Comparison between orientations and moduli of the field.
(PDF)

All individual data points of the BACI dataset.
One can clearly see higher density on the diagonal, and a decreasing number of points in the upper left and lower right quadrants of the RCLP plane.
(TIF)

Both w→ and v→ fields, to scale with each other.
The figure refers to the BACI dataset.
(PDF)

“inward” velocity field w→.
To calculate this field, considered all products that entered a given box, and averaged their displacements. The vectors are not to scale with those shown in the rest of the paper. To allow comparison, we present a depiction of both w→ and v→ fields in the next figure. The figure refers to the BACI dataset.
(PDF)

A scatter-plot of the Herfindahl index vs. the entropy value for individual products’ market share distributions, from the BACI dataset.
(PDF)

Left: Relation between values of the S and H field in the BACI dataset. Left, in black, a linear fit, which is good enough for the fields, that aggregate many individual points. The actual relationship between entropy and Herfindahl index of a distribution is not linear, as we show in the next section. Right: The field obtained by using S.
(PDF)

Panel a) Maximum value of the H field. Given that the less points one has per box, the higher the standard deviation for H in that box is, one expects to find higher values of H where the density of points is lowest on the RCLP plane. We check whether the null model can produce a H˜ field with the same highest value found in H, which we will call m...

v→ and -∇→kH fields for the BACI dataset.
The results are extremely similar to those shown in the paper for the Feenstra dataset, with great consistency across datasets. Panel a). In red, the average velocity field of the products on the RCLP plane, v→. The asymptotic zone is marked by the orange line, together with the bootstrap result for the 5%...

Occupation number timeseries for each box onb the RCLP plane.
Each of the plots in this figure shows the evolution in the number of products contained in each box. The horizontal axis of the plots represents time in years, and the vertical axis the number of points. The plots are to scale relative to each other. The figure refers to the BACI datase...

Average RCA weights per box, BACI dataset.
For each box b we plot a histogram showing the average R˜ values of countries exporting the products contained in b. Each bar of the histograms shows the average RCA of countries with a fitness value between two consecutive deciles of the fitness distribution. We remind that R˜ represents the share of a pr...

Inward (blue) and outward (red) fluxes for each box on the RCLP plane.
Each of the plots in this figure shows the yearly change in the number of products contained in each box. The horizontal axis of the plots represents time in years, and the vertical axis the difference in number of points (for the outward flux the difference is negative). The pl...

Comparison between the 1-year average velocity field, v→, and the field obtained from the gradient of HR.
Calculations done on the Feenstra dataset. Top: comparison between the horizontal and vertical components of the field Bottom: Comparison between orientations and moduli of the field.
(PDF)

Dataset, Basic Solow Model of Growth and Growth Accounting, Robustness checks with the 1995–2010 database.
This appendix has three roles: i) to describe the characteristics of the datasets used; ii) to explain a basic Solow Model, necessary to understand how to empirically account the growth components, dividing the total GDP growth in growth due t...

CIRD Values.
The values of the CIRD index for all the nations and years used in the paper.
(CSV)

We analyze global export data within the Economic Complexity framework. We couple the new economic dimension Complexity, which captures how sophisticated products are, with an index called logPRODY, a weighted average of the Gross Domestic Products per capita of a product's exporters. Products' aggregate motion is treated as a 2-dimensional dynamic...

Hydrogen-rich compounds are extensively explored as candidates for a
high-temperature superconductors. Currently, the measured critical temperature
of 203 K in hydrogen sulfide (H3S) is among the highest over all-known
superconductors. In present paper, using the strong-coupling Eliashberg theory
of superconductivity, we compared in detail the ther...

By borrowing methods from complex system analysis, in this paper we analyze the features of the complex relationship that links the development and the industrialization of a country to economic inequality. In order to do this, we identify industrialization as a combination of a monetary index, the GDP per capita, and a recently introduced measure...

We present a robust tight-binding description, based on the Slater-Koster
formalism, of the band structure of H$_3$S in the $Im\bar{3}m$ structure,
stable in the range of pressure $P = 180-220$ GPa. We show that the interatomic
hopping between the 3$p$ sulphur orbitals is fundamental to capture the
relevant physics associated with the Van Hove sing...

## Citations

... For instance, having in mind an urban system, its elements would be the cities composing it and as size one can use different measures ranging from population to light emission [7]. Also most financial and economical systems are generally described by Zipf 's scaling, which is found for instance in the distribution of returns, of stock prices, of the cryptocurrency market and of people' wealth [5,[8][9][10][11][12], this last being the first system where power law distributions were observed [13]. This explains a number of very counter-intuitive properties typical of financial and economics data. ...

... In such a way, two products can be defined as close in the sense that they share many of the capabilities needed in order to export them in a competitive way. Co-occurrences based approaches have however a low predictive performance, and this fact favors machine learning approaches as better tools to measure relatedness both at country (18)(19)(20) and firm level (21,22). In (16,23,24), the authors proposed approaches to explicitly model the relationship among products, capabilities, and development. ...

... In order to address some of the perceived weaknesses of the Economic Complexity framework's 'linear' approach, an enhanced method for the measurement of economic complexity was developed by Pietronero, Tacchella, Calderalli and their collaborators [9][10][11][13][14][15]. This research [13] makes use of crude oil as an anecdotal example of the inconsistency of Economic Complexity as a measure to rank the competitiveness of countries. ...

... Many studies recently relied on network-based techniques to unfold the complex interplay among patents, technological codes and geographical reference areas. Network science techniques allowed to analyse economic activities of countries 34 , regions [35][36][37][38][39] , cities 2,40-42 or firms 43,44 . ...

... Recent studies have shown that it is helpful to calculate the Fitness of sub-national actors using the complexity that comes from the national systems 54,79,80 . This measure is called exogenous Fitness and overcomes the issue of the limited capabilities of sub-national entities, such as cities or MAs in our case. ...

... Zipf 's law is a characteristic feature of complex systems and is often considered as a footprint of complexity [1][2][3]. Such scaling law is observed in an astonishing number of natural and social systems characterized by emerging complex behaviors, such as solar flares, cosmic structures, earthquakes, language, urban systems and many more [2,[4][5][6]. ...

... Zipf 's law is a characteristic feature of complex systems and is often considered as a footprint of complexity [1][2][3]. Such scaling law is observed in an astonishing number of natural and social systems characterized by emerging complex behaviors, such as solar flares, cosmic structures, earthquakes, language, urban systems and many more [2,[4][5][6]. Denoting by S(k) the size of the kth largest element in the system, Zipf 's law reads ...

... In (16,23,24), the authors proposed approaches to explicitly model the relationship among products, capabilities, and development. These frameworks naturally lead to the concepts of product progression (16,19,25) and arrow of development (26): the relationship between products is often not undirected, or symmetric, as in the product space (10), but directed: countries starts their development from simple products and gradually enter in more sophisticated markets, following well defined paths of development (16). Obviously, the identification of the specific products enabling countries to competitively export a given target product is a key element to design industrial policies and strategic patterns of development. ...

... Obviously, these modelling exercises face two very difficult challenges: they have to work in different country contexts, and they have to work at all times, hence predicting growth potentials and not only recoveries from shocks (Cristelli et al. 2017). In the wake of COVID-19, institutional forecasters turned to their models and tried to adapt them to the rapidly changing circumstances. ...

... Technological innovation is often highlighted as one of the main drivers for evolution and change in cities, and it has been shown that complex economic activities flourish in large urban areas 11 , although remote working and dispersed research teams can mitigate the concentration of innovation in urban areas 12-15 . In parallel, many studies recently focused on how innovation diffuses temporally and spatially [16][17][18] , also with a particular focus on cities 19 . In this paper, we focus on technological innovation, and we investigate how the technological fingerprints of cities can affect their development and potential. ...