Abstract
Virtually anything can be and is ranked; people, institutions, countries, words, genes. Rankings reduce complex systems to ordered lists, reflecting the ability of their elements to perform relevant functions, and are being used from socioeconomic policy to knowledge extraction. A century of research has found regularities when temporal rank data is aggregated. Far less is known, however, about how rankings change in time. Here we explore the dynamics of 30 rankings in natural, social, economic, and infrastructural systems, comprising millions of elements and timescales from minutes to centuries. We find that the flux of new elements determines the stability of a ranking: for high flux only the top of the list is stable, otherwise top and bottom are equally stable. We show that two basic mechanisms — displacement and replacement of elements — capture empirical ranking dynamics. The model uncovers two regimes of behavior; fast and large rank changes, or slow diffusion. Our results indicate that the balance between robustness and adaptability in ranked systems might be governed by simple random processes irrespective of system details.
Introduction
Rankings are everywhere. From country development indices, academic indicators, and candidate poll numbers to music charts and sports scoreboards, rankings are key to how humans measure and make sense of the world^{1,2}. The ubiquity of rankings stems from the generality of their definition: reducing the (often highdimensional) complexity of a system to a few or even a single measurable quantity of interest^{3,4}, dubbed score, leads to an ordered list where elements are ranked, typically from highest to lowest score. Rankings are, in this sense, a proxy of relevance or fitness to perform a function in the system. Rankings are being used to identify the most accomplished individuals or institutions, and to find the essential pieces of knowledge or infrastructure in society^{1}. Since rankings often determine who gets access to resources (education, jobs, and funds), they play a role in the formation of social hierarchies^{5,6} and the potential rise of systematic inequality^{7}.
The statistical properties of ranking lists have caught the attention of natural and social scientists for more than a century. A heavytailed decay of score with rank, commonly known as Zipf’s law^{8,9}, has been systematically observed in the ranking of cities by population^{10,11}, words and phrases by frequency of use^{12,13,14,15,16,17}, companies by size^{18,19,20}, and many features of the Internet^{21}. Zipf’s law appears even in the scorerank distributions of natural systems, such as earthquakes^{22,23}, DNA sequences^{24}, and metabolic networks^{25}. Rankings have also proven useful when analyzing productivity and impact in science and the arts^{7,26,27,28,29}, in human urban mobility^{30,31,32}, epidemic spreading by influentials^{33}, and the development pathways of entire countries^{34}. Recently, studies of language use^{17,35}, sports performance^{36}, and many biological and socioeconomic rankings^{37} have strengthened the notion of universality suggested by Zipf’s law: despite microscopic differences in elements, scores, and types of interaction, the aggregate, macroscopic properties of ranking lists are remarkably similar throughout nature and society.
The similarity of scorerank distributions across systems raises the question of the existence of simple generative mechanisms behind them. While mechanisms of proportional growth^{38}, cumulative advantage^{39}, and preferential attachment^{40} are often used to explain the heavytailed distributions of ranking lists at single points of time^{41,42}, they fail to reproduce the way elements actually move in rank^{43}, such as the sudden changes in city size throughout history^{44,45}. Here, we report on the existence of generic features of rank dynamics over a wide array of systems, from individuals to countries, and spatiotemporal scales, from minutes to centuries. By measuring the flux of elements across ranking lists^{46,47,48}, we identify a continuum ranging from systems where highly ranked elements are more stable than the rest, to systems where the least relevant elements are also stable. We show that simple mechanisms relying on fluxes generated by displacement and replacement of elements can account for all observed patterns of rank stability. A model based on these ingredients uncovers two regimes in rank dynamics, a fast regime driven by long jumps in rank space, and a slow one driven by diffusion.
Results
We gather 30 ranking lists in natural, social, economic, and infrastructural systems. Data include human and animal groups, languages, countries and cities, universities, companies, transportation systems, online platforms, and sports, with no selection criteria other than having enough information for analysis (for data details see Supplementary Information [SI] Section S2 and Table S1). Elements in each list are ranked by a measurable score that changes in time: scientists by citations, businesses by revenue, regions by a number of earthquakes, players by points, etc. Size and temporal scales in the data vary widely, from the number of people in 636 station entrances of the London Underground every 15min during a week in 2012^{49}, to the written frequency of 124k English words every year since the 17th century^{50}. Following an element’s rank through time reveals systematic patterns (Fig. 1). For example, in the Academic Ranking of World Universities (ARWU)^{51}, published yearly since 2003, institutions like Harvard and Stanford maintain a high score, while institutions down the list change rank frequently (Fig. 1a).
Ranking lists typically have a fixed size N_{0} (e.g., the Top 100 universities^{51}, the Fortune 500 companies^{52}), so elements may enter or leave the list at any of the T observations t = 0, …, T − 1, allowing us to measure the flux of elements across rank boundaries^{42,46,47} (for the observed values of N_{0}, T see SI Table S1). We introduce two timedependent measures of flux: the rank turnover o_{t} = N_{t}/N_{0}, representing the number N_{t} of elements ever seen in the ranking list until time t relative to the list size N_{0}, and the rank flux F_{t}, representing the probability that an element enters or leaves the ranking list at time t. Rank turnover is a monotonic increasing function indicating how fast new elements reach the list (Fig. 1b left; all datasets in SI Fig. S5). In turn, flux shows striking stationarity in time despite differences in temporal scales and potential shocks to the system (SI Fig. S3). By averaging over time, the mean turnover rate \(\dot{o}=({o}_{T1}{o}_{0})/(T1)\) and the mean flux F = 〈F_{t}〉 turn out to be highly correlated quantities that uncover a continuum of ranking lists (Fig. 1b right; values in SI Table S2). In one extreme, the most open systems (\(F,\dot{o} \sim 1\)) have elements that constantly enter and leave the list. Less open systems (\(F,\dot{o} \sim 0\)) have a progressively lower turnout of constituents. Five out of 30 ranking lists are completely closed (\(F=\dot{o}=0\)), meaning no single new element is recorded during the observation window.
The measures of rank turnover and flux reveal regularities in the stability of ranking processes^{43,53}. We follow the time series of the rank R_{t} occupied by a given element at time t^{44} (Fig. 1c; all datasets in SI Fig. S2). In most systems, highly ranked elements like Harvard University and the English word ‘the’ never change position, showcasing the correspondence between rank stability and notions of relevance like academic prestige^{7,27}, grammatical function^{17,50}, and underlying network structure^{53}. As we go down the ranking list of open systems, rank trajectories increasingly fluctuate in time. In the least open systems where turnover and flux are low, however, low ranked elements are also stable. In the ranking of British cities by population, for example, both Birmingham and Nairn remain the most and least populated local authority areas throughout the 20th century^{54}. These findings uncover a more finegrained sense of rank stability: most systems have a stable top ranking, but only the least open systems feature stable bottom ranks as well. The rank change C, measured as the average probability that element at rank R changes between times t − 1 and t, varies between an approximately monotonic increasing function of R for open systems to a symmetric shape as systems become less open (Fig. 1d; all datasets in SI Fig. S6).
Since the stability patterns of an empirical ranking list (as measured by rank change C) can be systematically connected to the number of elements flowing into and out of the list, we build a model of rank dynamics based solely on simple generative mechanisms of flux (Fig. 2). Without assuming systemspecific features of elements or their interactions, there are at least two ways to implement flux in rank space. Smooth (but arbitrarily large) changes in the score of an element might make it larger or smaller than other scores, causing elements to move across ranks (the way some scientists gather more citations than others^{27}, or how population size fluctuates due to historical events^{44}). Regardless of the score, elements might also disappear from the list and be replaced by new elements: young athletes enter competitions while old ones retire^{36}; new words replace anachronisms due to cultural shifts^{50}. We implement random mechanisms of displacement and replacement in a simple model by considering a synthetic ranking list of length N_{0} embedded within a larger system of size N ≥ N_{0}. At each time step of length Δt = 1/N, a randomly chosen element moves to a randomly selected rank with probability τ, displacing others. At the same time, a randomly chosen element gets replaced by a new one with probability ν, leaving other ranks untouched. The dynamics involve all N elements, but to mimic real ranking lists, we only consider the top N_{0} ranks when comparing with empirical data (Fig. 2a; model details in SI Section S4).
We solve the model analytically by introducing the displacement probability P_{x,t} that an element with rank r = R/N gets displaced to rank x = X/N after a time t (Fig. 2b; uppercase/lowercase symbols denote integer/normalized ranks). Since for small Δx = 1/N the probability that at time t an element has not yet been replaced is e^{−νt}, we have
Here, L_{t} = (1 − e^{−τt})/N is the (rankindependent) probability that up until time t an element gets selected and jumps to any other rank. The length of jumps is uniformly distributed, so they can be thought of as a Lévy random walk with step length exponent 0^{55} (full derivation in SI Section S4). The probability D_{x,t} = D(x, t)Δx that the element in rank r gets displaced to rank x after a time t (due to Lévy walks of other elements) follows approximately the diffusionlike equation
where α = τ/N. Since ∑_{x}D_{x,t} = e^{−τt}, both D_{x,t} and D(x, t) are not conserved in time. Instead of a standard diffusion equation, Eq. (2) is equivalent to the WrightFisher equation of random genetic drift in allele populations^{56,57}. The solution D(x, t) of Eq. (2) is well approximated by a decaying Gaussian distribution with mean r and standard deviation \(\sqrt{2\alpha r(1r)t}\), i.e., a diffusion kernel (Fig. 2b). Overall, local displacement makes elements slowly diffuse around their initial rank, while Lévy walks and the replacement dynamics reduces exponentially the probability that old elements remain in the ranking list.
An explicit expression for the displacement probability P_{x,t} allows us to derive the mean flux
and the mean turnover rate
where p = N_{0}/N is the length of the ranking list relative to system size (see SI Section S4). In order to fit the model to each empirical ranking list, we obtain N_{0} from the data and approximate N = N_{T−1} as the number of distinct elements ever seen in the list during the observation period T, thus fixing p (values for all datasets in SI Tables S1–S2). The remaining free parameters τ and ν (regulating the mechanisms of displacement and replacement) come from numerically solving Eqs. (3)–(4) with F and \(\dot{o}\) fixed by the data (Figs. 1b and 2c; for model fitting see SI Section S5). The approximations in Eqs. (3)–(4) introduce a small bias in the estimation of τ (SI Fig. S18). Despite this bias, the simple generative mechanisms of flux in the model are enough to recover the behavior of ranking lists as quantified by P_{x,t} and C (Fig. 2d and SI Fig. S19): When rank flux is low, both the top and bottom of the list are similarly stable and rank dynamics is mostly driven by an interplay between Lévy walks and diffusion. As systems become more open, however, this symmetry gets broken due to a growing flux of elements at the bottom of the ranking list (see SI Fig. S4). Regardless of whether we rank people or animals, words or countries, the pattern of stability across a ranking list is accurately emulated by random mechanisms of flux that disregard the microscopic details of the individual system.
The characterization of flux in ranking lists with mechanisms of displacement and replacement of elements reveals regimes of dynamical behavior that are not apparent from the data alone (Fig. 3). By rescaling the fitted parameter values of the model as
most open ranking lists (\(F,\dot{o}\ { > }\ 0\)) are predicted to follow the universal curve
which suggests that ranking dynamics are regulated by a single effective parameter [Fig. 3a; derivation in SI Section S5; for a discussion of the role of fluctuations on the validity of Eq. (6) see SI Figs. S18–S20]. Even if, potentially, displacement and replacement could appear in any relative quantity, adjusting the model to observations of rank flux and turnover (Fig. 1b) leads to an inverse relationship between parameters regulating their generative mechanisms. Realworld ranking lists lie in a spectrum where their dynamics is either mainly driven by score changes that displace elements in rank (high τ_{r} and low ν_{r}, like for GitHub software repositories^{58} ranked by daily popularity), or by birthdeath processes triggering element replacement (low τ_{r} and high ν_{r}, like for Fortune 500 companies^{52} ranked by yearly revenue). While the symmetry (or lack thereof) in rank change C may seemingly imply two distinct classes of systems (see Figs. 1d and 2d), Eq. (6) reveals the existence of a continuum of open ranking lists, which can be captured by a single model with a single effective parameter.
Data on empirical ranking lists is constrained by the average time length ℓ between recorded observations, which varies from minutes to years depending on the source and intended use of the rankings (ℓ for all datasets is listed in SI Table S1). We explore such scoping effect by subsampling data every k observations (for details see SI Section S5 and SI Figs. S21–S22). Longer times between snapshots of the ranking list lead to an increase in rank flux, turnover, and fitted parameters, such that the rate of element replacement ν/kℓ stays roughly constant (Fig. 3b top). A conserved replacement probability per unit time, robust to changes in sampling rate, is yet another measure of rank stability: online social systems exchange elements frequently (e.g., the ranking lists by daily popularity of both GitHub software repositories and of online readers of the British newspaper The Guardian^{59}), followed by sports, while languages are the most stable (values for k = 1 in SI Table S2).
The universal curve in Eq. (6) displays three regimes in the dynamics of open ranking lists, as measured by the average probabilities that, between consecutive observations in the data, an element performs either a Lévy walk [W_{levy} = e^{−ν}(1 − e^{−τ})], changes rank by diffusion [W_{diff} = e^{−ν}e^{−τ}], or is replaced [W_{repl} = 1 − e^{−ν}], with W_{levy} + W_{diff} + W_{repl} = 1. In systems with the largest rank flux and turnover (GitHub repositories and The Guardian readers), elements tend to change rank via long jumps, following a Lévy walk, where W_{levy} > W_{diff}, W_{repl} (Fig. 3c). Here, longrange rank changes take elements in and out of a short ranking list within a big system (low p), thus generating large mean flux F (see SI Table S2). Most datasets, like the yearly rankings of scientists by citations in American Physical Society journals^{27,60} and of countries by economic complexity^{34,61}, belong instead to a diffusion regime with W_{diff} > W_{levy}, W_{repl} (Fig. 3d). In this regime, local, diffusive rank dynamics is the result of elements smoothly changing their scores and overcoming their neighbors in rank space. Under subsampling, ranking lists move downwards along the universal curve, going from a state with a certain number of Lévy walks to one more driven by diffusion (Fig. 3b bottom; all datasets in SI Fig. S22).
The model also predicts a third regime dominated by replacement (W_{repl} > W_{levy}, W_{diff}; Fig. 3e), where elements are more likely to disappear than change rank. Such ranking lists replace most constituents from one observation to the next, forming a highly fluctuating regime that we do not observe in empirical data. To showcase the crossover between regimes, we simulate the model along the universal curve of Eq. (6) while keeping p and \(\dot{o}\) fixed in Eq. (5) (lines in Fig. 3c–e). These curves show how close systems are to a change of regime, i.e., from one dominated by Lévy walks to one driven by diffusion. When a ranking list is close to a regime boundary, external shocks (amounting to variations in parameters τ and ν) may change the main mechanism behind rank dynamics, thus affecting the overall stability of the system.
Discussion
Ranking lists reduce the elements of highdimensional complex systems into ordered values of a summary statistic, allowing us to compare seemingly disparate phenomena in nature and society^{2,42}. The diversity of their components (people, animals, words, institutions, and countries) stands in contrast with the statistical regularity of scorerank distributions when aggregated over time^{9,37}. By exploring the flux of elements of 30 ranking lists in natural, social, economic, and infrastructural systems, we present evidence of generic temporal patterns of rank dynamics. While open systems (large flux) keep the same elements only in top ranks, less open systems (lower flux) also have stable bottom ranks, forming a continuum of ranking lists explained by a single class of models. The model reveals two regimes of dynamical behavior for systems with nonzero flux. Realworld ranking lists are driven either by Lévy random walks^{55} that change the rank of elements abruptly or by a more local, diffusive movement similar to genetic drift^{56,57}, both alongside a relatively low rate of element replacement independent from the frequency at which the ranking list is measured.
Our results suggest that, even though score distributions differ across systems depending on what type of elements and interactions they have (SI Fig. S1), ranking lists have similar stability features. What are the underlying properties of the system that enhance this similarity? An extension of our model explicitly considering the links between score and rank may help further understand the experimental evidence in this area, like the recent observation that the stability of crowdsourced rankings depends on the magnitude difference between quality scores^{62}. It is also interesting to consider the observed deviations from the predictions of our model, even at the level of ranks. Rank flux for languages is not constant but decreases over time (SI Fig. S3), arguably due to the long observation period (over three centuries; see SI Table S1), variations in word sampling across decades, or even cognitive distortions at the societal scale^{63}. The rankdependence of flux for very open systems (SI Fig. S4) and the slow decay of inertia with long times we observe in most datasets (SI Fig. S7) might be better reproduced by a nonuniform sampling of elements in the mechanisms of displacement and replacement of the model. Finally, deviations in the data (indicating a departure from the assumptions of randomness and stationarity built into our model) could be used to detect shocks to the system larger than expected statistical fluctuations, such as the sudden increase in the rank flux of Fortune 500 companies during financial crises (SI Fig. S3).
A more nuanced understanding of the generic features of ranking dynamics might help us limit resource exhaustion in competitive environments, such as information overload in online social platforms and prestige biases in scientific publishing^{64}, via better algorithmic rating tools^{65}. The observation of a systematic interplay between “slower” and “faster” ranking dynamics^{66} (see SI Section S6) can be refined by exploring the relationship between ranking lists associated with the same system, or by incorporating networked interactions that lead to macroscopic ordering^{44,67}, which may provide a deeper understanding of network centrality measures based on ranking^{68}. Given that rankings often mediate access to resources via policy, similar mechanisms to those explored here may play a role in finding better ways to avoid social and economic disparity. In general, a better understanding of rank dynamics is promising for regulating systems by adjusting their temporal heterogeneity.
Data availability
For data availability, see SI Section S2. Nonpublic data is available from the authors upon reasonable request.
Code availability
Code to reproduce the results of the paper is publicly available at https://github.com/iniguezg/Farranks^{69}.
References
Érdi, P. Ranking: The Unwritten Rules of The Social Game We All Play. (Oxford University Press, 2019).
Langville A. N. & Meyer, C. D. Who’s #1?: The Science of Rating and Ranking. (Princeton University Press, 2012).
Diamond, J. Guns, Germs, and Steel. (W. W. Norton & Company, 1997).
Turchin, P. et al. Quantitative historical analysis uncovers a single dimension of complexity that structures global variation in human social organization. Proc. Natl Acad. Sci. USA 115, E144–E151 (2018).
Pósfai, M. & R. M., D. ’Souza Talent and experience shape competitive social hierarchies. Phys. Rev. E 98, 020302 (2018).
Kawakatsu, M., Chodrow, P. S., Eikmeier, N. & Larremore, D. B. Emergence of hierarchy in networked endorsement dynamics. Proc. Natl Acad. Sci. USA 118, e2015188118 (2021).
Clauset, A., Arbesman, S. & Larremore, D. B. Systematic inequality and hierarchy in faculty hiring networks. Sci. Adv. 1, e1400005 (2015).
Zipf, G. K. Human Behavior and the Principle of Least Effort: An Introduction to Human Ecology. (AddisonWesley Press, Cambridge, MA, USA, 1949).
Newman, M. E. J. Power laws, Pareto distributions and Zipf’s law. Contemp. Phys. 46, 323–351 (2005).
Auerbach, F. Das gesetz der bevölkerungskonzentration. Petermanns Geogr. Mitt. 59, 74–76 (1913).
Rosen, K. T. & Resnick, M. The size distribution of cities: An examination of the Pareto law and primacy. J. Urban Econ. 8, 165–186 (1980).
Booth, A. D. A “law” of occurrences for words of low frequency. Inform. Control 10, 386–393 (1967).
Ha, L. Q., SiciliaGarcia, E. I., Ming, J. & Smith, F. J. “Extension of Zipf’s law to words and phrases,” in Proc. 19th International Conference on Computational Linguistics, pp. 1–6, (2002).
Ferrer i Cancho, R. & Solé, R. V. Least effort and the origins of scaling in human language. Proc. Natl Acad. Sci. USA 100, 788–791 (2003).
CorominasMurtra, B., Fortuny, J. & Solé, R. V. Emergence of Zipf’s law in the evolution of communication. Phys. Rev. E 83, 036115 (2011).
Dodds, P. S., Harris, K. D., Kloumann, K., Bliss, C. A. & Danforth, C. M. Temporal patterns of happiness and information in a global social network: hedonometrics and twitter. PloS ONE 6, e26752 (2011).
Cocho, G., Flores, J., Gershenson, C., Pineda, C. & Sánchez, S. Rank diversity of languages: generic behavior in computational linguistics. PLoS ONE 10, e0121898 (2015).
Lucas Jr, R. E. On the size distribution of business firms. Bell J. Econ. 9, 508–523 (1978).
Stanley, M. H. et al. Scaling behaviour in the growth of companies. Nature 379, 804–806 (1996).
Axtell, R. L. Zipf distribution of US firm sizes. Science 293, 1818–1820 (2001).
Adamic, L. A. & Huberman, B. Zipf’s law and the internet. Glottometrics 3, 143–150 (2002).
Ogata, Y. & Katsura, K. Analysis of temporal and spatial heterogeneity of magnitude frequency distribution inferred from earthquake catalogues. Geophys. J. Int. 113, 727–738 (1993).
Sornette, D., Knopoff, L., Kagan, Y. Y. & Vanneste, C. Rankordering statistics of extreme events: application to the distribution of large earthquakes. J. Geophys. Res. 101, 13883–13893 (1996).
Mantegna, R. N. et al. Systematic analysis of coding and noncoding DNA sequences using methods of statistical linguistics. Phys. Rev. E 52, 2939 (1995).
Wagner, A. & Fell, D. A. The small world inside large metabolic networks. Proc. R. Soc. Lond. 268, 1803–1810 (2001).
Radicchi, F., Fortunato, S., Markines, B. & Vespignani, A. Diffusion of scientific credits and the ranking of scientists. Phys. Rev. E 80, 056103 (2009).
Sinatra, R., Wang, D., Deville, P., Song, C. & Barabási, A.L. Quantifying the evolution of individual scientific impact. Science 354, 6312 (2016).
Fraiberger, S. P. R., Resch, M., Riedl, C. & Barabási, A.L. Quantifying reputation and success in art. Science 362, 825–829 (2018).
Janosov, M., Musciotto, F., Battiston, F. & Iñiguez, G. Elites, communities and the limited benefits of mentorship in electronic music. Sci. Rep. 10, 1–8 (2020).
González, M. C., Hidalgo, C. A. & Barabási, A.L. Understanding individual human mobility patterns. Nature 453, 779–782 (2008).
Noulas, A., Scellato, S., Lambiotte, R., Pontil, M. & Mascolo, C. A tale of many cities: universal patterns in human urban mobility. PloS ONE 7, e37027 (2012).
Alessandretti, L., Sapiezynski, P., Sekara, V., Lehmann, S. & Baronchelli, A. Evidence for a conserved quantity in human mobility. Nat. Hum. Behav. 2, 485–491 (2018).
Gu, J., Lee, S., Saramäki, J. & Holme, P. Ranking influential spreaders is an illdefined problem. Europhys. Lett. 118, 68002 (2017).
Hidalgo, C. A. & Hausmann, R. The building blocks of economic complexity. Proc. Natl Acad. Sci. USA 106, 10570–10575 (2009).
Morales, J. A. et al. Rank dynamics of word usage at multiple scales. Front. Phys. 6, 45 (2018).
Morales, J. A. et al. Generic temporal features of performance rankings in sports and games. EPJ Data Sci. 5, 33 (2016).
MartínezMekler, G. et al. Universality of rankordering distributions in the arts and sciences. PLoS ONE 4, e4791 (2009).
Simon, H. A. On a class of skew distribution functions. Biometrika 42, 425–440 (1955).
Price, D. A general theory of bibliometric and other cumulative advantage processes. J. Am. Soc. Inform. Sci. 27, 292–306 (1976).
Barabási, A.L. & Albert, R. Emergence of scaling in random networks. Science 286, 509–512 (1999).
Maillart, T., Sornette, D., Spaeth, S. & von Krogh, G. Empirical tests of Zipf’s law mechanism in open source Linux distribution. Phys. Rev. Lett. 101, 218701 (2008).
Dodds, P. S. et al. Allotaxonometry and rankturbulence divergence: a universal instrument for comparing complex systems. Preprint at arXiv https://arxiv.org/abs/2002.09770 (2020).
Blumm, N. et al. Dynamics of ranking processes in complex systems. Phys. Rev. Lett. 109, 128701 (2012).
Batty, M. Rank clocks. Nature 444, 592–596 (2006).
Verbavatz, V. & Barthelemy, M. The growth equation of cities. Nature 587, 397–401 (2020).
Gerlach, M., FontClos, F. & Altmann, E. G. Similarity of symbol frequency distributions with heavy tails. Phys. Rev. X 6, 021009 (2016).
Pechenick, E. A., Danforth, C. M. & Dodds, P. S. Is language evolution grinding to a halt? the scaling of lexical turbulence in English fiction suggests it is not. J. Comput. Sci. 21, 24–37 (2017).
GarciaZorita, C., Rousseau, R., MaruganLazaro, S. & SanzCasado, E. Ranking dynamics and volatility. J. Informetr. 12, 567–578 (2018).
Murcio, R., Zhong, C., Manley, E. & Batty, M. Identifying risk profiles in the London’s public transport system. In Proc 14th International Conference on Computers in Urban Planning and Urban Management (2015).
Michel, J.B. et al. Quantitative analysis of culture using millions of digitized books. Science 331, 176–182 (2011).
Liu, N. C. & Cheng, Y. The academic ranking of world universities. High. Educ. Eur. 30, 127–136 (2005).
Zhu, J. Multifactor performance measure model with an application to Fortune 500 companies. Eur. J. Oper. Res. 123, 105–124 (2000).
Ghoshal, G. & Barabási, A.L. Ranking stability and superstable nodes in complex networks. Nat. Commun. 2, 1–7 (2011).
Edwards, R. & Batty, M. City size: Spatial dynamics as temporal flows. Environ. Plann. A 48, 1001–1003 (2016).
Shlesinger, M. F., Zaslavsky, G. M. & Klafter, J. Strange kinetics. Nature 363, 31–37 (1993).
Chen, L. & Stroock, D. W. The fundamental solution to the Wright–Fisher equation. SIAM J. Math. Anal. 42, 539–567 (2010).
Epstein, C. L. & Mazzeo, R. Wright–Fisher diffusion in one dimension. SIAM J. Math. Anal. 42, 568–608 (2010).
Vedres, B. & Vasarhelyi, O. Gendered behavior as a disadvantage in open source software development. EPJ Data Sci. 8, 25 (2019).
Thurman, N. Forums for citizen journalists? Adoption of user generated content initiatives by online news media. N. Media Soc. 10, 139–157 (2008).
Sinatra, R., Deville, P., Szell, M., Wang, D. & Barabási, A.L. A century of physics. Nat. Phys. 11, 791–796 (2015).
Hidalgo, C. A., Klinger, B., Barabasi, A.L. & Hausmann, R. The product space conditions the development of nations. Science 317, 482–487 (2007).
Burghardt, K., Hogg, T., D’Souza, R., Lerman, K. & Posfai, M. Origins of algorithmic instabilities in crowdsourced ranking. Proc. ACM Hum. Comput. Interact. 4, 1–20 (2020).
Bollen, J. et al. Historical language records reveal a surge of cognitive distortions in recent decades. Proc. Natl Acad. Sci. USA 118, e2102061118 (2021).
Sekara, V. et al. The chaperone effect in scientific publishing. Proc. Natl Acad. Sci. USA 115, 12603–12607 (2018).
Ciampaglia, G. L., Nematzadeh, A., Menczer, F. & Flammini, A. How algorithmic popularity bias hinders or promotes quality. Sci. Rep. 8, 1–7 (2018).
Oka, M. & Ikegami, T. Exploring default mode and information flow on the web. PloS ONE 8, e60398 (2013).
Krapivsky, P. L. & Redner, S. Statistics of changes in lead node in connectivitydriven networks. Phys. Rev. Lett. 89, 258703 (2002).
Pósfai, M., Braun, N., Beisner, B. A., McCowan, B. & D. ’Souza, R. M. Consensus ranking for multiobjective interventions in multiplex networks. N. J. Phys. 21, 055001 (2019).
Iñiguez, G., Pineda, C., Gershenson, C. & Barabási, A.L. Dynamics of ranking. Farranks, https://doi.org/10.5281/zenodo.5910806 (2022).
Acknowledgements
In memory of Jorge Flores and Germinal Cocho. We acknowledge José A. Morales and Sergio Sánchez for data handling at the start of the project. We are grateful for data provision to Gustavo Carreón, Syed Haque, Kay Holekamp, Amiyaal Ilany, Márton Karsai, Raj Kumar Pan, Roberto Murcio, and Roberta Sinatra. G.I. thanks Tiina Näsi for valuable suggestions. G.I. acknowledges support from AFOSR (#FA86552017020), project EU H2020 Humane AInet (#952026), and CHISTERA project SAI (#FWF I 5205N). C.P. and C.G. acknowledge support by CONACyT (#285754) and UNAMPAPIIT (#IG100518, IG101421, IN107919, and IV100120). C.G. was also supported by the PASPA program from UNAMDGAPA. A.L.B. was supported by an EU H2020 SYNERGY grant (#810115DYNASNET), the John Templeton Foundation (#61066), and AFOSR (#FA95501910354).
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G.I., C.P., C.G., and A.L.B. designed the study. G.I. performed data analysis and model fitting. G.I. and C.P. derived analytical results and performed numerical simulations. G.I., C.P., C.G., and A.L.B. wrote the paper.
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A.L.B. is the founder of Foodome, ScipherMedicine, and Datapolis, companies that explore the role of networks in health and urban environments. G.I. is the founder of Predify, a data science consulting startup in Mexico. C.P. and C.G. declare no competing interest.
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Iñiguez, G., Pineda, C., Gershenson, C. et al. Dynamics of ranking. Nat Commun 13, 1646 (2022). https://doi.org/10.1038/s4146702229256x
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DOI: https://doi.org/10.1038/s4146702229256x
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