Generic temporal features of performance rankings in sports and games
- José A Morales^{1},
- Sergio Sánchez^{1},
- Jorge Flores^{2},
- Carlos Pineda^{2, 4}Email authorView ORCID ID profile,
- Carlos Gershenson^{3, 4, 5, 6, 7},
- Germinal Cocho^{2},
- Jerónimo Zizumbo^{1},
- Rosalío F Rodríguez^{2} and
- Gerardo Iñiguez^{3, 8, 9}
DOI: 10.1140/epjds/s13688-016-0096-y
© Morales et al. 2016
Received: 15 June 2016
Accepted: 17 November 2016
Published: 25 November 2016
Abstract
Many complex phenomena, from trait selection in biological systems to hierarchy formation in social and economic entities, show signs of competition and heterogeneous performance in the temporal evolution of their components, which may eventually lead to stratified structures such as the worldwide wealth distribution. However, it is still unclear whether the road to hierarchical complexity is determined by the particularities of each phenomena, or if there are generic mechanisms of stratification common to many systems. Human sports and games, with their (varied but simple) rules of competition and measures of performance, serve as an ideal test-bed to look for universal features of hierarchy formation. With this goal in mind, we analyse here the behaviour of performance rankings over time of players and teams for several sports and games, and find statistical regularities in the dynamics of ranks. Specifically the rank diversity, a measure of the number of elements occupying a given rank over a length of time, has the same functional form in sports and games as in languages, another system where competition is determined by the use or disuse of grammatical structures. We use a Gaussian random walk model to reproduce the rank diversity of the studied sports and games. We also discuss the relation between rank diversity and the cumulative rank distribution. Our results support the notion that hierarchical phenomena may be driven by the same underlying mechanisms of rank formation, regardless of the nature of their components. Moreover, such regularities can in principle be used to predict lifetimes of rank occupancy, thus increasing our ability to forecast stratification in the presence of competition.
Keywords
complex systems sports data analysis rank distribution rank diversity1 Introduction
Sports and games can be described as hierarchical complex systems due to the myriad of factors influencing the dynamics of competition and performance in them, including networked interactions, human and environmental heterogeneities, and other traits at the individual and group levels [1–4]. In particular, the performance of players and teams is influenced by a variety of causes: Economical, political and geographical conditions determine their rankings and may thus be used for predicting performance. Moreover, the (relatively) simple rules of competition and measures of performance associated with sports and games allow us to explore basic mechanisms of interaction leading to hierarchy formation, which may be common to many systems driven by competition, not only leisure activities but other social, biological and economic systems. With this goal in mind, the availability of a large corpus of data related to sports, teams, and players allows researchers to perform multiple statistical analyses, in particular with respect to the structure and dynamics of performance rankings [5–7].
Data availability has made it possible not only to study the distribution of scores determining rankings, but also its time evolution [8]. In a recent paper, Deng et al. present a statistical analysis of 12 sports and report a universal scaling in rankings, despite the fact that the sports considered have very different ranking systems [9]. Here, we focus on the temporal trajectories of player and team performances, meaning the evolution of rank, with the objective of finding statistical regularities that indicate how competition shapes hierarchies of players and teams. In principle, rankings may be affected in time by events as apparently insignificant as a bad breakfast prior to an important event, or the weather during a competition [10]. Since these factors are inherently present for all activities, we would expect the evolution of rank to have generic features across sports and games.
We propose to quantify such evolution by means of a recently introduced measure, the rank diversity. With the help of the Google n-gram dataset [11], rank diversity has been used before to study how vocabulary changes in time [12]. That work shows that rank diversity has the same functional form for all languages studied, and is able to discriminate the size of the core of each language. Thus, here we concentrate on the temporal features of rank distributions corresponding to several sports and games with different ranking schemes. We consider data where an appropriate time resolution is available, and limit the analysis to six activities only: tennis, chess, golf, poker and football (both national teams and clubs). We find that all rank diversities have the same functional form as languages, despite having differences in their rank frequency distributions. Finally, we introduce a random walk model that, tuned by the parameter values of each dataset, reproduces qualitatively the diversity of all sports and games considered. Overall, our goal is to use rank diversity as a tool to understand rank dynamics in sports, games, and other hierarchical complex systems, thus enabling us to identify the dependence on rank of a change in the hierarchy of the system. By using this analysis, we may be able to estimate how well can a change in rank be predicted, regardless of the particularities of the phenomenon under study.
The article is organized as follows. In Section 2 we describe the datasets used. We then analyse ranking distributions in Section 3 and compare them with several models. In Section 4 we study the rank diversity for each sporting activity and compare it with a random walk model. The main conclusions of our analysis are included in Section 5. In Appendix A we discuss in detail the Kolmogorov-Smirnov index, which measures the goodness of fit for a given dataset. Finally, in Appendix B we describe the generic relation between rank diversity and the cumulative rank distribution in the random walk model.
2 Ranking data
We use ranking data on players and teams from six sports and games: (a) Tennis players (male), ranked by the Association of Tennis Professionals (ATP) [13]; (b) Chess players (male), ranked by the Fédération Internationale des Échecs (FIDE) [14]; (c) Golf players, ranked by the Official World Golf Ranking (OWGR) [15]; (d) Poker players, ranked by the Global Poker Index (GPI) [16]; (e) Football teams, ranked by the Football Club World Ranking (FCWR) [17]; and (f) national football teams, ranked by the Fédération Internationale de Football Association (FIFA) [18].
Summary of ranking data for each sport and game considered in this study
Sport/game | Data source | Time period | Ranking resolution | #players/teams |
---|---|---|---|---|
Tennis players (male) | Association of Tennis Professionals (ATP) [13] | May 5 2003-Dec 27 2010 | Weekly | 1,600 |
Chess players (male) | Fédération Internationale des Échecs (FIDE) [14] | Jul 2012-Apr 2016 | Monthly | 13,500 |
Golf players | Official World Golf Ranking (OWGR) [15] | Sept 10 2000-Apr 19 2015 | Weekly | 1,000 |
Poker players | Global Poker Index (GPI) [16] | Jul 25 2012-Jun 10 2015 | Weekly | 1,799 |
Football teams | Football Club World Ranking (FCWR) [17] | Feb 1 2012-Dec 29 2014 | Weekly | 850 |
National football teams | Fédération Internationale de Football Association (FIFA) [18] | Jul 2010-Dec 2015 | Monthly | 150 |
3 Comparison with ranking models
Parameter values for fitting process between sports data and ranking models
Model \(\boldsymbol{m_{1}}\) | Model \(\boldsymbol{m_{2}}\) | Model \(\boldsymbol{m_{3}}\) | Model \(\boldsymbol{m_{4}}\) | Model \(\boldsymbol{m_{5}}\) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(\boldsymbol {\log\mathcal{N}}\) | a | \(\boldsymbol{\log\mathcal{N}}\) | a | b | \(\boldsymbol{\log \mathcal{N}}\) | a | q | \(\boldsymbol{\log\mathcal{N}}\) | a | b | q | \(\boldsymbol{\log\mathcal{N}}\) | a | \(\boldsymbol{a'}\) | \(\boldsymbol{k_{c}}\) | |
ATP | 4.51 | 1.04 | 4.11 | 0.626 | 3.18 ×10^{−3} | −1.46 | 0.816 | 1.79 | 4.01 | 0.628 | 3.13 ×10^{−3} | 3.12 ×10^{−2} | 4.2 | 0.746 | 3.004 | 3.19 ×10^{2} |
FIDE | 3.46 | 0.0252 | 3.46 | 2.01 ×10^{−2} | 6.25 ×10^{−6} | 3.32 | 2.21 ×10^{−2} | 3.28 ×10^{−2} | 3.46 | 0.0202 | 6.25 ×10^{−6} | 5.05 ×10^{−9} | 3.45 | 0.016 | 0.036 | 1.97 ×10^{2} |
OWGR | 1.35 | 0.702 | 1.05 | 0.383 | 2.68 ×10^{−3} | −0.928 | 0.53 | 0.703 | 1.023 | 0.385 | 2.65 ×10^{−3} | 1.18 ×10^{−2} | 1.09 | 0.452 | 1.73 | 2.03 ×10^{2} |
GPI | 3.75 | 0.234 | 3.66 | 0.144 | 6.63 ×10^{−4} | 2.54 | 0.193 | 0.358 | 3.66 | 0.144 | 6.63 ×10^{−4} | 3.64 ×10^{−9} | 3.65 | 0.133 | 0.437 | 1.08 ×10^{2} |
FCWR | 4.52 | 0.529 | 4.24 | 0.218 | 3.06 ×10^{−3} | 2.19 | 0.341 | 0.732 | 2.93 | 0.269 | 1.39 ×10^{−3} | 0.458 | 4.35 | 0.371 | 3.47 | 4.26 ×10^{2} |
FIFA | 3.37 | 0.473 | 3.24 | 0.142 | 1.02 ×10^{−2} | 2..30 | 0.262 | 0.456 | 3.15 | 0.148 | 9.58 ×10^{−3} | 4.16 ×10^{−2} | 3.26 | 0.227 | 0.8622 | 33.86 |
Goodness of fit measures
\(\boldsymbol{m_{1}}\) | \(\boldsymbol{m_{2}}\) | \(\boldsymbol{m_{3}}\) | \(\boldsymbol{m_{4}}\) | \(\boldsymbol{m_{5}}\) | ||
---|---|---|---|---|---|---|
ATP | \(\langle R^{2} \rangle\) | 0.222 | 0.982 | 0.879 | 0.982 | 0.964 |
〈D〉 | 0.433 | 0.044 | 0.08 | 0.038 | 0.077 | |
\(\sigma_{R^{2}}\) | 0.0969 | 0.01652 | 0.009 | 0.0124 | 0.0288 | |
\(\sigma_{D}\) | 0.211 | 0.0126 | 0.0672 | 0.0128 | 0.0287 | |
p | 0.01 | 0.17 | 0.0 | 0.12 | 0.0 | |
FIDE | \(\langle R^{2} \rangle\) | 0.777 | 0.936 | 0.657 | 0.936 | 0.991 |
〈D〉 | 0.477 | 0.2 | 0.188 | 0.2 | 0.141 | |
\(\sigma_{R^{2}}\) | 0.0071 | 0.0053 | 0.0028 | 0.0054 | 0.0035 | |
\(\sigma_{D}\) | 0.0072 | 0.0048 | 0.0166 | 0.0048 | 0.0005 | |
p | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |
OWGR | \(\langle R^{2} \rangle\) | 0.631 | 0.981 | 0.943 | 0.982 | 0.97 |
〈D〉 | 0.316 | 0.046 | 0.088 | 0.043 | 0.088 | |
\(\sigma_{R^{2}}\) | 0.0264 | 0.0388 | 0.0138 | 0.0381 | 0.0391 | |
\(\sigma_{D}\) | 0.1292 | 0.0165 | 0.0192 | 0.0152 | 0.0104 | |
p | 0.0 | 0.92 | 0.0 | 0.89 | 0.0 | |
GPI | \(\langle R^{2} \rangle\) | 0.791 | 0.978 | 0.937 | 0.978 | 0.985 |
〈D〉 | 0.531 | 0.201 | 0.149 | 0.201 | 0.202 | |
\(\sigma_{R^{2}}\) | 0.01029 | 0.0115 | 0.0044 | 0.0115 | 0.0459 | |
\(\sigma_{D}\) | 0.01612 | 0.0039 | 0.0048 | 0.0039 | 0.00533 | |
p | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |
FCWR | \(\langle R^{2} \rangle\) | 0.727 | 0.986 | 0.981 | 0.997 | 0.947 |
〈D〉 | 0.295 | 0.115 | 0.057 | 0.055 | 0.172 | |
\(\sigma_{R^{2}}\) | 0.0186 | 0.0183 | 0.0098 | 0.0112 | 0.0268 | |
\(\sigma_{D}\) | 0.02833 | 0.0046 | 0.0052 | 0.00128 | 0.0104 | |
p | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | |
FIFA | \(\langle R^{2} \rangle\) | 0.833 | 0.993 | 0.981 | 0.996 | 0.979 |
〈D〉 | 0.387 | 0.076 | 0.071 | 0.041 | 0.155 | |
\(\sigma_{R^{2}}\) | 0.0277 | 0.0324 | 0.0135 | 0.0114 | 0.0413 | |
\(\sigma_{D}\) | 0.02888 | 0.004 | 0.007 | 0.002 | 0.0147 | |
p | 0.0 | 0.99 | 0.0 | 0.99 | 0.02 |
Table 3 shows the mean values \(\langle R^{2} \rangle\) and \(\langle D \rangle\) (and their associated standard deviations \(\sigma_{D}\) and \(\sigma_{R^{2}}\)), averaged over all time slices available, for the fitting process between the six datasets and five models \(m_{i}\) used here. We also include values of p for the single time slice of Figure 1. Higher \(\langle R^{2} \rangle\) and lower \(\langle D \rangle\) imply better fits. Since \(\sigma_{D}\) and \(\sigma_{R^{2}}\) are small, the fits shown in Figure 1 are representative of the entire datasets. We observe that none of the models are a good fit for all sports and games, although \(m_{4}\) and \(m_{5}\) are the most appropriate (in terms of \(R^{2}\)). However, in three cases (FIDE, GPI, and FCWR) we have \(p=0\) for model \(m_{4}\), and no model fits well, meaning that the theoretical distribution is not followed by the data. We stress again that Zipf’s law (\(m_{1}\)) is the worst fit among all considered, except for FIDE. It is interesting to notice that \(R^{2}\) and p lead to different criteria of what is a ‘good’ or a ‘bad’ model. This is due to both the amount of available data and the number of parameters in the model. The larger the data, the easier it is to distinguish the best available model from a good (but not accurate enough) approximation. On the other hand, the more parameters the model has, the easier it is to fit any data. Both of these aspects are taken into account in the definition of p, but not in \(R^{2}\).
4 Rank diversity in sports and games
The previous analysis of the functional form of the rank distribution in several sporting activities (even when the goodness of fit has been averaged over time) is restricted by the fact that the rank distribution is inherently an instantaneous measure, in the sense that it captures ranking at a given point in time and does not take into account the dynamics of players and teams changing rank as time goes by. In order to overcome this issue, here we contribute to the analysis of ranking in sports and games by computing the rank diversity, a measure of the number of elements occupying a given rank over a length of time. From previous [12] and current work, it appears that rank diversity has the same functional form, not only for sports but also for other complex systems, such as countries classified by their economic complexity, the 500 leading enterprises ranked by Fortune magazine, or a set of millions of words in six Indo-European languages.
We should stress that \(d(k)\) and \(f(k)\) measure different aspects of the hierarchical structure of a complex system. First of all, the rank diversity includes information on how elements change rank throughout time in a single function, while the rank distribution captures the hierarchy in the system for a single time interval. Secondly, the rank diversity disregards any information on the scores of elements beyond their order, and thus the same \(d(k)\) may be obtained for several shapes of \(f(k)\) (power-law, Gamma, Beta, etc.). As an example, consider any transformation in time of the scores of elements in the system, such that their ranking order stays the same; then \(f(k)\) could interpolate between different functional shapes as time goes on, while \(d(k)\) would stay constant. The inverse case is also possible, and any rank distribution may produce a wide variety of rank diversities. For example, we could construct several dynamics of scores that keep the number of elements with a given score constant, but that change the amount of time an element holds certain score, thus keeping \(f(k)\) fixed and changing \(d(k)\). Overall, both \(d(k)\) and \(f(k)\) measure some aspects of the structure and dynamics of hierarchy in a complex system, but only the rank diversity captures the way elements change their positions in the hierarchy, beyond minor changes in scores that could be attributed, for example, to different ways of measuring performance.
From Figure 3 we see that the empirical curves for rank diversity are (roughly) monotonic and have a single shoulder. The cumulative of a square-integrable function with a single bump would have these properties, and a Gaussian is arguably the simplest choice. Moreover, an analytical argument (see Appendix B) suggests that this may be an appropriate ansatz under very general conditions, at least qualitatively. In a large variety of physical systems composed of alike elements with similar interactions between them, the macroscopic response of the system is usually determined by general laws such as equations of state. However, in different empirical realisations of the same dynamics there may be differences associated to the law of the large numbers or the central limit theorem. These differences across realisations follow a normal Gaussian distribution, according to the Gaussian theory of errors. However, for complex systems with competitive dynamics, there may be generic features described by the Gamma (\(m_{2}\)) and Beta (\(m_{3}\)) distributions [31, 32], and there may also be differences across realisations that follow a multiplicative dynamics. This is indeed the case for several Indo-European languages [12] and for the games and sports datasets considered here (See Figure 1). In Appendix B we introduce the non-trivial idea that there are two different dynamics associated with so-called generic and contingent features, which may be described in terms of a one-step Markovian, Gaussian process. This allows us to establish an explicit relation between the diversity \(d(k)\) and the cumulative of the rank distribution, \(S(t)\).
4.1 A random walk model
From Figure 2 and Figure 4 we see that players and teams with low ranks change very slowly or not at all, while those with higher k have a larger rank variation in time. This intuition is clear from recent experience in sports like tennis and football: According to the analysed datasets, Hewitt, Nadal, Roddick, Ferrero, Agassi and Federer have been the only number one tennis players from May 2003 till December 2010. The same holds for football clubs: Real Madrid, Atlético Madrid, Barcelona, and Bayern München have been the best-ranked teams from January 2012 till December 2014. In other words, players and teams with small k tend to have a small rank diversity.
5 Discussion and conclusions
Competition and heterogeneous performance are characteristic of the elements of many complex systems in biological, social and economic settings. Despite the fact that these systems show a large variation in the definitions of their constituents and in the relevant interactions between them, it remains to be seen whether the emergence of hierarchical structure is mostly determined by the particularities of each phenomenon, or if there are mechanisms of stratification common to the temporal evolution of many systems. We have explored this notion by considering a set of relatively controlled and simplified systems driven by competition: Human sports and games, where the rules of engagement and measures of performance are well defined, in contrast to, say, the ranking of physicists (the question of whom is the ‘best’ physicist would have an ambiguous answer, to say the least). This allows us to characterise the emergence of hierarchical heterogeneity by comparing the temporal features of rankings of individuals and teams across activities in a clear way. Explicitly, we analysed the statistical properties of rank distributions in six sports and games, each with different number of members and rules for calculating scores (and, therefore, ranks). By comparing rank distributions with several ranking models, we find that the Zipf law (model \(m_{1}\)) does not provide a suitable fit for the empirical data. Even if the more generic ranking model \(m_{4}\) (a combination of the Gamma and Beta distributions) tends to offer good fits, it is not always the best.
Furthermore, we studied the temporal features of rankings explicitly by calculating the rank diversity \(d(k)\), a measure of the number of individuals or teams occupying a given rank over a length of time. We found that \(d(k)\) has the same sigmoid-like functional form, even for relatively small systems like FIFA (with only 150 elements per time slice). Coupled to the fact that a sigmoid rank diversity has also been found in the way vocabulary changes in time [12], our results suggest that the emergence of hierarchical complexity - as measured by \(d(k)\) - may have traits common to many systems. This claim is underlined by the fact that a simple model (the scale-invariant random Gaussian walk) can reproduce the diversity of the sports and games studied here, and also of languages [12]. One could initially suspect that rank changes depend on the intrinsic strength or qualities of players and teams. However, given the fact that our random walk model reproduces relatively well the rank dynamics of several sports and games, it seems that rank change can instead be characterised as a random process. This does not imply that rank change is random, but that the specific mechanisms associated with each activity and ranking system are irrelevant for the calculation of rank diversity.
A natural direction to follow in the near future is to study the behaviour of rank diversity in other competitive phenomena beyond sporting activities and language, such as physical, social and economic processes of stratification. If indeed a certain universality in the temporal features of rankings is present in other complex settings, it would indicate that hierarchical phenomena may be driven by the same underlying mechanisms of rank formation, regardless of the nature of their components. Potentially, we may exploit such regularities to predict lifetimes of rank occupancy, thus increasing our ability to forecast stratification in the presence of competition.
Declarations
Acknowledgements
Financial support from CONACyT under projects 212802, 221341, and UNAM-PAPIIT IN111015 is acknowledged.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Duch J, Waitzman JS, Amaral LAN (2010) Quantifying the performance of individual players in a team activity. PLoS ONE 5(6):10937 View ArticleGoogle Scholar
- Ben-Naim E, Vazquez F, Redner S (2007) What is the most competitive sport? J Korean Phys Soc 50:124 View ArticleGoogle Scholar
- Merritt S, Clauset A (2013) Environmental structure and competitive scoring advantages in team competitions. Sci Rep 3:3067 View ArticleGoogle Scholar
- Merritt S, Clauset A (2013) Social network dynamics in a massive online game: network turnover, non-densification, and team engagement in halo reach. Eprint. arXiv:1306.4363
- Albert J, Bennett J, Cochran JJ (2005) Anthology of statistics in sports, vol 16. SIAM, Philadelphia View ArticleMATHGoogle Scholar
- Radicchi F (2011) Who is the best player ever? A complex network analysis of the history of professional tennis. PLoS ONE 6(2):17249 View ArticleGoogle Scholar
- Yucesoy B, Barabási A-L (2016) Untangling performance from success. EPJ Data Sci 5:17 View ArticleGoogle Scholar
- Merritt S, Clauset A (2014) Scoring dynamics across professional team sports: tempo, balance and predictability. EPJ Data Sci 3:4 View ArticleGoogle Scholar
- Deng W, Li W, Cai X, Bulou A, Wang QA (2012) Universal scaling in sports ranking. New J Phys 14(9):093038 View ArticleGoogle Scholar
- Klaassen FJ, Magnus JR (2001) Are points in tennis independent and identically distributed? Evidence from a dynamic binary panel data model. J Am Stat Assoc 96(454):500-509 MathSciNetView ArticleGoogle Scholar
- Michel J-B, Shen YK, Aiden AP, Veres A, Gray MK, Team TGB, Pickett JP, Hoiberg D, Clancy D, Norvig P, Orwant J, Pinker S, Nowak MA, Aiden EL (2011) Quantitative analysis of culture using millions of digitized books. Science 331(6014):176-182 View ArticleGoogle Scholar
- Cocho G, Flores J, Gershenson C, Pineda C, Sánchez S (2015) Rank diversity of languages: generic behavior in computational linguistics. PLoS ONE 10(4):0121898 View ArticleGoogle Scholar
- ATP World Tour. http://www.atpworldtour.com/. Accessed 4 April 2016
- World Chess Federation. http://ratings.fide.com/. Accessed 6 April 2016
- Official World Golf Ranking. http://www.owgr.com/. Accessed 6 April 2016
- Global Poker Index. http://www.globalpokerindex.com/. Accessed 6 April 2016
- Football Club World Ranking. http://www.clubworldranking.com/ranking-clubs.aspx. Accessed 6 April 2016
- Fédération Internationale de Football Association. http://www.fifa.com/. Accessed 6 April 2016
- Elo AE (1978) The rating of chessplayers, past and present. Arco Pub., London Google Scholar
- Gerlach M, Altmann EG (2013) Stochastic model for the vocabulary growth in natural languages. Phys Rev X 3:021006 Google Scholar
- Katz JS, Katz L (1999) Power laws and athletic performance. J Sports Sci 17(6):467-476 View ArticleGoogle Scholar
- Alvarez-Ramirez J, Rodriguez E (2006) Scaling properties of marathon races. Physica A 365(2):509-520 View ArticleGoogle Scholar
- Visser M (2013) Zipf’s law, power laws and maximum entropy. New J Phys 15(4):043021 View ArticleGoogle Scholar
- Baek SK, Bernhardsson S, Minnhagen P (2011) Zipf’s law unzipped. New J Phys 13(4):043004 View ArticleGoogle Scholar
- Wikipedia: Gamma distribution. https://en.wikipedia.org/wiki/Gamma_distribution. Accessed 1 March 2016
- Wikipedia: Beta distribution. https://en.wikipedia.org/wiki/Beta_distribution. Accessed 1 March 2016
- Jóhannesson G, Björnsson G, Gudmundsson EH (2006) Afterglow light curves and broken power laws: a statistical study. Astrophys J Lett 640(1):L5-L8 View ArticleGoogle Scholar
- Li W, Miramontes P, Cocho G (2010) Fitting ranked linguistic data with two-parameter functions. Entropy 12(7):1743 View ArticleGoogle Scholar
- Kolmogorov AN (1933) Sulla determinazione empirica di una legge di distribuzione. G Ist Ital Attuari 4(1):83-91 MATHGoogle Scholar
- Clauset A, Shalizi CR, Newman ME (2009) Power-law distributions in empirical data. SIAM Rev 51(4):661-703 MathSciNetView ArticleMATHGoogle Scholar
- Alvarez-Martinez R, Cocho G, Rodríguez RF, Martínez-Mekler G (2014) Birth and death master equation for the evolution of complex networks. Physica A 402:198-208 MathSciNetView ArticleGoogle Scholar
- Martínez-Mekler G, Martínez RA, del Río MB, Mansilla R, Miramontes P, Cocho G (2009) Universality of rank-ordering distributions in the arts and sciences. PLoS ONE 4(3):4791 View ArticleGoogle Scholar
- Van Kampen NG (2007) Stochastic processes in physics and chemistry. North Holland, Amsterdam MATHGoogle Scholar
- Wheeler JC, Gordon RG, Baker GA, Gammel JL (1970) The Padé approximant in theoretical physics. Academic, New York Google Scholar
- Perline R (1996) Zipf’s law, the central limit theorem, and the random division of the unit interval. Phys Rev E 54(1):220 MathSciNetView ArticleGoogle Scholar
- Perline R, Perline R (2016) Two universality properties associated with the monkey model of Zipf’s law. Entropy 18(3):89 View ArticleGoogle Scholar