Understanding the development of talent has been a major challenge across the arts, education, and particularly sports. Here, we show that a dynamic network model predicts typical individual developmental patterns, which for a few athletes result in exceptional achievements. We first validated the model on individual trajectories of famous athletes (Roger Federer, Serena Williams, Sidney Crosby, and Lionel Messi). Second, we fitted the model on athletic achievements across sports, geographical scale, and gender. We show that the model provides good predictions for the distributions of grand slam victories in tennis (male players,

In 1869, Francis Galton published his work on the genetics of genius, in which he claimed that eminent individuals are born with the potential to excel in the future. He based this conclusion on his observation that elite performance tends to run in families at much higher rates than could be expected based on chance [

In order to define the kind of model that captures the process of talent development, the first important step is the conceptualization of talent and related concepts. Talent can be defined as an individual’s

So far, research on talent development has primarily centred around the question: How much do particular genetic and nurturing factors contribute to the development of elite performance [

Following the standard model, scientific projects across countries and types of sports have put a major focus on finding the physical, technical, tactical, psychological, practice, and environmental variables that distinguish groups of elite athletes from groups of sub- or nonelite athletes [

To exemplify the four properties mentioned above, first, evidence for the dynamic development of talent can be derived from research tracking athletes’ performance histories [

Finally, the fact that the distribution of talent across the population is not normal has been stressed repeatedly, mostly by Simonton [

To advance the modelling of talent development, one should define the principles that can explain the properties above, driven by assumptions about the definition of talent and the nature of developmental processes. This means that talent should be modelled as a potential that develops through complex nature-nurture interactions [

Fictive illustration of a talent network. In this case, the network includes an imaginary tennis player’s ability and other personal and environmental supporting and inhibiting factors that may differ across individuals. Green arrows represent positive influences, and red arrows represent negative influences. The sizes of the components reflect their level at a certain moment, and the thickness of an arrow is proportionate to the strength of influence. Note that the displayed network is a simple, speculative snapshot and that the network is dynamic and idiosyncratic in reality.

In the 2000s, applications of different kinds of network models have become prominent across different scientific domains, including physics, economy, biology, and the social sciences [

The specific network model we present here is inspired by dynamic systems applications to human developmental processes [

In order to account for events, such as a transition from youth to professional, it should be possible to model a singular perturbation to an athlete’s ability level around the transition and expose him or her to new challenges and environments [

In this study, we aimed to test whether a dynamic network model provides a valid theoretical foundation of talent development. Therefore, we simulated athlete-networks based on (

To empirically check the validity of the dynamic network model, we compared the model predictions based on computer simulations with data we collected from two major individual sports (i.e., tennis and golf) and two major team sports (i.e., (ice) hockey and soccer). More specifically, we compared the model predictions with cases of professional athletes (Federer, Williams, Crosby, and Messi) and with the distributions of performance attainments across sports, gender, and geographic scale (from worldwide to local).

For this study, we collected archival data from elite tennis players, golf players, (ice) hockey players, and soccer players. In tennis, the number of tournament victories is a direct indicator of a player’s achievements. In order to secure an even level of competition across the tournaments and to have comparable datasets for male and female players, we focused on the grand slam tournaments. Comparable to winning a grand slam in tennis is winning a major in golf. Major tournaments also host the highest-ranked players at the given point in time. Another parallel with tennis grand slams is that we can consider both male and female athletes for this sport.

Hockey is a team sport, in which six players are on the field for each team. Of these six players, one is the goaltender and the other five are so-called skaters. Due to the dynamic of the game and the relatively small rink size, each skater is involved in attacking as well as defending. This provides every skater with the opportunity to score goals. Since a team needs to score goals in order to win, scoring is a measurable expression of a player’s ability. We focus on the National Hockey League (NHL), USA, which is the highest level hockey competition worldwide. Similar to hockey, to determine performance achievements in soccer, we focus on the goals scored by field players.

To examine individual achievement trajectories, we zoomed into a few elite athletes with exceptional (measurable) achievements. These athletes were Roger Federer, who won an exceptional number of 18 grand slam titles in male tennis at the time of data collection, Serena Williams, who won an exceptional number of 23 grand slams in female tennis, Sidney Crosby, who scored an exceptional number of 338 goals in the NHL, and Lionel Messi, who scored an exceptional number of 312 league goals for FC Barcelona. In addition, we determined the population distributions of performance achievements in tennis, golf, hockey, and soccer. For tennis, we examined the distributions of grand slam titles for male (

The data for the different sports were retrieved from the sports’ official websites or the official website tracking the statistics of that sport. The data for tennis were collected through the Association of Professional Tennis’ website (

The dynamic network model was implemented in Visual Basic that runs under Microsoft Excel, which allowed us to simulate developmental trajectories of individual athletes. Table

Default parameter values used for the dynamic model simulations.

Parameter | Average | Standard deviation |
---|---|---|

0.05 | 0.01 | |

0 | 0.02 | |

1.00 | 0.15 | |

Connection probability with other variables | 0.25 | — |

Minimum | Maximum | |

0 | 0.05 | |

Time of initial emergence of a variable | 1.00 | 350.00 |

10.00 | 25.00 |

In addition to the default settings that suffice to run simulations of the basic network, we inserted a transition from youth to professional. In order to model this, we applied a “perturbation” to the ability variable (i.e., node 3) at step 300, which in the simulation marks the transition point. More specifically, we modelled a drop around the transition (

Mathematically speaking, the parameters that we defined are dimensionless numbers. This means that they are numbers that do not directly correspond with the dimensionality of specific physical or psychological properties. The parameters are ratio numbers that specify a particular ratio or proportion of effect of one component on other components and on itself. The population described by the model is represented in the form of hypothetical distributions of these parameter values. An individual in this population is represented by any combination of parameter values randomly drawn from these distributions. The empirical verification of the dynamic network model is then based on the following predictions: (1) in any representative sample of parameter combinations, we will find resulting individual trajectories that correspond with observed individual trajectories of athletes, and (2) any representative sample of parameter combinations will generate a population of individual trajectories, the general properties of which correspond with the properties of an observed population of athletes.

In order to model the athletes’ achievements, we connected the dynamic network model with a product model. The likelihood that an achievement was generated for an athlete was based on the ability level, level of tenacity, and a likelihood parameter

The default parameter settings that we used for the simulations of populations of tennis players, hockey players, and soccer players corresponded to those used for the individual simulations of Federer, Williams (tennis), Crosby (hockey), and Messi (soccer). For golf, we used the same parameter settings as for tennis. In order to compare the actual distributions with predictions of the dynamic network model, we simulated the accomplishments for the number of athletes that corresponded exactly to the number of athletes in the actual data samples (i.e., 1528 male tennis players, 1274 female tennis players, 1011 male golf players, 1183 female golf players, 6677 hockey players, and 585 soccer players).

In line with the literature on talent development, and with the fact that the extended logistic growth equation typically generates nonlinear developmental patterns, simulations of the dynamic network model revealed different trajectories of talent development for different athletes. Figure

Results of the simulations of three athletes’ talent networks. The black solid lines in the graphs correspond to the ability variable, represented by node 3 in the network. The other lines reflect the changes in the dynamic network variables that have supportive, competitive, or neutral relationships with the ability. The meaning of these variables differs among individuals and constitutes an individual’s idiosyncratic network. The starting values of the parameters were drawn from the distributions as defined in Table

In order to check whether the model provides predictions that fit with the archival data we collected, we first determined whether the performance accomplishments generated by the model are in agreement with the data of specific athletes. To model these accomplishments, we assumed that athletes may accomplish an achievement (e.g., winning a tournament or scoring a goal) from the moment they transition from youth to professional. The probability that at a particular moment in time an achievement is accomplished is a function of the ability-tenacity model (

Our first simulation corresponds to an athlete who reaches an ability level of 20.00, which is 17.74 standard deviations above the mean ability level (

Trajectories of performance accomplishments for Roger Federer and Serena Williams. Graph (a) corresponds with Federer’s actual trajectory of grand slam titles per year, and graph (c) with a simulated trajectory; graph (b) corresponds with Williams’ actual trajectory of grand slam titles per year, graph (d) with a simulated trajectory. In graphs (c) and (d), one year corresponds to 20 simulation steps.

To compare the model predictions with hockey, in which athletes’ performances could be measured based on the number of goals they scored, we increased the value of the

Trajectories of performance accomplishments for Sidney Crosby and Lionel Messi. Graph (a) corresponds with Crosby’s actual trajectory of goals scored for in the NHL per year, and graph (c) with a simulated trajectory; graph (b) corresponds with Messi’s actual trajectory of goals scored for FC Barcelona per year, and graph (d) with a simulated trajectory. In graphs (b) and (d), one year corresponds to 20 simulation steps.

To test whether the distribution of athletes’ achievements follows a power law, in which very few athletes accomplish exceptional achievements across sports, gender, and geographical scale, we conducted our analyses on: grand slam titles in tennis for male and female players, major wins in golf for male and female players, goals scored in the National Hockey League (NHL) competition, and goals scored by FC Barcelona players. Then, we simulated these achievements for populations of tennis, golf, hockey, and soccer players.

For all analyses on the archival data, we found patterns close to a power law in the log-log plots for tennis, golf, hockey, and soccer (see Figures

Log-log plots of the number of victories +1 against the number of athletes in the individual sports. The graphs correspond to actual and simulated grand slam titles by male players (a); actual and simulated grand slam titles by female tennis players (b); actual and simulated major titles by male golf players (c); and actual and simulated major titles by female golf players (d). Displayed simulated results are based on one simulation round of the population. For plots showing the raw actual and simulated data, see our research materials at

Log-log plots of the number of goals +1 scored against the number of athletes in the team sports. The graphs correspond to actual and simulated goals scored by National Hockey League (NHL) players (a), and actual and simulated goals scored by soccer players from FC Barcelona (b). For plots showing the raw actual and simulated data, see our research materials at

Achievements in individual sports according to archival and simulated data.

Sport | Measure | Actual titles | Simulated titles |
---|---|---|---|

Tennis (m) | Athletes with 0 titles | 1439 | 1417.60 ± 9.38 |

Maximum number of titles | 18 | 21.96 ± 7.62 | |

Beta coefficient ( |
−3.32 | −3.59 ± 0.21 | |

Tennis (f) | Athletes with 0 titles | 1231 | 1183.10 ± 8.90 |

Maximum number of titles | 22 | 20.46 ± 8.99 | |

Beta coefficient ( |
−3.26 | −3.58 ± 0.24 | |

Golf (m) | Athletes with 0 titles | 911 | 937.92 ± 8.54 |

Maximum number of titles | 14 | 16.40 ± 7.73 | |

Beta coefficient ( |
−3.40 | −3.64 ± 0.26 | |

Golf (f) | Athletes with 0 titles | 1098 | 1099.74 ± 16.08 |

Maximum number of titles | 10 | 20.62 ± 8.89 | |

Beta coefficient ( |
−3.64 | −3.59 ± 0.25 |

Achievements in team sports according to archival and simulated data.

Sport | Measure | Actual goals | Simulated goals |
---|---|---|---|

National Hockey League | Athletes with 0 goals | 1456 | 1327.40 ± 27.99 |

Maximum number of goals | 894 | 899.42 ± 2.96 | |

Beta coefficient ( |
−1.08 | −1.16 ± 0.01 | |

FC Barcelona | Athletes with 0 goals | 244 | 193.02 ± 10.24 |

Maximum number of goals | 312 | 463.68 ± 25.19 | |

Beta coefficient ( |
−1.27 | −1.25 ± 0.05 |

Simulating the performance accomplishments based on the dynamic network model, we find the same kinds of distributions as in the archival data. This is implied by the results that (i) the simulated number of players with zero accomplishments is close to the actual number of players with zero accomplishments, (ii) the simulated maximum number of accomplishments for an athlete within a given athletic population is close to the actual maximum number of accomplishments by an individual athlete, and (iii) the regression slopes (beta coefficients) of the log-log plots, which provide an estimate of the power parameter, show close resemblances between the simulated and archival data. Table

The results for hockey and soccer are shown in Table

Here, we proposed a dynamic network model of talent development and tested whether it explains the individual developmental patterns and achievements of elite athletes, as well as the distributions of achievements across populations of athletes in different sports. We therefore (i) defined the model principles based on the definition of talent and the literature on human developmental processes; (ii) ran simulations of the defined dynamic network model; (iii) collected performance attainments of specific cases in tennis (Federer and Williams), hockey (Crosby), and soccer (Messi) and compared their data with the patterns generated by simulations of our dynamic network model; and (iv) collected performance attainments across the population of elite athletes in tennis, golf, hockey, and soccer and compared the population distributions with those generated by the dynamic network model.

Regarding the ability-level trajectories, the dynamic network model generates nonlinear patterns that differ per individual athlete. This is in accordance with previous studies on talent development in sports [

However, we also went beyond general description of the trajectories of ability development and connected the dynamic network model to an ability-tenacity product model to examine athletes’ simulated performance attainments. Doing this, we were able to replicate the qualitative pattern of achievements of some exceptional athletes in different sports (i.e., Federer, Williams, Crosby, and Messi). Together, these results indicate that the dynamic network model can explain the individual trajectories of talent development, which would not be possible using traditional linear models, such as regression models applied to samples of athletes [

Furthermore, in line with previous research [

The resemblances between the performance accomplishment distributions based on the archival data and the model predictions were more evident for the individual sports than for the team sports. In particular, the predictions in hockey provided a distribution that was more curved than the actual distribution, although qualitative similarities were still apparent. An interesting question is whether there is any comparably general alternative model of talent development that provides an even better qualitative and quantitative fit with the data in team sports. Regarding the soccer data we collected, one may criticize that we took the goals scored by all Barcelona players rather than only the attacking players. We decided to do so, because it is difficult to draw a line defining which players clearly have (no) attacking tasks on the field. Interestingly, if one would only take only the attackers of FC Barcelona, one would again find a strongly skewed distribution. This supports the claim that distributions of the power-law kind hold across all kinds of scales of analysis (see the research materials at

A dynamic network model seems to underlie the development of talent in sports, which ultimately results in exceptional achievements for very few athletes. This conclusion has important implications at both a theoretical and practical level. At a theoretical level, an important step is to move away from a focus on unravelling the underlying variables of talent development and to embrace the complex interactions that exist across performer, environment, practice, and training [

With respect to the point of posing network-oriented research questions, the focus should be on the

From an applied perspective, talent detection programs in research and practice around the world are still largely based on the assumption that talent can be detected in certain variables “in the individual” and that it can be discovered at an early age [

The dynamic network model provides a comprehensive framework to understand the theoretical principles underlying the development of talent. The model suggests that talent emerges from intra- and interindividual variations in the composition of individual dynamic networks. Having demonstrated that the foundation of the dynamic network model explains empirical observations across a variety of sports, it is now time to explore and test the variety of practical applications of the dynamic network perspective.

The basic dynamic network model, the manual of the model, the archival data, and the simulated data are available at

The authors declare that there is no conflict of interest regarding the publication of this paper.

Publication of this article was funded by the Heymans Institute for Psychological Research, University of Groningen, and a research grant awarded to Yannick Hill by the Sparkasse Bank.