Early career wins and tournament prestige characterize tennis players’ trajectories

We study the evolution of the careers of male professional tennis players from 2000 to 2019. We obtained data from the official ranking of the ATP [30, 31], together with the match results of different tournaments: Grand Slam (the competitions with the highest value in terms of winner points), Masters 1000, ATP 500 and 250, Challenger (the competitions with the lowest value in our dataset) [31]. We selected players who started their careers in the timespan of our dataset and had at least two years of activity. We consider 3455 players and 651 tourneys, specifically 4 Grand Slams, 11 Masters 1000, 98 ATP 500 and 250, and 538 Challengers.

To distinguish between top and less successful tennis players, we group them according to the maximum amount of points they reached in the ATP ranking, which ranks players based on the score points they accumulate during a season [30]. We can conceive the ATP ranking as a first proxy of success, as it might weigh similar outcomes of players’ performance, which would be winning or losing one or more matches, in very different ways. For instance, winning a match in the round of 32 awards 5 points in a Challenger and 90 points in a Grand Slam. Thus, rather than relying on popularity to quantify success in sports [16], we explore the dynamics of success embedded in tennis rules, neglecting the influence of subsequent elements such as prize money, income, popularity, sponsors, etc. Moreover, we use the highest number of points players reach in the ATP ranking (namely, their career peak) instead of ranking placements. The reason is that the point totals of players with consecutive ranks can vary significantly. For example, consider three players ranked 1, 2, and 3, with point totals of 12,000, 10,000, and 9995, respectively. Although players ranked 1 and 2 are only one position apart from each other, as well as players ranked 2 and 3, there is a greater point difference (2000 points) between players ranked 1 and 2 than between players ranked 2 and 3 (5 points). Therefore, using point totals instead of placements allows us to assess differences between players more accurately. Also, it lets us compare rankings with varying numbers of players over the years.

We split male tennis players into three groups: Top players are defined by those with a career peak above the 95th percentile (top 5%); bottom players are within the 40th percentile (bottom 40%); the 55% left composes the middle group. Individual timelines of players within each of these three groups and their respective ranking points are reported in panel A of Fig. 1. Because players can start their careers at different times, we aligned their trajectories by the first appearance in the ATP rankings. Thus, a player’s trajectory is a time series composed of the sequence of ATP ranking updates. These updates occur weekly, with the exception of the weeks of the Grand Slams and few other tournaments which last longer (consequently, the ranking is run approximately 45 times a year) [30]. As long as a player is still active (i.e., an ATP professional player), he will appear in the ranking.

Our analysis is based on the peak of professional tennis players’ careers, prompting the question of whether this peak is obtained at a consistent time across all individuals within our dataset. To answer this question, we look at the time distribution of career peak, first considering the aggregated data, then each group separately (respectively, panels B and C of Fig. 1). To avoid right-censoring bias [35], we exclude active players from Fig. 1B-C (more details are provided in the Supplementary Information, where we report the case without the right-censoring correction in Figure S1, see Additional file 1). To deal with different career lengths, we normalize the time of the career peak of each player according to their career duration. We find a common tendency for the peak to occur after the first half of players’ careers in all three groups. This result, previously observed only for the top players [36, 37], suggests that peak time is not closely related to individual success.

Observing the individual sequences of the three groups in Fig. 1A, we notice marked differences between them, both in ranking appearances and accumulated points. In particular, the bottom players have shorter careers compared to the other groups. We further investigate this by looking at the survival rate [38] of tennis players in our dataset, bearing in mind that in this context “surviving” at time t means still playing or, in other words, being in the ATP ranking. The results are shown in panel A of Fig. 2. The bottom players’ survival curve (in blue) is the shortest and goes rapidly to zero (decay starts before 100 appearances), followed by the middle players’ (in yellow), which starts decaying a few rank updates later but at a slower rate. Conversely, the top players’ curve (in red) starts falling much later (at around 400 rank updates) and at a slower decay rate, meaning that they have longer professional careers, in line with previous work [39]. We also reported the survival rate of all players (in gray) for reference.

To highlight when the group differences in accumulated points appear, we take the average of the sequences shown in Fig. 1A, which results in the trend of Fig. 2B: We can observe that the top players have more ranking points compared to the others. Such a discrepancy in the number of points could be interpreted as a mere artifact of our definition of top/middle/bottom players. Yet, this difference emerges from the beginning, as indicated in panel C of Fig. 2, which zooms in on the points cumulated in only the first ten appearances of a player in the ATP ranking. Note that here we consider all players, thus including active players. See Figure S2 of the SI for an analysis that considers only those players who started and ended their careers in the dataset, checking for the eventual effects of right-censored data on Fig. 2B.

The initial gap in the average amount of points between the top players and the others may arise from different mechanisms. A first explanation for such a gap may lie in the differences in players’ performance. That is, top players may win more matches from the early stages of their careers, leading to the gap forming. To compare performance across the groups, we first consider the number of competitions in which a player wins at least one match. Panel A of Fig. 3 shows the probability P that a player, with at least one match won, reports a win in more than a given number of tourneys, within the first ten (see Methods for the mathematical definition). Top players (red dots) have higher chances of winning more matches, once they have won their first, at the beginning of their career. However, if we look at the probability P of players winning their first match in a certain tourney t after they turned professional (Fig. 3B, see Methods for the mathematical formulation), we do not see top players emerge. On the contrary, top players tend to win their first match later than the others.

The results of Fig. 3 show that, even though top players achieve more victories after their initial one, they have difficulty in winning their very first match during the early stages of their career. This counterintuitive behavior points out that players’ performance is not enough to explain the formation of the gap between top and less accomplished athletes. Hence, other mechanisms might be at play. For instance, our analysis so far has not taken into account the prestige of the different tournaments that players can attend at the beginning of their careers. Therefore, having illustrated the scenario of the initial stages of players’ careers in men’s professional tennis, we investigate the influence of the first tournaments they can access, together with their results. We aim to untangle the importance of early participation in more prestigious competitions from how players perform in those competitions.

Early access to prestigious tournaments could affect the career trajectories of players in the ATP circuit in a non-trivial way. Those trajectories then create complex relationships between players and tourneys. Indeed, players do not have the possibility to participate in all available tournaments and choose which tourney to sign up for based not only on their own skills but also on the characteristics of tourneys themselves (e.g., the court), their relevance during a season (preceding or succeeding famous events, for example), and their prestige. One could quantify tournament prestige from their prizes in terms of awarded points. Yet, assessing the tourney level based only on prizes does not capture the prestige perceived by the players and determined by their choices. Following Ref. [7], we propose a new method to assess the level of a certain ATP tournament, which not only captures its historical prestige but also includes the reciprocal influence of players and the reputation of a given competition. More importantly, this method does not require any previous knowledge about the tournaments or their prize points. We define a network where nodes are ATP tourneys and links depend on players’ careers. A directed link \((i, j )\) is created when a player first competes in tourney i, then in tourney j, and is weighted by the number of players who have the same attendance sequence [7] (see panel A of Fig. 4 for an example). Note that we connect tourney i to all consecutive tournaments attended by a player and not only to the tourney attended immediately after i. Thus, we consider the effects of all the competition choices that players made during their careers in the data. In this way, the movements of players link competitions far in space and time, and recurrent movements signal that those competitions tend to co-occur in players’ careers.

The resulting network has 659 nodes and 255,055 edges. We focus on the largest strongly connected component of the original network, which has 651 nodes and 254,583 links; from now on, we refer to the largest component as our network (see Table S1 in the SI for more details on the features of the network, such as its density and clustering coefficient).

From the co-attendance network, we can extract a measure of tourney prestige that correlates with the importance of tournaments in terms of their points. The prestige of a tournament can be derived from the topology of the network, using the eigenvector centrality [7, 40] (see also Methods for the mathematical definition we used). This definition captures the historical level of the competitions (Fig. 4B), expressed by the maximum number of points assigned to each tournament category (the allocation of points awarded per tournament is explained in the Methods section and summarized in Table S2 of the SI). Note that identifying tournaments based on their awarded points still remains a categorical definition, as tourneys belonging to the same categories award the same points in each round, in general (see Table S2 of the SI).

We divide competitions into three groups based on their centrality: The most prestigious tournaments are in the top 10% (above the 90th percentile), and we refer to them as high-level tourneys; the bottom 50% (below the 50th percentile) of the competitions are labeled as low-level tournaments; the others fall into the medium-level group of events. Tournaments belonging to the same category in the ATP can have vastly different centralities, suggesting that the network topology allows a fine-grained distinction between them that cannot be obtained by looking at categories only (see Fig. 4C). Having defined the level of the tourneys, we now focus on the possible connections between those levels and the success of players in the ATP circuit. These connections are crucial to determine whether the opportunity of competing in a given tournament could be more relevant than, or at least as relevant as, players’ abilities.

We check the possible differences in tournament attendance within the first ten tournaments of athletes’ professional careers on the ATP circuit (left panels of Fig. 5). First, in Fig. 5A, we observe the eigenvector centrality of the first ten tournaments for each group of players based on their career peak. We find that, at the beginning of their career, players attend competitions with comparable centrality, having median values in between the thresholds (dashed lines) of tournament splitting (we consider the median due to the asymmetric distribution of the centrality in our network, as shown in Figure S3 of the SI). Only after a considerable number of tourneys do top players attend only high-level tournaments, which means that they consistently participate in events having a central position in the co-attendance network (see Figure S4 in the SI). Then, we inspect the fraction of players who enter a certain tourney of a given level at the beginning of their career (panel C of Fig. 5). We do not observe a pronounced prevalence of future top players in high-level competitions (red bars in Fig. 5C).

Interestingly, we find no significant differences in the prestige of tournaments players can access when their careers start. One might argue that the seasonality of tournaments plays a role, hence affecting the centrality of players’ first competitions: In other words, if the centrality of the first tourneys of the season is around the median value, we should expect the trend observed in panel A of Fig. 5. Nonetheless, professional players can start their career on the ATP tournament circuit at any time during a competitive season, coincident with the calendar year. Therefore, the centrality of the opening tournaments of the season (in other words, the tournaments organized in January/February) does not determine the entire trend of Fig. 5A. Consequently, players’ first attended tournaments can vary widely from athlete to athlete. It is also worth mentioning that we neglect the influence of the junior circuit on players’ professional development. According to some studies [41‐43], the youth career could impact the future success of an athlete in tennis. Even if that impact were not a prerequisite for professional success [37, 44], young players who performed well at the junior level could be favored to access more prestigious ATP venues. However, such an effect, if present, does not create a significant gap among players in terms of the level of the first attended tourneys. We specify that we do not differentiate players by age or other factors like country of origin or physical characteristics (e.g., height, left-handed or right-handed, etc.).

Since no patterns emerge when looking at tournament attendance, one might ask if there are differences related to performance. In our framework, tennis performance is expressed by the outcome of matches. Thus, we check for patterns linking the victory of matches and the start of players’ professional careers in the ATP circuit. To do so, we focus on the first win of a match at the beginning of tennis players’ careers. Specifically, we are interested in the first victory in the main draw (i.e., the starting lineup of a tourney after the qualification rounds) of the first ten tournaments they attended. We consider the first match win because it allows us to directly compare the outcome of players’ performance for all the competitors. To visualize the relationship between the first win and the tourney level, in terms of centrality, we refer to the right panels of Fig. 5. In Fig. 5B, the eigenvector centrality of the top players is the only one above the threshold of high-level competitions. Note that the trend remains constant irrespective of the time of the first win, indicating that there is no distinction between winning earlier (within the first five tourneys) or later (after the sixth tourney). We find that most of the top players (around 70%) have their first win in the main draw of a high-level tournament (Fig. 5D). Furthermore, only top players can be identified by looking at the prestige of their first match win. Both middle and bottom players have similar behavior (Fig. 5D), and their first victory rarely occurs in high-level competitions.

To better understand the discrepancy in the behavior of top players, either when we consider only their attendance or when we add their first win, we compare the boxplots of the centrality of the players’ first tournaments with that of their first win (see panels E and F of Fig. 5, respectively, while a fine-grained visualization is available in the SI, Figure S5). In this way, we observe a clear difference between the two situations. In Fig. 5E, there is no distinction between the top, middle, and bottom players, with respect to the network-based prestige of the first tournaments they attended. Panel F of Fig. 5, instead, shows that the average centrality of the first win of the top players crosses the threshold of high-level tourneys. In particular, the top players’ boxplot is the only one that changes from panel E to F, which means that the higher prestige of the top players’ first win cannot be explained by the average level of the first attended tourneys. Panels E and F of Fig. 5 highlight the relationship between the prestige of the tournament in which the top players win their first match in a main draw and their future success. To further validate our analysis, we computed the correlation between (1) the player’s career peak and the median centrality of the first tournaments they played; and (2) the player’s career peak and the centrality of the tourney where they got their first win in a main draw. We use Spearman’s correlation coefficient and find that \(r_{s,1} = -0.008^{\mathrm{ns}}\) and \(r_{s,2} = 0.47^{***}\), where ^ns and ^∗∗∗ indicate the level of significance of the two values, that is, the p-value is greater than 0.05 (not significant) and less than 0.001 (significant), respectively.

Whether comparing players in groups or directly through their maximum number of points, we conclude that the prestige of the tournament where they first win a match in the main draw is a revealing factor for the future career of male tennis players. In the Supplementary Information, we show two examples of individual career progression before and after their first win, each time comparing a middle and a top player having won their first match in a tournament of the same ATP category but different centrality level (Figure S6). It should be noted that taking the qualification rounds into account does not appreciably change our findings (see Figure S7 of the SI). Furthermore, we do not assume that players should attend at least ten tournaments to be in the dataset, and we do not exclude active players, but adding these constraints does not significantly alter our results (see Figures S8-S9 in the SI). Lastly, we check the eventual relationship between ranking points and tournament centrality. We observe that the increase in ranking points that players had a week after winning their first match is only weakly correlated either with the centrality of the tourney in which they had their first win or with their future success (see Figure S10 in the SI).

To assess the significance of our findings, we build two distinct null models for the network of co-attendance of ATP tournaments. Building on previous work [1, 7], we proceed as follows (Fig. 6A): In the first model, we reshuffle the careers of each player individually so that the events they play are the same but have a different temporal order; in the second model, we reshuffle all the competitions attended by the players, so that each player’s career has the same number of events, but it can consist of different tourneys. In both cases, all temporal correlations are destroyed. We choose these two randomizations because they focus on different aspects: The first randomization preserves not only the number of competitions but also the actual events players attended; the second randomization preserves the number of tournaments of each player and the distribution of competitions among all players, destroying, however, the possible player-tourney correlations.

We repeat these two randomizations multiple times to create an ensemble of 500 random realizations. We specify that in both configurations we preserve the information about the time of the first win as given by the data. Therefore, the randomizations would only change the corresponding tournament in the sequence, but not when the first win of a player occurred (as illustrated by the asterisk in Fig. 6 A). For each realization, we build the correspondent co-attendance network and evaluate tournament centrality, considering the prestige of competitions as done in the data.

We analyze the average distributions of tournament centrality per group of players, thus keeping the possible relationships between players’ success and prestige of their initial ten competitions. We follow the order of tourney attendance and define tournament levels based on their importance in the null models. Panels B to E of Fig. 6 show that the null models cannot reproduce at the same time both the prestige of the tournaments attended and that of the first match win in the early stage of top players’ professional career in tennis (Fig. 5E-F).

The reshuffle of the individual sequences of tournaments per player increases the gap between top and middle-bottom players: given the cyclic nature of individual sports, where competitions repeat themselves every year around the same week, players are encouraged to attend the same tourneys season after season, to preserve or improve their amount of points. It follows that reshuffling the careers of top players only anticipates those tournaments they start to play once they have already reached the top. In contrast to empirical data, consequently, top players tend to compete more in high-level tournaments from the beginning of their professional careers, so that they are more likely to win their first match in highly central competitions (panels B-C of Fig. 6).

On the other hand, the randomization of all tourneys destroys the cyclic trend of sports based on seasonal tournaments. Thus, we do not expect significant differences in the level of competition among players or any eventual correlation between their career peak and their results. Indeed, we observe in Fig. 6D-E that there is no distinction between the groups and no patterns emerge in the prestige of their tournaments.

We also compare the mean value of Spearman’s correlation coefficients for both null models over all configurations. As described for the data, we computed the correlation between (1) the player’s career peak and the centrality of the first tournaments they played, and (2) the player’s career peak and the centrality of the tourney where they got their first win in a main draw. The results of both randomizations are summarized in Table 1. When reshuffling individual sequences, we observe that the athlete’s career peak is slightly positively correlated with both the centrality of the first tournaments attended and the centrality of the first win. Instead, when we randomize all possible tournaments among the players, we find almost zero correlation in both cases.

Whereas we find a significant difference between these two correlation coefficients in the data, we observe that such a discrepancy is not significantly different from zero for both null models. This means that the behavior we observe in the data cannot happen by chance, i.e., the discrepancy between the centrality of the first tourneys top players attend compared to where they first win a match in their professional career is not random. Thus, the prestige of the tournament where male tennis players have their first win is a predictor of their future careers.

Professional male tennis players accumulate points in the ATP ranking during a season (52 weeks). Any new result cancels out the corresponding one from the previous year, if present, so the rankings are updated approximately every week [30]. Tournaments of different ATP categories award different point scales. Within each category, players generally compete for the same (fixed) number of points in each round. Among the competitions we considered in this work, Challengers are the less prestigious, as awarded points vary from 3 (to the loser of the first round of qualifications) up to 125 (winner of the tourney), whereas Grand Slams are the most prestigious, as players’ awards range from 8 to 2000 points. The other tourneys fall in between: Masters 1000 points vary from 8 to 1000; ATP 500 points scale from 4 to 500; ATP 250 points range from 3 to 250. Detailed scales per tournament are available in the SI (Table S2).

In Fig. 3A, we show that players belonging to a group i have a probability \(P_{i}(T \geq t)\) of attending at least t tourneys where they win a match, within the first ten tournaments of their career in the ATP. This probability results from the cumulative distribution of the function \(p_{i}(t)\): Where \(p_{i}(t)\) is the fraction of players of a group \(i = \{ \mathrm{top, middle, bottom} \}\) with a win w in exactly t attended tourneys, with \(1 \leq t \leq 10\), namely:

We also consider the probability P that players have won their first match since becoming professional in a given tourney t (Fig. 3B). Equation (3) reports the fraction of players who have their first win at time t, that is, at the tournament \(1 \leq t \leq 100\), grouped by their career peak. Given the players in the group i with their first win \(w^{*}\) at the time t, we can write the following: Where \(P_{i}(t,w^{*})\) is the fraction of players with their first win \(w^{*}\) at time t, that is the ratio of the number of athletes \(N_{i}(t,w^{*})\) with their first win \(w^{*}\) at time t, divided by the number of players who competed at time t, \(N_{i}(t)\).

The co-attendance network of tennis tournaments is based on the career trajectories of the players in our data. This results in a weighted directed network, where nodes are tourneys and links \((i, j )\) are created when players first attend tournament i, then j. Link weights are obtained by the number of times different players generate the same link. Specifically, every link \((i, j )\) has a weight \(\tilde{\omega}_{ij} = \frac{\omega _{ij}}{\omega _{\mathrm{max}}}\), normalized to the maximum possible weight found in the network, i.e., \(\omega _{\mathrm{max}}= \mathrm{max} ( \omega _{ij} )\). We use the topology of the co-attendance network to assess the prestige of tourneys. Specifically, we rely on the eigenvector centrality \(x_{i}\) [34], defined for a node i in a directed network as proportional to the centralities of the nodes that point to i [40]: Where the term \(\kappa _{1}\) represents the largest eigenvalue of the adjacency matrix A whose elements are \(A_{ij}\).