Defining a historic football team: Using Network Science to analyze Guardiola’s F.C. Barcelona

The application of Network Science to social systems has introduced new methodologies to analyze classical problems such as the emergence of epidemics, the arousal of cooperation between individuals or the propagation of information along social networks. More recently, the organization of football teams and their performance have been unveiled using metrics coming from Network Science, where a team is considered as a complex network whose nodes (i.e., players) interact with the aim of overcoming the opponent network. Here, we combine the use of different network metrics to extract the particular signature of the F.C. Barcelona coached by Guardiola, which has been considered one of the best teams along football history. We have first compared the network organization of Guardiola’s team with their opponents along one season of the Spanish national league, identifying those metrics with statistically significant differences and relating them with the Guardiola’s game. Next, we have focused on the temporal nature of football passing networks and calculated the evolution of all network properties along a match, instead of considering their average. In this way, we are able to identify those network metrics that enhance the probability of scoring/receiving a goal, showing that not all teams behave in the same way and how the organization Guardiola’s F.C. Barcelona is different from the rest, including its clustering coefficient, shortest-path length, largest eigenvalue of the adjacency matrix, algebraic connectivity and centrality distribution.


Results
Average passing networks. Figure 1 shows an example of a football passing network, in this case the average network of FCB against Real Madrid in the season 2009/2010. Note that links are unidirectional (from player A to player B) and weighted according to the number of passes between players. In the figure, nodes (i.e., players) are placed in the average position from where their passes were made and the width of the links is proportional to the number of passes between players. Also note that both the x and y coordinates of the field are bounded between [0,100] and are measured in "field units" (f.u.), since not all fields have exactly the same dimensions. Finally, the radius of the nodes is proportional to their importance in the passing network, quantified by means of the eigenvector centrality (see Methods).
First, we analyzed the average passing networks of all matches played by FCB during season 2009/2010 (38 in total), obtaining the networks of FCB and their rivals. Specifically, we obtain 2 average passing networks for each match (1 per team), both of them including all passes and positions along the match and projecting them into a single network for each team. See the Methods section for details about the construction of average passing networks. Previous literature about average passing networks has shown that they reveal information about the way a team is organized 50 and are also related with team performance 51 . Figure 2 shows the comparison between 8 different parameters obtained for FCB and its rivals. Four of them, (a) the number of passes L, (b) the number of shots to goal M shots , (c) the number of goals M goals and (d) the number of points M points (at the end of the season) are classical metrics of the team performance. Note that, in order to compute these 4 variables, there is no need to obtain and analyze the network structure of each team, despite some of them (i.e., the number of passes) can affect the organization of the passing networks. The other 4 parameters of Fig. 2 are related to the spatial properties of the networks: (e) x-coordinate of the network centroid 〈X〉, (f) y-coordinate of the network centroid 〈Y〉, (g) dispersion of the position of the players around the network centroid NC disp and (h) average ratio between the passing distance parallel and perpendicular to the opponent's goal 〈Δ y 〉/〈Δ x 〉 (see Methods for details). Left bars in all plots correspond to the average values of these metrics for all matches of FCB along the season, while right bars are the same metrics obtained for the rivals at the same matches. FCB is always averaged with itself, while all other teams are averaged together, the reason being that we are only interested in observing differences between the FCB and all other teams. Error bars account for the standard deviations of each metric. Plots in yellow highlight statistically significant differences (see Methods for details about the statistical analysis).
As we can see in Fig. 2A, the number of passes made by FCB is much higher than the average of their rivals. This fact is a consequence of Guardiola's playing style, focused on keeping the ball as much as possible ("In football, I am very selfish: I want the ball for me", "take the ball, pass the ball" 52 ). The high number of passes unavoidably leads to passing networks with links that have higher weights and, as we will see, this fact will have consequences on the network parameters. The number of shots to goal is also higher in FCB (Fig. 2B), leading to a higher number of goals ( Fig. 2C) and, ultimately to a high number of points accumulated during the analyzed matches (Fig. 2D). In fact, FCB won the league with 99 points (31 wins, 6 ties and only 1 loss). Note that these four metrics (passes, shots, goals and points), specially the last three, are traditionally considered as indicators of the team performance, thus revealing that FCB was the best team during season 2009/2010. Bottom plots of Fig. 2 are related with the spatial features of Guardiola's team. The 〈X〉 and 〈Y〉 average coordinates of all passes made during the match define the network centroid (or the network center of mass). We can observe in Fig. 2E how FCB played closer to the opponents goal (〈X〉 FCB > 〈X〉 rivals ), while no differences are found at the 〈Y〉 coordinate (Fig. 2F), indicating no preference for any of the sides of the pitch. Interestingly, the dispersion of the position of the players around the centroid (see Methods) is slightly higher for FCB, which indicates that the area covered by the initial position of the passes made by all players is wider (Fig. 2G). Finally, it is worth analyzing the ratio of advance 〈Δ y 〉/〈Δ x 〉, which is an indicator of the direction of the passes of a team, since the Δ y = y 2 − y 1 of a pass is the difference between the y-coordinates at the final (y 2 ) and initial points (y 1 ) of a pass, while Δ x is defined, accordingly, for the x-coordinate. In Fig. 2H, we can observe how FCB has a ratio of advance much higher than the rivals, which reveals that passes are more parallel to the opponent's goal than the rest of the teams. Note that this metric is independent from the number of passes, and it is an indicator of how "direct" the game of a team is. Clearly, FCB is not concerned about advancing directly towards the goal, but on moving the ball in parallel, probably to find the most adequate moment to advance.
But, how is the structure of the average passing networks? And, more importantly, are there differences between FCB and the rest of the teams? Figure 3 shows the comparison of 6 parameters directly related with the topological organization of the average passing networks (see Methods for a detailed description of all these network parameters). In Fig. 3A, we plot the clustering coefficient C, which is related to the amount of triangles created between any triplet of players. Clustering coefficient is an indicator of the local robustness of networks 31 , since when a triangle connecting three nodes (i.e. players) exists, and a link (i.e., pass) between two nodes is lost (i.e., not possible to make the pass), there is an alternative way of reaching the other node passing through the other two edges of the triangle. In football, the clustering coefficient mesures the triangulation between three players. As we can observe in Fig. 3A the value of C is much higher in FCB, which reveals that connections between  Figure 1. Schematic illustration of a football passing network. In the plot, players are represented by circular nodes, whose size is proportional to their eigenvector centrality, a mesure of importance in the network structure. The position of each player is given by the average of the positions of all passes made by the player along the match. The width of the links is proportional to their weights, which account for the number of passes between players. Note that links are unidirectional. In this example, we plot the average passing network of the match between F.C. Barcelona and Real Madrid, played during the season 2009/2010 at Santiago Bernabeu Stadium. Datasets leading to the passing network were provided by Opta.
three players are more abundant than at their rivals. The average shortest path d is an indicator about how well connected are players inside a team. It measures the "topological distance" that the ball must go through to connect any two players of the team. Since the links of the passing networks are weighted with the number of passes, the topological distance of a given link is defined as the inverse of the number of passes. The higher the number of passes between two players, the closer (i.e., lower) the topological distance between them is. Furthermore, since it is the ball that travels from one player to any other, it is possible to find the shortest path between any pair of players by computing the shortest topological distance between them, no matter if it is a direct connection or if it involves passing through other players of the team. Finally, the average shortest path d of a team is just the average of the shortest path between all pairs of players. As we can observe in Fig. 3B, the shortest path of FCB is much lower than their rivals, which reveals that players are better connected between them. As we will discuss later, note that this fact could be produced by the network organization or just being a consequence of having a higher number of passes, which reduces the overall topological distance of the links and, consequently, the value of d. Figure 3C shows the comparison between the largest eigenvalue λ 1 of the connectivity matrix A (also known as the weighted adjacency matrix), whose elements a ij contain the number of passes between players i and j 31 . The largest eigenvalue has been used as a quantifier of the network strength 53 , since it increases with the number of nodes and links (see Methods). As expected (due to the high number of passes), the largest eigenvalue λ 1 of FCB is much higher than the corresponding values of its rivals. This metric reveals the higher robustness of the passing network of Guardiola's team, which indicates that an eventual loss of passes would have less consequences in F.C. Barcelona than in the rest of the teams.
It is also worth analyzing the behavior of the second smallest eigenvalue λ  2 of the Laplacian matrix  L, also known as the algebraic connectivity (see Methods). The value of λ  2 is related to several network properties. In synchronization, networks with higher λ  2 require less time to synchronize 54 and in diffusion processes, the time to reach equilibrium also goes with the inverse of λ  2 . In the context of football passing networks, λ  2 can be interpreted as a metric for quantifying the division of a team. The reason is that low values of λ  2 indicate that a network is close to be split into two groups, eventually breaking for λ =  0 2 . In this way, the higher the value of λ  2 the more interconnected the team is, being a measure of structural cohesion. In Fig. 3D, we have plot the comparison of λ  2 , which reveals that FCB attacking and defensive lines are more intermingled, leading to a λ  2 higher than its rivals. www.nature.com/scientificreports www.nature.com/scientificreports/ Finally, Fig. 3E-F show how centrality (i.e., the importance of the players inside the passing network) is distributed along the team, a metric calculated by means of the eigenvector related to the largest eigenvalue of the connectivity matrix (see Methods). Figure 3E contains the average dispersion of centrality and Fig. 3F shows the highest value of a single player. In both cases, differences are not statistically significant to support evidences of a different centrality distribution between FCB and the rest of the teams.
Temporal evolution of the network metrics. As we have seen in the previous Section, average passing networks show differences between the organization of FCB and its rivals. However, these difference may be interpreted as a consequence of the higher number of passes between Barcelona players, which could lead to statistically significant differences in a diversity of network metrics, namely, a reduction of the average shortest path d and an increase of the clustering coefficient C, largest eigenvalue λ 1 and algebraic connectivity λ  2 . In view of these results, two questions must be addressed before any interpretation: (i) Is just the number of passes behind the differences of the network parameters? and (ii) is it enough to look at the average values of the network metrics? To address both issues, we have conducted a complementary study where passing networks are constructed in a different way. On the one hand, we are going to define passing networks as non-static entities, thus evolving in time, and we will track the evolution of their parameters. On the other hand, we are going to exclude the importance of the number of passes, in order to just focus on the topological organization of the networks. With these two objectives in mind, we construct the l−pass networks of a team as the networks containing l consecutive passes, with l ≪ L, being L the total number of passes during the match. In our study, we set l = 50, since it is a value low enough to allow a tracking of the network evolution along the match and, at the same time, high enough to guarantee the creation of a network between players (too low values of l would lead to networks with disconnected components). Therefore, we obtain the 50-pass networks in the following way: (i) we construct the network of the first 50 passes of a team since the beginning of the match, (ii) we calculate its parameters, (iii) we dismiss the oldest pass and include (sequentially) a new one, (iv) we recalculate the network parameters and (v) we repeat the procedure until the last pass of the match is included.
Note that 50-pass networks contain exactly the same number of passes for both teams and, thus, any difference between network metrics can not be attributed to the total number of passes. In addition, also note that metrics evolve in time and their values can be related to a certain moment of the match. However, it is also important to remark that the time required to construct a 50-pass network can differ from team to team.  shows an example of the evolution of 3 parameters of the 50-pass networks of two teams along a match, specifically, the 〈X〉 coordinate of the centroid (A), the ratio of advance 〈Δ y 〉/〈Δ x 〉 (B), and the dispersion of the network centrality (C). Parameters are calculated, for both teams, during the match between Real Madrid (red lines) and FCB (green lines), whose final score was 0-2. Vertical lines indicate the moment at which a goal was scored. Figure 4A shows how the position of the team moves forward and backward during the match. In this particular case, Real Madrid plays, most of the time, more advanced than FCB, which did not lead to an advantage in the result. Note how the centroid of FCB seems to be more stable, while Real Madrid has higher fluctuations, arriving to its maximum value around minute 63. Also note how FCB is the first team to construct the 50-pass network around minute 9, while Real Madrid required 20 minutes.
In Fig. 4B, we plot the ratio of advance of the 50-pass networks of both teams. Again we can see fluctuations of the parameter during the match. Specifically, FCB has a highest value during the first part of the match. However, we can observe how Real Madrid increases its advance ratio as time goes by, eventually overcoming FCB during the second half.
Finally, Fig. 4C shows the fluctuations of the centrality dispersion of the players of both teams. We can observe how Real Madrid has a strong increase of the centrality dispersion between minutes 50 and 70, which seems to be related with the period where the centroid of the team advances towards FCB's goal (see Fig. 4A). This change of the centrality distribution could be related to a change of the style of playing. Since centrality dispersion increases, there is a higher heterogeneity in the importance of the players in the passing networks, which could be related to the fact that a few players are taking the lead of the team. However, this change in the organization of the passing network does not seem to be effective, since the second goal of FCB comes around to the maximum of centrality dispersion.
The fact that network metrics change during the match increases the complexity of the study. It is expected that several factors may influence the fluctuations of the network parameters (a goal, a substitution, physical condition, etc…) and, furthermore, not all teams may behave in the same way. From the diversity of factors, here we are going to focus on the particular organization of each team before a goal. With this aim, we have analyzed the value of the network parameters, for all teams, before scoring/receiving a goal. Our purpose is to detect the existence of differences in the network metrics and identify those parameters that change before scoring or receiving a goal. Figure 5 shows the average values of 4 temporal and spatial metrics obtained before scoring/receiving a goal (during season 2009/2010). The diagonal line (y = x) helps to identify those metrics that behave differently when scoring or receiving a goal. In Fig. 5A we can observe how FCB is the team requiring less time to construct the 50-pass network, both when scoring or receiving a goal. In fact, as indicated by the diagonal line, it takes approximately the same time in both cases. On the opposite side, we find Athletic Club and Osasuna, both teams characterized by a direct game towards the opponents's goal. Concerning the 〈X〉 position of the centroid, we can observe in Fig. 5B that, despite having a high value, FCB is not the team that constructs its network closest to the www.nature.com/scientificreports www.nature.com/scientificreports/ opponent's goal, since it is overcome by Real Madrid and Tenerife. Note that Tenerife ended up the season in the last position, which indicates that playing forward it is not a sufficient condition to achieve good results. However, it is also worth noting that all teams, with the only exception of Osasuna, are placed above the line given by the function 〈X〉 scored = 〈X〉 received . This fact reveals that when a team scores a goal is, in average, playing more advanced than when it receives it. In Fig. 5C we have compared the ratio of advance 〈Δ y 〉/〈Δ x 〉 of all teams, showing that Barcelona is not only the team with the highest value (both when scoring and receiving a goal) but also the one deviated the most from the the diagonal line. In this way, FCB is the team that increases the most its probability of scoring a goal when increasing the ratio of advance. Finally, Fig. 5D shows the average dispersion of the position of the players around the centroid coordinates of the 50-pass network. Interestingly, we can observe how FCB is one of the teams with lower dispersion of La Liga and, furthermore, the dispersion increases before a goal is received, indicating that FCB performs better when players are closer to the network centroid. Figure 6 shows, in a similar way, the values of 6 different network parameters obtained for all teams (during the whole season). Interestingly, FCB has the highest values of the league at 4 of them: The clustering coefficient (Fig. 6A), the largest eigenvalue of the connectivity matrix (Fig. 6C), the centrality dispersion (Fig. 6E) and the highest centrality of a player (Fig. 6F). High values of these four metrics are related to strong and robust networks: (i) a high clustering coefficient is an indicator of local robustness 31,38 , (ii) the largest eigenvalue λ 1 is also an indicator of global robustness 53 ; when the number of nodes and links are the same, λ 1 increases when important players are, in turn, connected between them, (iii) a high centrality dispersion together with a high value of maximum centrality are indicators of heterogeneity in the network structure, and heterogeneous networks are know to have strong resilience against random failures 55 (i.e., the loss of weight of the links, due to lost passes, would have less impact on the overall structure).
At the same time, the analysis shows low values at other 2 metrics: the shortest-path length d (Fig. 6B) and the algebraic connectivity λ  2 (Fig. 6D). In this case, having a low shortest-path length is an indicator of a better connection between players, since the ball can travel from a player to any other in a lower number of steps. Finally, it is interesting to note that FCB has one of the lowest algebraic connectivities, which is an indicator of structural integration. Low values of λ  2 reflect that the team is more split into two different groups. Note that, when the algebraic connectivity λ  2 is calculated from the average connectivity matrix (Fig. 3D), FCB has a value higher than their rivals, reflecting a higher cohesion of the whole team. However, when it is computed from the 50-pass networks, FCB algebraic connectivity is one of the lowest. A possible explanation is that cohesion of the team may be grounded on a higher number of passes between players, and not on the topological organization of the network.

Discussion
What passing networks tell us about FCB. As we have seen, using Network Science to analyze football passing networks gives a new perspective that allows distinguishing between different teams and relating network properties to the teams particular style of playing. Here, we have made use of these metrics to characterize the passing networks of Guardiola's Barcelona, focusing on the season 2009/2010 of the Spanish national league,  www.nature.com/scientificreports www.nature.com/scientificreports/ one of the years where FCB was considered to reach its top in terms of playing style and trophies. When passing networks are constructed as a simple addition of all passes made between players during the match, statistically significant differences between the passing networks of FCB and its rivals arise. Specifically, the clustering coefficient, the shortest-path, the largest eigenvalue of the connectivity matrix and the algebraic connectivity, always have "better" values in the Catalan team. The term "better" refers to the fact that differences in these network properties are related with a higher local resilience against the loss of passes (due to a higher clustering), a lower number of steps to connect any two players of the teams (due to a lower shortest-path length) and a higher connectedness between the whole team, as indicated by a higher largest eigenvalue of the connectivity matrix and a higher algebraic connectivity.
However, it is worth looking beyond the differences in the network metrics and trying to find the reasons behind them. When focusing on the number of passes made by FCB we can, first, observe that it is much higher than their rivals and, second, that the advance ratio, measuring the percentage of distance that the ball advances parallel to the opponent's goal is also much higher. Concerning the latter, note that the advance ratio is not related to the number of passes and, therefore, there is not an obvious reason why it should influence network parameters. However, the number of passes has, indeed, crucial consequences on any quantitative analysis using Network Science. The fact that we are comparing networks with the same number of nodes (eleven) but links with different weights (number of passes) has unavoidable consequences on the network parameters. For example, since the "topological" distance between two directly connected players is given by the inverse of the number of passes between them, the higher the average number of passes of a team, the lower topological distance between their players. Despite being obvious, a reasonable conclusion of the study is that increasing the number of passes benefits the general properties of passing networks. However, comparing the properties of two networks with different number of passes hinders the role played by the network topology itself, i.e., we can not say that a network is better organized, since we can not separate the effect of the number of passes ("quantity") from that of the topology of the network ("quality").
A second issue related to the number of passes is possession. Note that the number of passes is intimately related to the possession a team has. A team with higher possession will unavoidably have more passes and that is exactly what FCB, under the guidance of Guardiola, is doing. But to what extend can we relate possession to the particular organization of FCB passing networks? Are the reported values of its network parameters just a consequence of having the ball more time?
A third issue arises when trying to interpret the results of the averaged passing networks. Since, as we have seen, network organization and, consequently, network parameters, are continuously evolving during the match, considering the sum of all passes may hide interesting information about how different crucial events influence a team's style of playing, such as a scored/received goal, a substitution or, simply, the fatigue of the players as time goes by. www.nature.com/scientificreports www.nature.com/scientificreports/ In order to overcome these three issues and, particularly, to exclude the influence of the number of passes (or possession) and, at the same time, accounting for the evolution of the network topology we studied the properties of the 50-pass networks. A part from the benefits of tracking their temporal evolution, 50-pass networks contain exactly the same number of nodes and links for the two teams playing a match, which allows a direct comparison of the network organization, no matter what the final number of passes of each team is. However, the tracking of the parameters of the 50-pass network shows that network parameters are in continuous evolution, which increases the complexity of the analysis. Here, we focused on the state of the passing network just before scoring/ receiving a goal, which allows to extract information about what are the network properties associated to the ability of a team to score/receive a goal. With such an approach we were able to complement the information extracted from the averaged passing networks, obtaining a more detailed profile of Guardiola's team. These results reinforced all the conclusions drawn by analyzing average passing networks and included the following additional information about FCB: 1. It is the team that requires the shortest time to construct 50-pass networks, and this time remains unaltered when scoring/receiving a goal, 2. It is the team with the highest advance ratio (i.e., the team that plays the most horizontal to the opponent's goal) and this metric is specially high before scoring a goal, 3. The dispersion of the players around the network centroid is the lowest but significantly increases before receiving a goal, 4. The clustering coefficient is higher when receiving goal than when a goal is scored, 5. The shortest-path is one of the lowest and does not depend on scoring/receiving a goal, 6. The largest eigenvalue of the adjacency matrix, measuring the strength of the network is the largest, and significantly increases before receiving a goal, 7. The algebraic connectivity, measuring the cohesion between groups of players, decreases before receiving a goal (i.e., the interplay between groups is reduced), 8. The highest centrality acquired by a single player and the centrality dispersion are the highest, which indicates that the importance of players in the FCB network is not evenly distributed, with one player, Xavi, being the hub of the passing networks.
Note that all these patterns refer exclusively to FCB, while passing networks corresponding to other teams behave in its own way. Therefore, one of the conclusions we can draw from Figs 5 and 6 is that variables of each team before scoring/receiving a goal behave in a particular way. For example, as we can see in Fig. 5D, the dispersion of FCB's players around the position of the network's centroid is higher when a goal is received, indicating that when players are more separated from the centroid, the risk of receiving a goal increases. However, if we look at the same parameter for Valencia CF, we can observe that the behaviour is just the opposite, and higher dispersions around the team's centroid (note that in this case players are occupying more field) are reported when scoring a goal.
Finally, it is worth mentioning the limitations and risks of our study. As we have seen, computing the parameters related to the average passing networks gives interesting, but limited, information about the way a team is organized. As shown in Fig. 4, there exist strong fluctuations on the network parameters during a match and defining 50-pass networks is a reasonable option to capture the evolution of the structure of passing networks. However, there are associated issues and alternatives that may be explored in further studies. For example, the length of the 50-pass networks could be adapted to capture the "momentum" of the match, which may change from team to team or just due to the events occurring during the match. Therefore, it would be interesting to find a way of defining the most adequate time windows and how the length of these windows are related to the particular style of playing a team has. Another limitation is related to the causes of the parameter fluctuations, since they can have different origins (a goal, a substitution, fatigue, etc…). It would be extremely useful to identify all possible variables affecting the network organization and compute the network parameters after these particular events occur, trying to identify what are those variables that crucially change the style of playing of a given team.

An interpretation within a football framework.
Going beyond passing networks, the strategy of having the ball most of the time leads, in general, to controlling the game by creating a dynamical context to which the opposing team needs to adapt and, in particular, gave FCB a systematic superiority that led to an increase of the scoring opportunities.
Firstly, FCB defense can lengthen or shorten the space by moving the line of defenders forwards or backwards. In other words, it can play with the occupied length of the field and use the off-side area in its favor. The fact that FCB network centroid was advanced (in average) compared to its rivals (see Fig. 2E), left the opponents a smaller area of the pitch and fewer playing options.
Furthermore, Barcelona organized the team into "situational areas" around the ball, which comprised the commitment of five or six players. Inside these areas, the team must overcome a challenge, i.e., either play the ball or recover it, leading to a division of the game into two phases. Players organized spontaneously inside these situational areas (and, as we will discuss, after training these situations), communicating with each other and exchanging physical, verbal, and motor-related signals. This way of modulating the playing field into building blocks leads to more playing patterns, resulting in more different options to overcome the rival.
Specifically, during the attacking phase (see Fig. 7), the player with the ball had a helping area #1 (also known as the helping zone) with two players forming possible triangles within a distance d 1 of 10 to 15 meters. At the same time, there was a co-operation area (#2) with two more players (one slightly forward and the other covering the back) occupying a wider radius d 2 (around 20 meters). During this phase, passes were promoted between players www.nature.com/scientificreports www.nature.com/scientificreports/ inside the situational area, which, from the network perspective, resulted in a higher clustering coefficient (Fig. 3A) and a lower shortest-path distance between them (Fig. 3B). In addition, trying to keep the game inside a situational area promoted the creation of short passes, reducing the risks of losing the ball, as opposed to long passes.
On the other hand, the defensive phase started as soon as the opponent had the ball and was based on the creation of a large "interception space" to increase the chance of recovery. Similarly to the attacking phase, two regions were organized and coordinated over a radius of 20 meters. The fact that FCB played more advanced towards the opponent's goal than their rivals, (see Fig. 2A) together with the coordinated pressure, made any eventual recovery more dangerous, increasing the probability of subsequent shooting actions.
In addition to dividing the field into various areas, a distinctive factor of FCB was the role players adopted and their area of specialization. The fact that FCB promoted generalist players is linked to what is known as "total football" 46 . In effect, with the FCB's playing style, all players could play the ball, recover it, and score. As a consequence, a possible interpretation could be that more generalist profiles tend to generate more connections, take more advantage of the space, and generate different game options, leading to more complex passing networks. In addition, a team based on generalist players forces opponents to spend more energy and work harder at coordination. For example, a team without an obvious centre-forward player generates ambiguity and uncertainty for the three or four opposing defenders. One consequence of the promotion of generalist players is the the fact that, from the 18 field players that played more than 1000 minutes during the season, only 3 of them did not score a goal (Milito, Maxwell and Abidal).
Furthermore, a number of generalist players promoted the arousal of spontaneous playing patterns, that is, different ball flows and/or positions for players who are successfully scoring goals, passing, or recovering the ball. By managing the right trajectories and the right supporting positions, the opponent was forced to cover more ground running and increasing his fatigue. Controlling the ball while the opponent run out of energy leaded to much better positions for gaining superiority, creating surprises, and obtaining opportunities to score.
At the same time, it is worth noting the existence of a core group of players whose participation in the passing networks was higher than the rest. This fact is indicated by the high eigenvalue λ 1 (Fig. 6C), the high heterogeneity in the centrality of the players (Fig. 6E) and the existence of a player with the maximum centrality higher than the other teams (Xavi) (see Fig. 6F). In addition, the existence of this core could be related to the fact that the algebraic connectivity is reduced when analyzing 50-pass networks (an indicator of the existence of groups), since an underlying core-periphery structure, combining "leading" players with "follower" players, may lead to the existence of two identifiable communities. In this way, the existence of 4-5 players that carried and passed the ball most often could be translated into the existence of a certain distributed leadership in the different situational areas and phases of the game -while the other players followed, coordinately, the game carried out by these leaders.
Finally, we have to remark that the tactical organization was carefully planned and trained by Guardiola and his technical staff, and it was not a matter of serendipity 46 . In fact, this style of playing was one of the FCB's signatures and it was promoted at lower categories of the team. In this way, seven out of the ten players that played more than 1000 minutes during the 2009/2010 were raised up at La Masia, the FCB youth academy. In addition, three of them (Xavi, Iniesta and Messi) were designed as the three finalists of the Ballon D'Or at the end of that season, which is given to the best football player along the whole season.
Summarizing, we have identified a series of particular network properties that make Guardiola's Barcelona a team different from the rest, allowing the interpretation of the reported network parameters. We believe that further studies taking into account the spatiotemporal evolution of football passing networks, together with recent approaches including the construction of network-of-networks 53,56 , multilayer networks 57,58 or hypernetworks 59 could further enhance the understanding of how football teams, in particular, and sport teams in general, organize and evolve along a match and what are the key factors that determine their performance. Furthermore, despite our results are focused on team performance, they can be adapted to evaluate single players and their contribution to the team. This change of "scale", would imply some collateral issues, such as the difference in the number of matches played by each player or the fact that the position a player has in the team unavoidably affects his/her network properties. However, we believe that this kind of new approaches will be incorporated, in the years to come, to complement classical metrics of player performance.  Table 1 for details). Since we are concerned about the game of FCB, we focused on all matches played by this team, and analyze the passing networks of FCB and its rivals. We construct networks in two different ways. On one hand, we obtain the match average passing networks, where nodes are players and links represent the number of passes between them. Note that links are unidirectional and weighted according to the number of passes between players. To ease comparison between networks, each titular player is assigned a node at the beginning of the match. If a player is changed, the new player occupies the node of the previous player. In this way, we assure that all networks have eleven players, focusing on the structure of the network as a whole instead of the performance of isolated players.
On the other hand, we construct the "50-pass networks" with the aim of accounting for the temporal evolution of the game. 50-pass networks contain only 50 consecutive passes and are assigned the time of the last of these passes. This way, when the match begins, we wait for the first l = 50 passes to occur and, at this moment called t 0 , we construct and analyze the 50-pass network G t0 . Next, each time a new pass is made, we disregard the oldest of the passes of the network and include the new one, assigning the time of the last pass t to the new network G t . This kind of networks has two advantages compared to the averaged ones: (i) it accounts for the fluctuations of the network parameters along the match and (ii) it has exactly the same number of nodes and links for both teams, which detaches the influence of the absolute number of passes and focuses only on the structural differences between networks. It is worth noting that the number of passes to construct the network could be modified to another quantity, however it should be low enough to account to the fluctuations occurring during the match (i.e., avoiding averages) but long enough to guarantee the connectivity between all nodes of the network. In our case, we analyzed the effects of using different number l of passes and chose l = 50 as a trade-off value.
Definition of network metrics. Centroid coordinates and dispersion. 〈X〉 and 〈Y〉 centroid coordinates correspond to the average position of all pases of the network, i.e., all passes of the match in the average network and only 50 of them in the 50-pass passing networks. Specifically, we only consider the position from where the pass is sent. Values are given in field coordinates, which, in both axis, range from 0.00 to 100.00. In this way, the center of the field corresponds to coordinates [50.00, 50.00] and the center of the opponent's goal is [100.00, 50.00] (being [0.00,50.00] the center of the own team's goal). The centroid dispersion Cent disp corresponds to the standard deviation of the distances of the players with regard to the position of the network centroid.
Clustering coefficient. In general, the local clustering coefficient of a node i is obtained as the percentage of the nodes directly connected to it that, in turn, are connected between them. This measure can be averaged along the N nodes of the network to obtain the average clustering coefficient. However, when the network is weighted, we can not simply account for the number of nodes connected between them but, also, how the link weights are distributed. This is the case of passing networks, where the number of passes between pairs of players is not constant. In this way, we use the weighted clustering coefficient C w (i) to measure the likelihood that neighbours of a given player i will also be connected between them 60 : where j and k are any two players of the team and w ij and w ik the number of passes between a third player i and both them. Finally, the clustering coefficient of the whole network is obtained by averaging C w (i) over all players, i.e., = ∑ = C C i ( ) . Note that, the weighted version of the clustering coefficient characterizes the tendency of the team to form balanced triangles between players and it is a measure of local robustness.
Shortest-path length. In a passing network, the shortest path length d is the minimum number of players that must be traversed by the ball to go from one player to any other. Since passing networks are weighted (i.e., the number of passes between players is different), we have to take into account the different weights of the links, considering that, the higher the weight, the shorter the topological distance between two nodes. The topological length l ij of the link between two players i and j is defined as the inverse of the link weight, l ij = 1/w ij . However, www.nature.com/scientificreports www.nature.com/scientificreports/ when computing d for weighted networks, the shortest-path length between a pair of players may not be a direct link, since there could exist a shorter path by combining two (or more) alternative links. Therefore, we compute the minimal shortest-path p ij between all pairs of players using the Dijkstra's algorithm 61 . Next, we define the average shortest path d of the whole team as: where N = 11 is the total number of players of the team.
Largest eigenvalue of the adjacency matrix. The largest eigenvalue λ 1 of the weighted adjacency matrix A of a network is a measure of the network strength 53 . The weighted adjacency matrix A is a N × N matrix whose elements a ij contain the number of passes going from player i to player j. The largest eigenvalue of A is bounded by the average number of passes between players 〈S〉, as λ 1 ≥ 〈S〉, and also by λ ≥ ≥ 〈 〉 s max S s ( , ) max m ax 1 62 , where s max is the maximum number of passes that a player has made to any other player of his team. As a rule of thumb, networks with higher number of links (passes) will have a higher λ 1 and networks with the important nodes connected between them (known as assortative networks) will also have higher λ 1 than networks where the hubs (i.e., important players) are not directly connected between them.
Algebraic connectivity. The algebraic connectivity λ  2 corresponds to the second smallest eigenvalue of the Laplacian matrix  L, which is defined as = −  L S A, with A being the weighted adjacency matrix and S a diagonal matrix whose i-elements are the sum of the passes made by player i. The algebraic connectivity is closely related to both structural and dynamical properties of networks 31,62,63 . On one hand, algebraic connectivity is an indicator of the modular structure of a network 13 : The lower the λ  2 , the clearer the existence of independent groups inside the network, with the limit value of λ =  0 2 indicating the existence of, at least, two disconnected groups in the network. In the framework of multilayer networks, one can interpret the value of λ  2 as a way to quantify structural integration and segregation of different network layers 64 . On the other hand, λ  1/ 2 is proportional to the time required to reach equilibrium in a linear diffusion process 65 . Additionally, the time t sync to reach synchronization of an ensemble of phase oscillators that are linearly and diffusively coupled is also proportional to λ  1/ 2 54 .
Eigenvector centrality: Maximum value and dispersion. The eigenvector centrality ec(i) of a player i is a measure of node importance that is obtained by calculating the eigenvector v 1 associated to the largest eigenvalue λ 1 of the weighted adjacency matrix A. The eigenvector centrality is a measure of node importance that takes into account the number of all directed connections a player (node) has. Furthermore, two factors contribute to increase the value of eigenvector centrality: (i) a higher number of direct connections to other players (note that connections are weighted) and (ii) to be connected to other nodes that, in turn, also have a high centrality. In this way, important players are those that are (highly) connected to other important players of the team.
50-pass network time. The 50-pass network time t diff is the time required to construct a 50-pass network. It is obtained subtracting the time of the first pass of the network from the time of the last pass. Teams with shorter t diff are those that generate more passes in less time.
Statistical analysis. All parameters of Figs 2 and 3 have been compared pairwise with the Wilcoxon ranksum test, as the number of observations to compare was small enough to prevent us from safely assuming normality (<40 in most of the cases). However, the t-statistics yielded the same rejections of the null hypothesis (central tendency equality, median or mean) in all parameters. As the number of comparisons (20) would raise type I errors, p-values have been corrected for multiple comparisons with non parametric false discovery rate (FDR) 66 for α = 0.01, which changed some results on the verge on significance. After FDR correction, α = 2.7132e −04 (0.004 if we originally set alpha at 0.01). <Y> was not statistically significant for any threshold, which is expected, and should not change with the number of observations. On the contrary, eigenvector centrality dispersion (p = 0.0042) should be considered significantly different only if we keep alpha unaltered. After FDR correction we cannot state any difference confidently. The centroid dispersion remains significant in any case, although just barely (p = 2.6e −4 ). In both cases, conclusions must be taken with caution, and we would need more statistical power (i.e., more data) to assert confidently that there are statistically significant differences in those parameters.