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Statistical Physics

The little homeland in soccer shoes

Only three degrees of separation are interposed between all Brazilian players, whether stars or duffers

The duffer calls the star’s cell phone.

– Hello, is that Romario? This is Cláudio Egg White.
– Cláudio who?
– Egg White, from the Juventude FC.
– I don’t know you. I’m going to hang up. The footvolley is just going to begin.
– No, wait a mo. I’m a friend of Rafa-Three-in-One, who played in the Criciúma with Fritz-German Dog, who is a friend of Rascal, who was your colleague in the Fluminense.
– Ah, Caiçara’s my mate.

The above dialog and the personages, with the exception of Romario, are fictitious. But work done by researchers from the São Carlos Physics Institute of the University of São Paulo (IFSC/USP) shows that the world of professional soccer in Brazil, where there are talents and real nobodies, is small, a little smaller, for example, than the universe of Hollywood actors. In the network of interpersonal relations that connect the national elite of footballers, any sportsman, famous or unknown, manages to establish contact with another colleague in the profession with the help of just another three players or former players. Similar studies carried out in the Mecca of the American cinema suggest that, on average, the social distance between two actors is 3.7 degrees of separation, slighter more than the level of separation between two players of the elite of Brazilian soccer, calculated as 3.3 degrees.

According to a classic work, done in 1967 by social psychologist Stanley Milgram, from Harvard University, a mere 6 degrees of separation – and no more than that – are interposed between all the people in the world. You and the Pope, Bush and Bin Laden, Brad Pitt and your sister.  The name small world effect is given to this exaggerated proximity between the 6.4 billion inhabitants of the planet. A kind of proximity that derives more from the network of friends and acquaintances than from geographical barriers. “We thought that the distance between Brazilian players would be less than 6 degrees, but we didn’t imagine that it would be half that”, comments physicist (a Corinthians fan) Roberto Nicolau Onody, the main author of the study, published in September in the Physical Review E magazine.

With the help of another physicist, Paulo Alexandre de Castro, his student for a doctorate and co-author of the scientific article, Onody collected and analyzed data on all the sportsmen and clubs that between 1971 and 2001 took part at least once in the first division of the Brazilian championship. To get this mountain of raw information, the researchers resorted to a CD-ROM published in 2003 by the Placar magazine, with the history of 32 editions of the tournament. “We tried to get this material on the Internet, but not even in the site of the Brazilian Football Confederation (CBF) did we find what we wanted”, says Castro, who roots for Santos FC.

In the period studied, 13,411 players and 127 teams contended for the championship. The article by the Onody-Castro focuses on the web of social relations existing between the sportsmen and between them and their (former) clubs. They worked on the matter from the point of view of complex networks, a field of study of statistical physics, which has been used for analyzing the organizational architecture of systems as diverse as the biochemical reactions involved in cell metabolism, the sexual contacts between persons, and the connections between the pages of the web.

A network is a set of interconnected vertices, also called knots or points. A law usually governs the connection between these knots. When all the vertices of a system always show the same number of connections, like the structure of a crystal, there is a homogeneous or crystalline network. If some vertices of a system show many links, while the majority of knots have few, there is a complex network. This is the case of the network of professional players in Brazil. In the world of soccer, the physicists studied the interrelations between the two kinds of vertices, the players and the clubs of the first nationwide division, with an emphasis on the first kind of knot. What makes a sportsman link himself, connect himself with another sportsman? Having contended in the elite division of the Brazilian championship for the same team in the same season.

It is not necessary for them to have played side by side in the same match, but they must have been part of the roster of the same team in a given year. This was the rule arbitrarily used by the researchers to find connections between the players. According to the rules proposed by the researchers, sportsmen who left clubs of Brazil’s first division for any reason – they were transferred to teams in a lower division, to teams abroad, or simply hung up their soccer shoes  – continue to be part of the network, but do not make any new connections until, when this happens, they return to the elite of Brazilian soccer.

Similar work done on the movie industry established a link between two actors when both acted in the same film. So footballers who played in several editions of the tournament for one team or who constantly changed clubs tend to show many connections. “They may even not have made many friends with their team mates, but they certainly maintained a social relationship with them while they played side by side”, Onody ponders.

The network made up by the elite of Brazilian players is complex because many (former) sportsmen have few links, while a few have many. In 2002, each sportsman inserted into the system had established, on average, connections with another 47 players. In other words, he had worked alongside this quantity of footballers. Like every average, the figure hides the extremes. The member of the network with most connections is the former forward Dario (Dadá Maravilha), a folkloric striker who brought his long career to a close in the mid-1980’s. Dadá went onto the pitch with 305 colleagues in the profession. “He was a soccer nomad”, explains Onody.

In a career of more than two decades, the forward wore the colors of 11 first division clubs, another record that belongs to King Dadá. At the other extreme, amongst the least interconnected of the network, ten obscure players appear, whose career was short, and least in teams of the first division. One of them is the former goalkeeper Vílson, who played one match for the Colatina team from Espírito Santo in 1979 and connected up with another 14 players.

Longest career
One fact from the study suggest that the duration time of the career of soccer players has increased over the last three decades. This is because, in 1975, each member of the network had played – had connections – with another 39 colleagues in the profession, a quantity 8% lower than was seen in 2002. “Either the players’ careers are getting longer, or they are changing teams more frequently”, Castro comments. As the majority of sportsmen who have passed through the first division of Brazilian soccer did not go so far as to work for two teams, the first hypothesis seems more reasonable. The physicists noticed that, in spite of the sportsmen showing more connections today than in the past, the network of soccer players is becoming more and more elitist.

Very popular sportsmen, with a high number of connections, are tending increasingly to relate with players with a similar profile, while those who show low connectivity basically circulate amongst equally little known colleagues. “Players from major clubs tend to be transferred to other big teams, and those from the small teams change preferably to modest clubs”, Onody sums up. That is, the average difference between all the players is small, only 3 degrees or steps, but this proximity is not sufficient to do away with the existence of castes, of well-defined groups in the world of the ball.

Besides scrutinizing mathematically the social relations between soccer players, the physicists produced some surprising revelations in other fields of the soccer world.   They discovered, for example, that a professional footballer becomes reasonably known – and thus guarantees their employability in the best teams of the country – after having taken part in a certain number of games. How many matches are necessary to guarantee the future of a sportsman on the playing fields of Brazil?  Forty games for any club, large or small. “If this critical number is exceeded, it becomes easier for the sportsman to carry on playing in the elite division”, Onody comments. The footballer acquires a certain stability in his occupation, which will guarantee him offers of employment.

Few score
There is no lack of curious statistics in the physicists’ study. In the period analyzed, the former attacker Tarciso, who worked for Grêmio in the 1970’s and 1980’s and made the Brazilian team, was the sportsman who went onto the pitch most often: he was in 336 matches of 18 championships, wearing the three-colored jersey from Rio Grande do Sul in 13 editions. The list of sportsmen who played only one match in the first division is enormous, and is made up of 2,160 anonymous footballers, a little more than 16% of those active in the championship between 1971 and 2002.

Goals are another theme that yields some interesting figures. Using this question as a parameter for comparing, two groups of players stand out in the network set up by Onody-Castro: those who scored less than ten goals, the vast majority, and those who scored more. At the least, two third of the sportsmen in soccer play in the defense or in the midfield. So they have less probability of filling the adversaries’ onion bag. Even so, the dearth of strikers is surprising. In the Brazilian championship, scoring goals is a privilege for few.

Almost 65% of the players – 8,709, to be more precise – never scored a goal, and about 30% – 4,089 – marked up only between one and ten goals in their careers. The rest, 618 sportsmen, can be considered intimate with the ball: they made the net shake 11 or more times. “The chance of any player, chosen from the network at random, having scored 13 goals is ten times greater than of having scored 36”, says Castro. Booking the data from the 32 editions analyzed of the championship, the former attacker Roberto Dinamite is the greatest goal scorer of the competition. He scored 186 goals in 20 championships contended for for Vasco da Gama, from Rio de Janeiro, and one for Portuguesa, of São Paulo. Which means, according to the study from USP, the players are not far away from each other, but the goal is a long way from the majority of them.