Complex network study of Brazilian soccer players

Although being a very popular sport in many countries, soccer has not received much attention from the scientific community. In this paper, we study soccer from a complex network point of view. First, we consider a bipartite network with two kinds of vertices or nodes: the soccer players and the clubs. Real data were gathered from the 32 editions of the Brazilian soccer championship, in a total of $13411$ soccer players and 127 clubs. We find a lot of interesting and perhaps unsuspected results. The probability that a Brazilian soccer player has worked at $N$ clubs or played $M$ games shows an exponential decay while the probability that he has scored $G$ goals is power law. Now, if two soccer players who have worked at the same club at the same time are connected by an edge, then a new type of network arises (composed exclusively by soccer player nodes). Our analysis shows that for this network the degree distribution decays exponentially. We determine the exact values of the clustering coefficient, the assortativity coefficient and the average shortest path length and compare them with those of the Erdös-Rényi and configuration model. The time evolution of these quantities are calculated and the corresponding results discussed.

Figure 1

Histogram of the number of clubs against the number of goals scored by match. Bins of size 0.1 were used. The full line corresponds to the fitted Gaussian curve. The average number of goals by match is equal to 1.00.

Figure 2

Probability $P\left(N\right)$ that a player has worked for $N$ clubs. The full line corresponds to the fitted curve $P\left(N\right)\sim {10}^{-0.38N}$. So, it is 190 times more probable to find someone who has played for only two clubs than for eight clubs. The inset is the degree distribution $P\left(S\right)$ for the clubs.

Figure 3

(a) Game probability $P\left(M\right)$ versus the number $M$ of disputed matches. (b) Cumulative distribution $P_c\left(M\right)$ built from $P\left(M\right)$. Fittings appear as full lines.

Figure 4

(a) The goals probability $P\left(G\right)$ that a player has scored $G$ goals. Choosing randomly, a Brazilian soccer player has ten times less chance to have scored 36 goals than 13 goals. The player with the highest score is Roberto Dinamite with $G=186$. (b) The corresponding cumulative probability distribution $P_c\left(G\right)$.

Figure 5

Degree distribution of the BSP network. The fitting curve (full line) has the exponential form $P\left(k\right)\sim {10}^{-0.011k}$.

Figure 6

Nearest-neighbors average connectivity for the ER, BSP, and Configuration model.