Professional football is famous for its scarcity of talents. Therefore, wealthy clubs are willing to pay millions of euros in exchange for one qualified player. Moreover, the average annual expense, revenue and volume, i.e., the sum of expense and revenue as a measure of the flow of transfer fees through a club, of clubs show an increasing trend from 2011 to 2015 (Fig 1). The top 10 clubs in terms of annual expense, revenue and volume are summarized in Table A in S1 File. The average expense of the clubs was over 5 million euros and the average transfer fee volume exceeded 10 million euros.

However, the financial resources of the elite clubs are not evenly distributed. Fig 2A shows the distribution of average annual expenses of the clubs in 5 years. The distribution is heavy-tailed, in which top spending clubs have the ability of raising an amount of money 10 times larger than most of the clubs. Fig 2B shows the proportion of clubs and their cumulative expenses in the transfer market in a percent scale. It is shown that 80% of the total transfer fee is spent by less than 20% of the clubs. Fig 2C shows the Gini coefficient [21] of transfer expenses of all clubs from 2011 to 2015. An increasing trend indicates that the inequality in financial ability of professional football clubs is gradually magnifying. Overall, we have observed substantial and increasing inequality in the financial ability of professional clubs.

In the 5 year period from 2011 to 2015, there were totally 8948 transfer actions, excluding loans, among the elite clubs, constituting a player transfer network of 410 nodes connected by 6316 directed edges, where the direction of edges denotes the direction of player movements. All the nodes in the network are strongly connected except for three nodes with only outgoing edges and no incoming edges. The mean shortest distance between all nodes is 2.8 and the clustering coefficient is 0.21. Here, the calculation of the mean shortest distance and the clustering coefficient both take the directions of the edges into account. For the clustering coefficient, we have adopted the simple definition proposed in [22] that it is the average ratio between all directed triangles actually formed by each node and the number of all possible triangles that the node could form. Comparing to the average mean shortest distance of 2.5 and the average clustering coefficient of 0.04 in random networks of the same size and connection density, the player transfer network exhibits the small-world phenomenon [23]. Fig 3A shows the distributions of in-degree k_in and out-degree k_out of the network. The distributions are highly skewed but not as unequal as that of the transfer expense in Fig 2A. Fig 3B and 3C show the correlation of in-degree and out-degree and the distribution of excess degree k_ex = k_in − k_out of all nodes respectively. It is shown that the numbers of clubs that a professional club “buys from” and “sells to” are basically equal.

The rich-club phenomenon refers to the tendency that well-connected nodes also connect to each other. This phenomenon is commonly found in social networks and transportation networks [24]. To evaluate this phenomenon in the player transfer network, we computed the rich-club coefficient, ϕ_norm(k), for all node degrees k (see Data and Methods for detailed information on this measure). A value ϕ_norm(k) > 1 for large k indicates that high degree nodes are organized as a rich club. Fig 4 shows that the rich-club phenomenon actually cannot be observed for the clubs with the highest degrees. On the contrary, it suggests that the clubs which exchange players with the largest number of peer clubs tend not to exchange players among themselves and that the global transfer market may have disparate “cores” that control compartmented transfer resources.

In this section, we will explore the relationship between the functionalities of a club and its network properties. The ultimate measure of the success of a commercial organization is its profitability, which also applies to a professional football club. Generally speaking, clubs with the highest achievements in prestigious competitions are also the ones that generate the largest revenue from various commercial activities [1]. Meanwhile, a club could also profit directly from the transfer market, by receiving more compensation from the players transferred out than it pays to acquire new players. Therefore, the club functionalities can be described either by its match performance or its transfer profit. Match performance includes the domestic and international match results. We quantify domestic performance of a club by the average game points in its domestic league matches from 2011 to 2015. On the other hand, the five year aggregate IFFHS Club World Ranking (CWR) point is employed to quantify the overall performance of a club in both domestic and international competition [25]. Table B in S1 File shows the top 10 clubs in terms of their match performance. Fig 5 shows that although the distributions of average league game points and aggregate IFFHS CWR points exhibit different characteristics, the two performance measures are positively correlated. The ability of profiting from player transfers are defined by two measures, i.e., the average annual transfer balance and the cumulative price overflow from player transits. If a player has transferred from club A to club B then to club C, we define that the player has transited through club B. The price overflow of this player in club B is the difference between the transfer fees payed by club C to club B and by club B to club A. Table C in S1 File shows the top 10 clubs in terms of profitability in the transfer market. Table 1 shows that the two categories of club functionalities are generally weakly or not correlated. Overall, match performance and transfer profitability can be considered as independent indicators of the functionalities of a professional football club.

Which factors of the transfer market affect the functionalities of professional football clubs? To answer this question, we first examine the relationship between the amount of money a club has spent at the transfer market and its functionalities. Table 2 shows Kendall’s Tau coefficients [26] between the average annual transfer fee quantities, i.e., average annual revenue, expense and total volume, and the functionality measures of the clubs. It is shown that the amount of transfer fee is positively correlated with club performance, which indicates that the more money a club spends or receives at the transfer market, the better match performance it will generally achieve. In addition, this “money effect” is more obvious in the clubs’ international performance than in their domestic performance. Meanwhile, Table 2 also shows that the expense and transfer fee volume are basically negatively correlated with the profitability from player transfers of a club. This indicates that for most clubs, their transfer expenses are larger than the transfer revenue.

Next, we study the relationship between the functionalities of the clubs and their properties in the player transfer network. The node properties of concern include two categories of metrics that measure the coreness and brokerage of the nodes, respectively. The coreness of a node measures the richness of its connections and is usually described by the number of direct connections combined with the number of indirect connections [27]. In this paper, eigenvector centrality and PageRank centrality are chosen as the coreness metrics. The coreness metrics could indicate the search breadth of the club scouts in the transfer market. Table D in S1 File shows the top 10 clubs in terms of coreness properties in the player transfer network. The brokerage of a node measures the extent to which it controls the network flow [20]. Clubs with large brokerage metrics tend to exclusively control certain transfer resources and act as brokers amongst other clubs. In this paper, effective size, closeness centrality and betweenness centrality are chosen as the brokerage metrics. Table E in S1 File shows the top 10 clubs in terms of brokerage properties in the player transfer network. Table 3 shows Kendall’s Tau between club functionalities and network properties. The match performance, especially international performance, of the clubs is positively correlated with the brokerage metrics, while clubs’ profitability from transfers is generally weakly or not correlated with the network properties. This result suggests that the brokerage power of a club plays the most significant role in determining its match outcome, while the relationships between other network measures and club functionalities are relatively weak. In the next sections, we will further explore the relationships between club functionalities and node properties from several different aspects.

International sports labor migration shows characteristics different from those of domestic sports labor movements [11, 12]. In this paper, the football player transfer network can also be separated into two subnetworks accordingly. The domestic transfer network contains only transfers within a same league and the international transfer network contains only international transfers between different leagues. Table 4 shows the correlation of network properties and club functionalities in both international and domestic transfer networks. It is shown that the node properties taking account of global network connections, i.e., eigenvector centrality, PageRank centrality, betweenness centrality and closeness centrality, are better indicators of both domestic and international match performance, while the node property taking account of local network connections, i.e., effective size, in the domestic transfer network is a better indicator of match performance.

The movement of a player can take two different forms, i.e., transfer and loan. In a transfer, the player terminates his contract with the former club and signs a contract with the new club, while in a loan, the player is allowed to temporarily play for a club other than the one he is currently contracted to. The player loan network can be constructed similarly to the player transfer network, in which the nodes are clubs and directed edges are player loans. However, loan network and transfer network show different characteristics. In the 24 leagues, only 394 out of 410 clubs are involved in player loans. The number of edges is 2509, with totally 3586 player loan records. All the nodes in the loan network are weakly connected but the network has 129 strongly connected components. The clustering coefficient in the loan network is 0.11 compared to 0.21 in transfer network. The average path length is 4.1, compared to 2.8 in the transfer network. Therefore, the loan network is a sparsely and weakly connected network with multiple weakly disconnected subnetworks. Moreover, the properties of nodes in the two networks are loosely related. Although the correlations of eigenvector centrality, effective size and betweenness centrality of the same nodes in the two networks are positive and strong (0.46, 0.75 and 0.51, respectively), those of the PageRank centrality and closeness centrality of the same nodes in both networks are weak (-0.02 and 0.08, respectively).

The correlations among network properties and club functionalities in transfer network and loan network are shown in Table 5. Similar to the transfer network, the node brokerage properties are found to be strongly related to match performance. However, the coreness properties in the loan network no longer have positive correlations with the match performance of the club. Actually, the differences between the correlations of coreness properties and the match performances in loan and transfer networks are quite large. The reason is that instead of players loaned-in, players loaned-out indicate the excessiveness of player resources of a club and hence the eigenvector centrality and the PageRank centrality actually cannot capture the attraction of clubs to players as they do in player transfer network. Nonetheless, our finding shows that the larger and more exclusive the player resources one club controls, the better overall performance it will achieve.

Table 1 has shown that the match performance and profitability of clubs are weakly or not related. However, a closer examination shows that the clubs in different leagues exhibit different characteristics in the relationship between the two kinds of club functionalities. In some leagues, the more transfer profit a club generates, the better match performance it achieves while in other leagues it is just the opposite (S1 Fig). We roughly classify the leagues into three categories, i.e., the leagues with negative correlation between the clubs’ annual balance and average league game points with adjusted R² > 0.25 are referred to as “money leagues”, as the more the clubs spend, the better match performance they achieve; the leagues with positive correlation and adjusted R² > 0.25 are referred to as “farm leagues”, as the more the clubs profit from the transfer market, the better match performance they achieve; the rest are referred to as “outlier leagues”. In money leagues, the clubs’ performances are strongly related to their abilities in raising transfer fund. In farm leagues, the club’s performances are strongly related to their abilities of profiting from player transfer. In the outlier leagues, there is little interdependence between match results and clubs’ profitability from player transfer. A summary of our classification is shown in Table 6.

The relationship between the node properties in the player transfer network and the functionality of clubs in different league categories is shown in Table 7. The parameter correlations of clubs in “money leagues” generally agree with the average nodes in the transfer network. However, in “farm leagues”, the correlation between eigenvector centrality, effective size, closeness centrality and club domestic match performance vanishes, but strong correlation emerges between node coreness and brokerage and club profitability. Especially, the brokerage of a club in the player network is strongly correlated to its annual balance. This phenomenon suggests that clubs in different financial environments actually have various strategies of achieving success, either by acquiring the best players at all costs, or by cultivating players with potential and profit from the reselling of these valuable assets. More importantly, the successes of both strategies are strongly related to the clubs’ network properties in the global player transfer network.

The transfer and loan records in and out of 410 clubs in 24 top-class professional football leagues (Table 6) across the world from 2011 to 2015 are retrieved from the online football database Soccerway (whose data provider is Opta Sports). The dataset contains 31128 transfer records and 13837 loan records, with totally 3660 clubs and 23765 players involved. Transfers and loans are recorded in various forms: transfer with an explicit transfer fee, free transfer, transfer with a undisclosed fee, swap and loan. Transfers with undisclosed fee and player swaps are treated as free transfers in this paper.

Each win in the domestic league game is counted 3 points, each draw 1 point and each lose 0 points. The IFFHS Club World Ranking points are composed by the International Federation of Football History & Statistics (IFFHS) based on a set of rules composing both domestic and international match results [25]. The CWR points are updated once per month, the aggregate CWR points used in this paper covers the period from January 2011 to December 2015.

The rich-club coefficient ϕ_norm(k) is a measure of connection density of high degree nodes in the network [24]. First, we calculate the value ϕ(k) = 2E(k)/k(k − 1), where E(k) is the number of edges among nodes with a degree ≥ k. Then we determine ϕ_rand(k) as the average value in 1000 random graphs with the same node degrees as in the empirical network. Finally, ϕ_norm(k) = ϕ(k)/ϕ_rand(k). A ϕ_norm(k) > 1 indicates that the rich-club phenomenon exists in the network. A ϕ_norm(k) < 1 indicates that “rich” nodes do not tend to form a dense clique among themselves.

The coreness properties refer to a class of node properties measuring the richness of its connections and of its neighbors [27]. Eigenvector centrality and PageRank centrality are instances that fall into this category. For a network G = (V,E) and A = (a_i,j) the adjacency matrix, a_i,j = 1 if there is a directed edge pointing from node j to node i. The eigenvector centrality E_i of node i is defined as the solution of E i = x i = 1 λ ∑ j a i , j x j , (1) where λ is a constant. The values x in E can be solved approximately by setting the initial values x(0) = 1 and repeatedly updating the values by x ( t ) = A t x ( 0 ) , (2) for a limited number of t times.

PageRank centrality is an extension of eigenvector centrality with a scaling factor. The PageRank centrality P_i of node i is defined as the solution of P i = x i = α ∑ j a i , j x j L j + 1 - α N , (3) where L_j = ∑_i a_j,i is the in-degree of node j and N is the number of nodes in the network. In this work, we set α = 0.85. Note that eigenvector centrality and PageRank centrality measure the centrality of a node by the density of the inbound connections to this node and its neighbors. Therefore, larger eigenvector and PageRank centralities indicate stronger attraction of a club to football players.

Network brokerage is used to describe the ability of a node to connect other nodes in the network. In this paper, we have chosen effective size, closeness centrality and betweenness centrality as measures of both local and global brokerage power of a node in the network. The effective size is defined by S i = 1 - k i - 1 k i C i , (4) where k_i is the degree of node i and C_i is its clustering coefficient [28]. S_i ranges between 0 and 1. It takes its smallest value when node i is part of a fully connected structure (hence, has no local brokerage power), and it takes its largest value when the neighbors of the node have no mutual connections (hence, the node achieves maximum local brokerage power).

Closeness centrality Cl_i is the reciprocal of the sum of geodesic distances to all other nodes Cli=1∑j≠i dij, (5) where d_ij is the geodesic distance pointing from nodes i to j. A large closeness centrality means that the node is close to all other nodes in the network, hence can easier act as a broker from a global perspective.

The betweenness centrality B_i of node i is defined by B i = ∑ s ≠ i , s ≠ t , i ≠ t g s t i g s t , (6) where g_st is the number of directed shortest paths from nodes s to t, and g_sti is the number of shortest paths between nodes s and t that pass through node i. A large betweenness centrality can also indicate large global brokerage power.

In this paper, the interrelations between network properties and club functionalities associated with the nodes in the global football player transfer network are studied. As shown in Fig 5, the distributions of the club functionality measurements are largely different from each other. Therefore, we have chosen Kendall’s Tau coefficient, which is an accepted non-parametric measure of concordance particularly suitable for detecting ordinal association between two measured quantities without assumptions of underlying distributions. Let (x₁,y₁), (x₂,y₂), …,(x_n,y_n) be a sequence of measurements to compare. Any pair of measurements (x_i,y_i) and (x_j,y_j), where i ≠ j, is said to be concordant if their ranks in the two sequences have the same order, i.e., if both x_i > x_j and y_i > y_j or if both x_i < x_j and y_i < y_j. They are said to be discordant, if x_i > x_j and y_i < y_j or if x_i < x_j and y_i > y_j. If x_i = x_j or y_i = y_j, the pair is neither concordant nor discordant. Kendall’s Tau is defined as: τ = n + - n - 1 / 2 n ( n - 1 ) , (7) where n₊ is the number of concordant pairs and n ₋ the number of disconcordant pairs. The coefficient τ ranges between 1 and -1, where a larger positive value indicates a stronger positive association, a larger negative value indicates a stronger negative association and a value closer to 0 indicates that the two measured quantities are weakly interrelated or independent.