Testing styles of play using triad census distribution: an application to men’s football

2.1 Passing networks in football

A passing network G is defined as the ordered triple ( V , A , W ) consisting of a set of 11 first team players v ∈ V , a set of arcs a ∈ A ⊆ V × V , consisting of all the passing relationships between them and a set of weights ω ∈ W , i.e., the number of passes occurred during the match, denoting the strength of the connection. We assume that both vertices V and arcs A are finite as well as | V | = n . Passing network can be expressed in the form of an adjacency matrix P = p i , j , with p _i,j = ω(a _i,j ) if ∃ a i , j ∈ A and p _i,j = 0 otherwise.

In Table 1 we present the team passing network of a match involving Arsenal against PSG in September 13th 2016 (first match of the Group Stage, group A). For example, during the match, goalkeeper Ospina passed successfully the ball to Monreal four times (p ₇) and 11 to Mustafi (p ₉), while he received once the ball from Bellerin (p ₁₀). Figure 1 represents the directed and weighted graph representation obtained through the adjacency matrix of Table 1; eleven agents (representing the players) are connected between arrows and the strength of relationship, i.e., the number of passes, is expressed through the size of them.

Figure 1:

Graph of adjacency matrix of the team passing network of Arsenal illustrated in Table 1.

We need to remark that values on the main diagonal in Table 1 are constrained to be null since we assume that a player cannot pass the ball to himself. Thus the so-called “loops” are not allowed in this framework.

To summarize, passing networks in football share some peculiarities: they are not high dimensional, involving a small and generally fixed number of vertices, but may also present a certain degree of sparseness. In fact, the number of interactions may depend on (i) position on the pitch (ii) individual skills and team cooperation attitudes and, finally, (iii) team strategies and/or style of play.

In order to formally uncover the qualitative attributes of these links, the analysis of triads represents a possible way to detect patterns in the passing distribution and, under some mild assumptions, make a proper inference. Thus, observed triadic distribution can be used to compare connectivity of a team during competitions, in order to analyse changes in strategies and styles of play. In addition, the comparison could be also made through teams in the same tournament, helping to understand if different passing behaviours can have an impact on the football outcomes.

2.2 Triad census

Although the minimal structural element in a network is a dyad, defined as the interaction occurring between two agents, the minimal group is represented by a triad, involving a subset composed by three agents. In general, given a set of n nodes (agents), a k-subgraph is defined as the set containing all the possible subsets of dimension equal to k of a network. The number of possible distinct k-subgraph that can be examined are n k , with k ≤ n, and the number of possible unweighted and directed arcs occurring between k individuals is k(k − 1), spanning from the totally disconnected (null) to the full connected (clique) case. Each tie can be either present or absent; for each subgraph there are 2 ^k(k−1) possible states considering the exact identity of the individuals in the whole set.

Focusing on the structure elapsing between individuals, there are some recurrent states where the relational structure (i.e., in which way agents are linked between each other) is invariant under permutations of labels attached to the agents. Such categories are called isomorphism classes. The number of possible isomorphism classes of a k-subgraph depends on the number of elements belonging in each subgroup, in general this value can be exceedingly high even with modest subgroup dimensions, (Holland and Leinhardt 1976). For instance, when k = 2, 3 the number of isomorphism classes is 3 and 16, respectively, rapidly increasing even with k = 4 (218 classes).

In a purely theoretical point of view, both dyads and triads are two special cases of k-subgraphs. Figure 2 presents all the possible isomorphism classes derived from triad census, from the so called “003”, which involves no relationship between vertices, to the full connected triad “300”, presenting six edges. A summary of triads’ labels can be found in Figure 2.

Figure 2:

List of isomorphism classes belonging to the triadic subgraph. Each label consists in three digits representing the number of mutual (↔), asymmetric (→), and null links connecting two units. Some classes with the same MAN distribution have a letter representing the additional structural differentiation: C = cycle, T = transitive, U = up and D = down. Plot adapted from Faust (2008).

Main differences between dyads and triads have been surveyed in Krackhardt (1999) and he stated that the individuality of agents is preserved when the analysis is focused on dyads, helping to highlight the direction of interactions between couples of agents. The census of all possible triads has been introduced by Davis and Leinhardt (1967), while Wasserman and Faust (1994) and Faust (2010) help to develop the method in order to retrieve information about network structural properties regarding the level (or quality) of transitivity nodes between individuals (see also Lorrain and White 1971; Luczkovich et al. 2003; Cugmas, Ferligoj, and Žiberna 2017, for further details).

In order to analyse the probabilistic properties of triad census, we use the notation introduced in Wasserman and Faust (1994) starting from the general k-subgraph theory. Then we apply authors’ results shown in Theorem 14.1 and 14.2 to the triadic case (k = 3), in order to obtain moments of triad census. We remark that these quantities are based on the theoretical distribution related to the random structure.

Given a network having n agents, let us suppose to label each node with a number equal to 1, 2, …, n. We denote two distinct isomorphism classes for the k-subgraph η and ν, defined as two k-subgraphs having at least one vertex or edge that they do not share (while all other vertices and edges can overlap). We refer to generic elements of the two isomorphism classes as K = K i 1 , … , i k ∈ η and L = L i 1 ′ , … , i k ′ ∈ ν , where the subscripts are related to the labels attached to each vertex of the subgraph. Therefore, we define the following probabilities:

representing the probability that any single subgraph K or any two subgraphs K, L belong to classes η and ν, respectively. Since the above probabilities may vary for each configuration, it would be necessary to average the probability of each isomorphism class over all possible K and L structures available in the k-subgraph, as follows:

where 0 ≤ h ≤ k − 1.

The subgraph census’ random variable H _η is then defined as the number of k-subgraphs belonging to the isomorphism class η. Using the theorem introduced Holland and Leinhardt (1976), it is possible to define the moments of k-subgraph census random variables H _η for each admissible isomorphism classes that can be retrieved in the k-subgraph. Hence, such moments rely on a given probability structure.

3.1 Testing against random structure

Several possible random distributions can be used to compute moments for each isomorphism (see Holland and Leinhardt 1976; Faust 2007, for further details). Among others, the so called U|MAN (Uniform given Mutual Asymmetric and Null arcs) is one of the most relevant structures in the literature, and will be used as reference throughout the rest of the paper. U|MAN random digraph is the probability distribution on the set of all digraphs with n agents considering fixed and discrete values of mutual, asymmetric and null arcs present in the network. The motivation to use the U|MAN distribution can be found in the possibility to construct, imposing a set of assumptions, an asymptotic statistical test for each of the 16 isomorphism classes and, more importantly, a comprehensive asymptotic test involving all types of triads.

Thus, it is possible to define a random network structure starting from the number of mutual, asymmetric and null arcs in the network comparing the observed triads against those which are uniformly distributed among the agents. In order to make this comparison, our purpose is to formally test if the probabilistic structure of a team passing network is closer or not to the one defined in the conditionally uniform random case.

Since it holds a direct connection between averaged and single probabilities shown in Equations (1) and (2), the use of U|MAN distribution allows to simplify results of Equations (3) and (4) in the following way:

where

and 0 ≤ h ≤ k − 1. The first equality is a consequence of the invariance under permutations property of U|MAN distribution; to summarize, changing the node’ labels of a triad does not change the probability of belonging to an isomorphism class. On the same basis, the second equation indicates that the probabilities p _K,L (η, ν) depend only on η and ν and also on the number of nodes they share, i.e., h = |K ∩ L|.

Moreover, it is possible to derive a correspondence between the joint probabilities of two triads having either zero or one vertex in common, such as:

Basing on the above statements, it is possible to apply the results shown in Holland and Leinhardt (1976) under the condition that data generating process of links in a network is characterized by a U|MAN distribution. As mentioned, the total number of isomorphism classes of a triad census is 16 (as in Figure 2) and it is possible to define the triad census vector T = (T ₁, …, T ₁₆) from a random directed graph based on the U|MAN distribution. Expectation, variance, and covariance for each isomorphism class are directly retrieved combining the results in Holland and Leinhardt (1976) and are depicted in the Appendix A. Regarding the asymptotic distribution of vector T, it is worth to notice that it has an approximate multivariate normal distribution with an increasing number of vertices included in the graph (Wasserman and Faust 1994). Starting from results presented in the Appendix A, Equations (9)–(11), it is possible to compute the τ _η statistics for each isomorphism class

For each class η, τ _η represents the standardized departures of observed triads from the theoretical random distribution, allowing to formally test the presence of patterns that deviate from what is expected in the randomly distributed case. Thus, a test statistic denoted as τ max 2 can be used to build an omnibus test of randomness:

where μ is the vector of expectations and Σ the variance–covariance matrix computed using the U|MAN distribution. Σ⁻¹ is the pseudo-inverse of the variance–covariance matrix and can be obtained as terms of eigenvalue decomposition Σ⁻¹ = Γ D ⁻¹ Γ′, where Γ is the matrix of eigenvectors of Σ and the values λ _i of the inverse diagonal matrix D − 1 = diag 1 / λ 1 , … , 1 / λ p , 0 , … , 0 correspond to the non-null eigenvalues of Σ, for i = 1, …, p and p = rank Σ ≤ 16 . The above statistic can be used as test for detecting the possible deviation against a random network structure in the data.

Holland and Leinhardt (1978) prove that, given a random reference distribution and if the vector T is approximately Normal, τ max 2 asymptotically follows a χ ² distribution with degrees of freedom equal to the rank of Σ. Therefore, we can test if data exhibit a random structure compared to the alternative hypothesis of strong deviations against randomness.

3.2 Testing specific team strategies

The approach introduced in Holland and Leinhardt (1978) lacks of flexibility: it is not straightforward to manipulate the null hypothesis in order to test for deviation from different random structures or, more importantly, to test for significant changes in a subset of triad counts. Moreover, the τ max 2 test statistic may be more conservative in finite samples.

A possible workaround could be the use of univariate tests for each isomorphism class, but this alternative generally implies the fulfilment of strict assumptions (such as independence between the triadic configurations) that are not satisfied in this context. Then, given the multivariate nature of triadic configurations, we are interested in defining a set of multivariate hypotheses allowing to formally test a set of factors composed by non-independent counts.

Testing procedures of count data have been applied in other fields, such as the analysis of contingency tables or ratings data obtained from ordered categorical responses showing the presence of variability between samples (Ennis and Bi 1999), but mainly related with biology and genetics, i.e., olfactometer data (Ricard and Davison 2007) or DNA sequencing data (Wu et al. 2017). A comprehensive analysis of discrete multivariate distributions can be found in Johnson, Kotz, and Balakrishnan (1997) for a.

Despite the presence of interest in literature regarding this topic, to the best of our knowledge there are no applications to triad census analysis. In this context, due to the structure induced by isomorphism classes, the Dirichlet-Multinomial (hereafter DM) distribution (introduced in Mosimann 1962) suits properly the features belonging to triad census data. Throughout the rest of the paper we will refer to Equation (13) (see Appendix B) and then we will denote that an m-variate random variable follows a DM distribution as X ∼ DM _m ( π , θ). The main information regarding DM distribution useful for the purpose of the present work is depicted in the Appendix B.

Beta-Binomial distribution is a special case of the DM distribution when the number of modalities reduces to m = 2. It can also be shown that it approaches the Multinomial distribution as θ approaches to zero. Parameter estimation can be carried out through Maximum Likelihood or Method of Moments as described in Minka (2000) and Brier (1980).

Our proposal is to adopt DM family of distributions to the problem of formally testing for the presence of “structural equivalence” in a set of triad census data. The aim of the test is to determine if a sample belongs to a previously specified distribution, given that the data generating process is from a DM family.

Initially, Brier (1980) focused their effort on the study of fitting and testing of general log-linear models in contingency tables following DM distribution, then the problem of testing vector of proportions generated from different families of processes has been surveyed (Koehler and Wilson 1986), deriving general goodness-of-fit statistics based on generalized Wald statistic, also computing asymptotic results.

More recently, a goodness-of-fit statistic for testing the fit of a model based on DM family is also obtained in Johnson, Kotz, and Balakrishnan (1997): given that data have been generated by a DM distribution, a possible approach could be to compare the estimated frequencies with a reference distribution defined in null hypothesis by imposing theoretical frequencies π ₀. Nowadays, with the One-Sample DM test (OSDM) for grouped count data, La Rosa et al. (2012) extended this testing procedure to formally test for a set of g samples belonging to the same population measured on different subjects.

Consider a set { x i } j = 1 , … , g of g samples, each one consisting of m counts of distinct modalities where the count of modality i corresponding to the jth sample is denoted as x _i,j . Denote x 0 , • = ∑ i = 1 m ∑ j = 1 g x i , j as the sum of all frequencies over all samples, while x 0 , j = ∑ j = 1 g x i , j stands for the total of the jth sample. Then, to formally tests the hypothesis H ₀: π = π ₀ versus the alternative H ₁: π ≠ π ₀ the following test statistic is presented

where the operator (⋅)⁺ is the Moore–Penrose generalized inverse and the matrix Σ corresponds to

with ( π ̂ , θ ̂ ) unbiased estimators of distribution parameters. The test statistic X _obs asymptotically converges to a χ ² distribution with degrees of freedom equal to the rank of the matrix diag ( π 0 ) − π 0 π 0 ⊤ + .

In addition, due to the aggregation property (see, e.g., Johnson, Kotz, and Balakrishnan 1997, and references therein), the DM family allows for a flexible re-parametrization of the probability vectors: the joint distribution over sums of disjoint subsets of modality counts is also DM, i.e., if (x ₁, …, x _m ) ∼ DM _m (π ₁, …, π _m , θ) then, x 1 , … , x s , x 0 − ∑ i = s + 1 m x i ∼ DM s + 1 π 1 , … , π s , 1 − ∑ i = s + 1 m π i , θ . Hence, it is possible to “collapse” a subset of isomorphism classes into one containing the sum of their counts, e.g., the ones having low frequencies or structural zeroes.

We redesign the testing procedure in Equation (7) to triad census analysis to unveil if a triadic pattern, computed from a group of networks sharing similar features, has been generated from a reference DM distribution. To this end, we propose two sets of null hypotheses. Firstly, reference team strategies (based on the triadic configuration) are set to identify if and how teams follow a given strategy. Secondly, a more direct comparison between couple of teams can be performed to detect possible similarities.

Although a small-scale simulation of size has been performed considering either few m modalities and small total counts x _0,• (see La Rosa et al. 2012), there is no empirical evidence in literature of test statistic validity when the number of g samples is small, as it happens in our empirical application. For this reason, in order to give a further support to our testing strategy, we carried out a small scale Monte Carlo exercise computing the empirical size of the DM test with small sample sizes. All the details are given in Appendix.

4.1 Descriptive analysis

Network structure can be relevant in this kind of analysis, showing evidence of having relationship with the number of passes. For example, considering all networks, a positive and statistically significant large correlation can be found between the number of passes and the mutual dyads (0.77, p-value < 0.001 ). In addition, the asymmetric and null dyads were negatively correlated with the passes (−0.67 and −0.45 respectively), again with statistical significance (p-value < 0.001 for both variables).

Regarding the dyadic distribution, the football teams expressed a median of 37 mutual relationships per match, while unilateral links varied between 10 and 15 in 50 % of the teams. There was an average of 6 unconnected dyads for each team per match, varying between a minimum of 0 and a maximum of 19, as well as presenting higher variability (47 %) among the MAN. Differences between the three seasons seemed to be negligible for the three counts. We must remark that the sum of MAN for football teams (n = 11 players) is constrained to 55, i.e., (n(n − 1)/2).

Furthermore, generalizing to the triad census, we recognized specific triadic patterns in football passes. Table 4 shows the distribution of triads for passing networks of all teams involved in the group stage of the UCL 2016–2017 to 2018–2019 seasons (n = 576).

We highlighted noticeable differences in terms of mean and variability between 16 isomorphism classes. In particular, the most frequent triads were as follows: “300” (fully connected), which also presented the highest variability; “210”; “201” and “111D”. Additionally, classes with lower ties, involving sparse connections, seemed to be recurrent structures in these types of networks. For example, if we consider classes with “0” as first number (corresponding to all the isomorphism classes having zero mutual edges), we can also observe median values greater than zero and a certain amount of variability, expressed in terms of standard deviation. Moreover, several isomorphism classes appeared less frequently in football games, such as the types “030C” and “030T”, involving three directed links.

These findings revealed uncommon types of triadic relations in passing networks: type “030C” refers to cyclic ties, i.e., A → B → C → A, while “030T” represents transitive triads of the form A → B → C, A → C. To summarize, such classes mainly concern transitivity structure and share the presence of three directed links (arrows). The reasons behind these low counts remain somewhat unclear, but could be linked to the direction of the ball (clearly related to the main aim of the game) and positions on the pitch (initial formation). Another issue can be represented by the possibility of “structural” zero issue (Block 2015). Some triad counts can be constrained to zero in particular conditions (i.e., the “003” case if all players nearly make a pass to each individual other player). In general, we remark that, similarly to the dyad census case, the total number of triads for each team (n = 11 players) in each match was constrained to 165, i.e., n 3 .

Figure 4 shows the distribution of triad counts for the three UCL Seasons, confirming some considerations mentioned above.

Figure 4:

Distribution of isomorphism classes in Group Stage of Uefa Champions League (n = 576).

In Figure 5, the first 10 teams are plotted according to their average value for all 16 isomorphism classes. We immediately noticed that skilful teams such as Real Madrid, Barcelona and Manchester City presented an average of “300” counts, which is always greater than 70. On the other hand, teams such as Anderlecht, Dinamo Zagreb and Leicester were more likely to present counts of disconnected or partially non-connected triads (such as “003” and “012”).

Figure 5:

First 10 European Teams for all isomorphism classes (average values).

In keeping with this, triadic information could be summarized by factor analysis. We followed the approach of Faust (2006) to investigate the “resemblance” of triad census in empirical networks, although several approaches may be conducted to make dimensionality reduction (see Greenacre 1984, for a comprehensive overview). In practice, we applied the Correspondence Analysis (CA) to the original data, where the 16 isomorphism classes can be viewed as the modalities of a single variable by looking at data as a cross-tabulation between two categorical variables. Each row of the input matrix represents a team in a match that occurred during the Group Stage of the UCL, while the counts of all the isomorphism classes are arranged in columns.

Considering all networks, the first two dimensions of the CA explain the more than 60 % “triadic” variability, with 38.4 % for the first dimension and 22.9 % for the second. The third dimension also explains the 6.3 % “triadic” variability.

The scree plot of Figure 6 suggests the presence of at least three factors (67.6 % of the overall variability) according to the conventional Kaiser criterion (Kaiser 1960). According to the above results, we focused on these three dimensions. Furthermore, Figure 7 shows both individuals (passing networks of teams) and isomorphism classes on the factor map in first two dimensions. The first dimension is mainly characterized by the presence or absence for connections, with opposing classes such “012C”, “021D” and “021U” against the fully connected type, i.e., “300”. Hence, the second dimension is mainly driven by differences between “210”, i.e., including two transitive connections and two disconnected vertices, and “210”, presenting five links over six.

Figure 6:

Scree plot of the CA on triad census. Red dashed line indicates the reference value 1/(m − 1), where m is the number of modalities.

Figure 7:

Biplot of first two CA factors. Red arrows are isomorphism classes and grey points are the networks (matches). Blue supplementary points represent coordinates of four k-means centroids explained in the text.

Then, in Figure 8, we can observe the contribution of all 16 isomorphism classes for the first three CA dimensions in terms of squared cosines (or squared correlations). These quantities directly express the acceptability of the triad projections along the three dimensions; that is, these quantities can be used to measure the quality of the representation of each isomorphism class on the factor map. For the first dimension, the “300” class is the best represented in terms of squared cosine, reaching a high value of 91 %, followed by “012C”, “012” and “111U” with values higher than 45 %. In general, only “201” and “210” appear to be almost orthogonal with respect to the first factor. Regarding the second dimension, satisfyingly represented classes in terms of squared cosine were “102” (including only a single mutual link between three agents), as well as “120” and “210”, presenting two transitive links. Interestingly, the three classes “012D”, “012U” and “012C” are not correlated with the second factor. Only classes “003” and “201” appeared to be correlated with the third dimension.

Figure 8:

Heatmap of squared cosines related with triads in the CA dimensions.

We also graphically identify the isomorphism classes that presents the higher contribution to the first three CA dimensions (Figure 9). Classes “300”, “120”, “012” and “210” are greater than the empirical threshold of 1/(m − 1), while “030T” and “003” are almost equal to this value. Other triads appear redundant to explain the variability of the football passing network.

Figure 9:

Contributions of isomorphism classes to the CA dimensions. Red dashed line is the reference value 1/15.

Finally, starting from these results, we applied k-means clustering in order to establish a set of motivated profiles, defining a proper null hypothesis for the OSDM test. This allowed us to identify a set of triadic profiles that are related to different triadic behaviours in football. We chose κ = 4 clusters based on the total within-cluster sum of squares.

The four resulting centroids are plotted on the factor map as supplementary points in Figure 7, and their reconstructed triads are found in Table 5 and compared with the general median of the overall teams.

Table 5, shows that the first centroid T ₁ presents a number of “102” and “201” greater than the general medians, while the second one (T ₂) presents lower “201” and greater “210” and “120” (D, U, C) types. The centroid denoted as T ₃ shows the highest counts of fully connected triads, i.e., “300” (exceeding the general median of 22 units), while on the contrary, centroid T ₄ exhibits the lower value of this class and the higher counts of triads with a low number of links, i.e., “012” and “102”.

4.2 Testing the presence of a strategy

After performing a statistical comparison of triad census, we empirically evaluated differences between the observed triads of the football teams against their conditional random reference, i.e., the U|MAN, following the methodology presented in Section 3.1.

Figure 10 graphically shows the distribution over three UCL Seasons of τ _u statistics as defined in Section 3.1. As expected, such statistics did not seem to show substantial changes among seasons both in terms of location (mean or median) and scale (variability expressed through interquartile range), although there were some exceptions (see, for example, type “030C”). Moreover, most connected classes, i.e., “201”, “210” and “300” as well as “102” and “111D”, present evidence against the theoretical values of the random graph scenario, i.e., the null hypothesis discussed in Section 3.1 (here represented by a dashed red line) more often than others.

Figure 10:

Boxplot of τ _u statistics, showing discrepancies between triad frequencies and expected frequencies under U|MAN distribution per Season.

In order to carry out the formal test introduced in Section 3.1, we computed τ max 2 statistics for all available UCL matches. Table 6 contains the percentage of p-values associated with τ max 2 , which are lower than conventional nominal levels, i.e., 0.1, 0.05 and 0.01, even divided by the three Seasons (576 matches). Since we are considering multiple testing, we also included the rejection frequencies adjusted using the Bonferroni correction (i.e., p = α/G). The proportion of p-values lower than nominal levels can vary from 24 % (when the nominal level is 0.01) to 53 % with a greater I error type, and it is noticeable that first Season (2016–2017) presents the highest number of rejections. Using the Bonferroni correction for multiple testing, we observed lower rejection frequencies, and the null hypothesis of U|MAN structure can be rejected for one-tenth of the networks.

Despite these unexciting results, we should remark that (a) a test based on τ max 2 statistic may be more conservative in finite samples, mainly due to the well-known asymptotic approximation issue, and (b) the (conditional) distribution of τ max 2 under the null hypothesis of triadic randomness strictly depends on the linear dependence occurring in the triad census matrix (i.e., its rank). For example, in our football data, we found a mode of 13 for the ranking of the triad census matrix among all networks. As discussed in Section 3.1, this number corresponds to the degrees of freedom in the τ max 2 statistic under the null hypothesis of a random graph.

Ultimately, we can conclude that more than 40 % of the football teams involved in a match showed strong empirical evidence in terms of triadic behaviour against their random graph counterparts based on U|MAN.

Table 7 contains the rejection frequencies, at a 5 % nominal level, for European teams during the three Seasons. PSG rejected all the times in Season 2017–2018, followed by Real Madrid, Porto and Rostov also rejecting the randomness five times out of six in the previous Season. Considering all Seasons, Atletico Madrid rejected the null hypothesis two times out of three (12 matches), followed by PSG (9 matches over 18). Surprisingly, team such as Borussia Dortmund rejects the null for only one match over 18.

The reasons of those rejections can be significantly vary. In some cases, they are driven by high levels of isomorphism classes with low connection (i.e., “012” or “120”); in other cases, the teams are able to exhibit a large number of connected triads (i.e., types “300” or “210”), as well as providing the expected values of their MAN distribution. If we consider the adjusted error rate to be equal to 0.05/192, we can observe that the test was rejected two times out of six (in each Season) for the following teams: Manchester City and Benfica (2016–2017), Real Madrid and Anderlecht (2017–2018), and Valencia (2018–2019).

4.3 Triad census and style of play

We further tested for the presence of a “structural equivalence” in terms of triadic relationships applying the OSDM test outlined in Section 2.2. First, the OSDM test was applied considering the six matches of each team in the three Seasons. Thus, we formalized four null hypotheses based upon the four cluster centroids reconstructed in Table 5 of Section 4.1.

In Tables 9 –12 (depicted in Appendix D) we rank the teams according to their OSDM estimated statistics and their associated p-value in the three UCL Seasons. Regarding the first system, denoted as T ₁, it can be seen that the null hypothesis cannot be rejected for almost all Dutch and French teams (excluding the first Season of PSG) and the Spanish teams of Atletico Madrid and Seville. Similar to this strategy are most of the Italian teams, such as Naples, Rome (2017–2018) and Inter.

If we consider the null hypothesis constructed upon T ₂, few teams were ultimately reflected in this strategy. Notably, however, all Russian teams (CSKA Moscow, Spartak and Lokomotiv Moscow) were unable to reject the hypothesis at a 5 % nominal level throughout the three UCL seasons. These teams are followed by other “Eastern” Teams such as Qarabag, Galatasaray, AEK Athens and Warsaw.

As mentioned, the third cluster (T ₃) contains teams that generally produce more closed triads between players. For all three years, we found that the OSDM could not be rejected for Manchester City, Juventus, Real Madrid, Bayern Munchen, Borussia Dortmund and Barcelona (except the preceding Season). Finally, almost all teams rejected the null hypothesis T ₄ based on the four centroids.

Another possibility is given by the use of “reference” teams as null hypothesis. This kind of test consists of imposing under the null hypothesis a specific parameter scheme derived from a set of chosen teams, e.g., by using the triadic median. In Tables 13 –15 (Appendix D), we perform the OSDM test for each of the proposed reference teams, ranking the teams according to their OSDM estimated statistic and its associated p-value in the three UCL Seasons.

Considering only the first examples for the sake of brevity, European teams that were closer to Real Madrid in a triadic point of view were Barcelona (2017–18 and 2018–19), Juventus (2017–18 and 2018–19) Bayern Munchen (2016–17 and 2018–19), PSG (first two Seasons), but also Naples and Borussia Dortmund (last Season) and Arsenal (first Season). Conversely, in terms of triadic distribution it can be found in Dynamo Zagreb, Rostov, Spartak Moscow, APOEL and PSV.

We would like to note that the results of the OSDM test discussed here are highly similar considering the adjusted type I error rate obtained through the Bonferroni correction. Indeed, observing the Tables in Appendix D, the null hypothesis can be rejected for all the teams such that p < α/192, approximately equal to 0.0003 when α = 0.05.

To conclude, network structure and triadic analysis can provide information for trainers, coaches and policy-makers in football. Triad census can highlight tactical and technical aspects, and may be able to separate out the cohesion of a team in terms of passing behaviour, clarifying types and the strength of relationships among players, as well as express the collectivity of sub-groups. Therefore, from a tactical perspective, knowledge of opposite triadic distribution may provide a specific improvement both at individual level and for the overall team, also becoming relevant to the outcome of a game, e.g., the probability of winning.