The Soccer Game, bit by bit: An information-theoretic analysis
Luis Ramada Pereira, Rui J. Lopes, Jorge Louçã, Duarte Araújo, João Ramos
TThe Soccer Game, bit by bit: An information-theoretic analysis
Luis Ramada Pereira , Rui J. Lopes
2, 3 , Jorge Lou¸c˜a , Duarte Ara´ujo , Jo˜ao Ramos , Abstract
We modeled the dynamics of a soccer match based on a network representation where players are nodesdiscretely clustered into homogeneous groups. Players were grouped by physical proximity, supportedby the intuitive notion that competing and same-team players use relative position as a key tacticaltool to contribute to the team’s objectives. The model was applied to a set of matches from a majorEuropean national football league, with players’ coordinates sampled at 10Hz, resulting in ≈ Complex systems, with time evolving interactionsamong its elements, abound in the social, biologicaland physical domains. In many of these systems,elements are clustered in groups that also undergochanges with time. A temporal, clustered networkcan be an appropriate representation of such a sys-tem.In this article we apply this representation tothe sport of soccer. Soccer, as many other competi-tive team sports, can be seen as a social-biologicalcomplex system [22]. The domain dynamics of thesesport modalities are neither fully random nor fullydesigned. This contributes decisively to their com-plexity. This is a property shared by many othercomplex systems that are subject to constrained ran-dom chance, therefore we believe the techniques andapproaches researched for this article have potential application beyond sports.We use the term “clustering” to mean the setof disjoint non-empty subsets of nodes observed inthe network at a given point in time. Some authorscall it a “partition”. These terms represent similarconstructs, clustering being semantically associatedwith an emerging, bottom-up aggregation of nodes,while partition conveys the idea of a top-down drivenprocess. In soccer there is not a single entity control-ling group formation [26], at least not directly andin real time, so the former seems more appropriate.The soccer match is here represented as a successionof network observations where clusters are subsetsof players, including the two football goal frames,resulting in a network with a maximum of 24 nodesconcurrently active, plus substitutes [21].While studying a soccer match as an evolvingclustered network, we start from the propositionthat players’ spatial distribution on the pitch is the1/18 a r X i v : . [ c s . S I] F e b he Soccer Game, bit by bitdetermining variable for clustering. Intuitively wecould think that an optimal assignment of players toclusters would require a physical distance measure,predicating link weights by player relative distance.However, there are complicating factors to the usageof such a precise measurement, as the importanceof inter-player distance is not independent of gameplay [19]. It varies with pitch location, ball position,game rules, environmentals (such as playing sur-faces or weather), or the relation between time anddistance in dynamic game settings. All these con-tribute to the actual player’s instantaneous graspof his performance environment and perception ofopportunity for action [1]. The network representa-tions we are using for the present analysis were builtwith a different approach. Instead of inter playerlinks weighed by distance, players were clusteredinto homogeneous and disjoint groups connectedby a single link [21], using the formalism of hyper-graphs [2]. A hypergraph is characterized by havingmultiple nodes connected by a single link in con-trast with a traditional graph where links have amaximum of two endpoints. A set of nodes thatshare a link is called a simplex. The process to iden-tify these sets is non parametric and is explainedin [21]. It guarantees that no node is closer to anode belonging to a different simplex than to itsclosest node in the same simplex. In the particularcontext of the present article, simplices are sets orclusters, and the collection of simplices observed ina single sample, a clustering. In the reminder ofthis document, we use the terms simplex and clusterinterchangeably, all referring to the same construct:a group of players in articulated interaction andproximity. An example of the clustering process isillustrated in figure 9 in the appendix.It could be argued that discretization and as-signment of nodes to a pairwise disjoint family ofsets, would lead to a distorted representation ofevents on the pitch. After all, players move freelyin an Euclidean space and in continuous real time,while in the proposed representation time is discreteand players move on a lattice, understood not as agrid that spans the pitch but as the configurationspace of all possible set arrangements [3, 11]. Fre-quent observation, however, mitigates these effects.For example, peripheral players in a simplex willmore easily transfer to a different simplex and, iffrequently observed, any simplex changes will bequickly captured. Due to the high frequency char-acteristic of the network (10Hz), errors will smoothout as player simplices form and dissolve, estab- lishing a bridge between the continuous domain ofgame play and the time sliced network representa-tion employed [10].This discretization carries with it a significantadvantage. We are no longer in a continuous domain,and the toolkit of information theory [4] becomesavailable to us. In a discrete domain, informationcan be quantified for complexity, such as in theKolmogorov complexity or the Shannon entropy[8, 12, 29]. Similarly, two pieces of information canbe compared for distance. We can determine how farapart or how close they are by the number of unitsof information that are needed to find one giventhe other. In this article the pieces of informationare the individual clustering samples of the soccermatch.Formally, a clustering is: C = { c , · · · , c k } :( c i ∩ c j = ∅ ∀ (1 ≤ i, j ≤ k ∧ i (cid:54) = j )) ∧ ∪ ki =1 c i = V (1)where c are the disjoint subsets, k the number ofsubsets, and V the set of all nodes.There are several methods to measure the inter-distance between clusterings, with varying proper-ties, such as the Rand Index [24], Adjusted RandIndex [9], the Normalized Mutual Information [5],the Van Dongen-Measure [6] and others. A thor-ough discussion of the major methods can be foundin [14, 31, 32]. We chose the Variation of Informa-tion ( VI ) [14] to measure the information distancebetween samples and thus evaluate the change aclustered network experiences as a function of time.In a nutshell, VI , measures the amount of informa-tion gained or lost on every new network observation.If no changes in the clusters are observed, then thereis no variation of information. As clusterings shiftfrom one another, VI increases. This is easy tovisualize when considering the so-called confusionmatrix [30] between clusterings at successive ob-servations. This matrix describes the node spread,where each element represents the number of nodesmoving from one cluster to another. If clusters areunchanged and keep their node affiliation, the con-fusion matrix will be a monomial matrix, VI = 0and we know exactly where each node ends up. Butas the number of non-zero entries in the confusionmatrix increases and their distribution tends to uni-form, the uncertainty about each node destinationalso increases. Consider as an example a clusterthat splits in half versus another that sheds a singlePage 2 of 18he Soccer Game, bit by bitnode. There is a higher uncertainty about each nodefinal destination in the former than in the latter.In simple terms, VI measures this uncertainty. VI has been applied in multiple contexts, for exampleto address the problem of clustering news publishedby online newspapers [27]. A practical illustrationof how to compute VI can be seen in tables 1 and2 in the appendix.We have selected VI as it is a true metric, re-specting the triangle inequality, meaning that noindirect path is shorter than a direct one. This isimportant in analyzing the rate of change at mul-tiple scales, avoiding the unreasonable possibilityof having a greater rate of change for a given timeinterval, when sampling the network at a lower rate. VI also increases when fragmentation and mergesoccur in larger clusters, which intuitively relatesto playing dynamics, given the rise in degrees offreedom experienced in larger groups of interactingplayers. Fundamentally, although in this articlewe consider VI as a proxy for game dynamics, VIitself is not a quantification of informational mean-ing or semantics, but simply, a quantification ofinformational variation, or as Shannon puts it “se-mantic aspects of communication are irrelevant tothe engineering problem” [29, p.1].In this article we consider a split of VI into twoterms. A clustering has a signature in the (multi)setof its clusters’ sizes. We call it a formation, as itvaguely captures the popular notion of team matchformation in soccer, although these concepts do notoverlap. A formation, using the previous notationis defined as: F = {| c | , · · · , | c k |} : k (cid:88) i =1 | c i | = | V | (2)Using this construct we split VI into two terms:1. VI f , which is the minimum amount of inherentchange resulting from the evolving formationas described above, and2. the compositional VI c computed as VI c = VI − VI f , which is the additional informationdistance accrued on top of the minimum im-plied by the evolving formation.To understand these constructs consider that fortwo consecutive clusterings to show a null VI it isnecessary, but not sufficient, that their formationsare equal. In fact, the formations can be equal(which implies that VI f = 0), but the clusterings’ transition can still show a positive VI , due to com-positional changes (in which case VI = VI c > n clusters. For simplicity, consider they are all of thesame size s , or formally C t = { c t , · · · , c tn }∧| c tk | = s .Its formation is F c t = { s n } . Comparing with an-other clustering C t + δ , also with F c t + δ = { s n } , wehave: (cid:40) VI = 0 ⇐⇒ ∀ i ∈ { · · · n } ∃ ! j ∈ { · · · n } | c ti = c t + δj VI > , otherwiseAnother example can be found on figure 9 in theappendix. There we can observe a transition froma formation { , , } in (a) to { , , , } in (b).As these formations are not equal, VI >
0, howeverit is not the minimum for this transition. We cansee that there is additional entropy, for instancein the restructuring of the 4-node simplex fromplayers {
12, 20, 22, 21 } to {
9, 10, 14, 21 } , that thesimple changes in formation would not necessarilyrequire. We consider the usefulness of such a splitanalysis, guided by the intuition that the interplayof strategy, play patterns, set pieces, and individualplayer initiative [1,25] may drive differently VI f and VI c . Depending on the represented system, thesetwo components can have different meanings. Thisis an open issue that we briefly touch upon but thatdeserves further research.While calculating the total VI is computation-ally trivial if the network partition into clusters isknown, finding VI f is not routinely tractable, aswe need to find the minimum node change for theformation transition, an NP-hard problem, meaningthat it will be at least as algorithmically complex tosolve precisely as any non-deterministic polynomialtime algorithm. We employ a heuristic developedpreviously to approximate it efficiently [18].In the reminder of this document, we discuss thecorrelation of VI and playing dynamics in section2, followed by a section 3 describing the resultsobtained. We discuss these results in section 4 andwe conclude with directions for future research insection 5.The main research question is whether ( VI ) canbe a faithful proxy for game dynamics, and weexpect to confirm a strong correlation. We arealso interested in how VI f and VI c contribute tototal VI and how it relates to game tactics and playdevelopment. Page 3 of 18he Soccer Game, bit by bit The proposed approach is applied to the analysisof a set of 9 soccer matches from the 2010-11 sea-son of the English Premier League. Based on aninformation stream collected from realtime pitch-located raw video feed, each match is modeled as ahigh-resolution (10Hz) temporal hypernetwork withsimplices as clusters [20, 21], parsed by player prox-imity. The whole network is made up of up to 30nodes (28 players and 2 football goals) of whichonly a maximum of 24 are present on the pitch atany given moment (11 players from each team and2 goals). These nodes are clustered into a variablenumber of simplices, 10 times a second based onthe location data. The method used for clusteringguarantees that a node and its closest node belongto the same simplex. This implies that the smallestsimplex has a minimum of 2 nodes, i.e., there areno isolated nodes. Although there maybe occasionswhere a player is side-lined, this will be an excep-tion, as the expectation at the top-level of sportsperformance is that every single player have an ac-tive role in-play, in relation to their teammates andtheir opponents. Although the football goals areobviously fixed on the pitch, there is no fixed frameof reference for the clustering process, and the rela-tion between players and football goals, especiallywith the goal keeper, are of particular importance,which justifies their inclusion.On average, considering a match, including extratime, we observed and measured the network ≈ VI , isa function that takes two clusterings as parametersand returns the information distance between theclusterings. VI is computed as: V I ( X ; Y ) = − k (cid:88) i =1 l (cid:88) j =1 r ij [log ( r ij p i ) + log ( r ij q j )](3)where X = { x , · · · , x k } and Y = { y , · · · , y l } areclusterings of a given set S , with n = | S | , k = | X | , l = | Y | , r ij = | x i ∩ y j | n , p i = | x i | n and q j = | y i | n .From this equation it is easy to see that when thesimplices in X and Y are the same, the result is zero,as r ij = p i = q j . This result expresses the fact thatno information is gained or lost when going fromone clustering to the other. For empty intersectionsof pairwise simplices, r ij = 0, and although log(0)is not defined, applying l’Hopital rule we get a null contribution from these intersections to theoverall VI . In summary, only pairwise non-disjoint,non-identical clusters contribute to the informationdistance. VI works as a distance metric for clusteringsof the same set of nodes. In the model used torepresent the soccer match, the set of nodes re-mains constant, except on substitutions and send-offs. However, the number of observations affectedby these events are so low, that we have ignoredtheir contribution in the model.Using base 2 logarithms, VI is measured in bitsand describes the balance of information neededto determine one clustering from another. VI isalgorithmically simple (it can be computed in O ( n + kl ))) and, as mentioned in section 1, it is a truemetric [13], respecting positivity, symmetry, andthe triangle inequality.Using the previous notation, for every individualplayer p ij ∈ { x i ∩ y j } his contribution to the overall VI is computed as: VI p ij = − r ij [log ( r ij p i ) + log ( r ij q j )] | x i ∩ y j | (4)which takes the contribution of pairwise simplices x i , y j to the overall VI , and divides it in equalparts among all players ∈ x i ∩ y j . Note that, inthe particular case of the network that we built, allnodes/players are present in all observations andare members of one and only one simplex in any oneobservation. Equation 4 registers the contributionsof players involved in their simplices when thesechange. The only exception is the case of a send-offor substitution, in which case the player no longercontributes to the dynamics of the match.The VI of two clusterings ( X, Y ) of S can onlybe zero if ∀ s ∈ S | s ∈ X ↔ s ∈ Y . If this con-dition is not met then min( VI ) ≥ n ∗ [14], where n ∗ = max( k, l ) still using the same notation. In thesoccer match representation proposed in this articlethe number of nodes is fixed at 24 (barring anyred cards), and thus, n ∗ = 12 and min( VI ) = every time there are any clustering changes. Thisis also min( VI f ) under those conditions. VI de-pends on the level of fragmentation on the pitchacross observations, which intuitively reflects thesituation of players jostling for position, but cannotexceed log ( n ) [14]. These extreme values of VI are, however, just boundaries that limit minima andmaxima given any set of clusterings. In the presentcase, we have a minimum of 2 nodes per cluster,which implies a maximum of 12 clusters, resultingPage 4 of 18he Soccer Game, bit by bitin max( VI ) = log (12) = 3 . VI ≈ . . s separation, or ˙VI ≈
12 bps( bits · s − ).As mentioned previously, the data used for thisarticle were captured 10 times a second. A signif-icant amount of sparsity, i.e. a large amount oftransitions without clustering changes, is observedat this frequency. This posits the question of theideal sampling rate [15], given the dynamics of asoccer game, the capturing technology and the clus-tering methodology. The observed sparsity lead usto adopt a set of measures in the results sectionahead, to enhance analysis and observability. Theseincluded: • the usage of differentials and measuring changein bps, denoted as ˙VI ; • the use of rolling averages for visualization andcompatibility with the rate of change and playof a soccer match. Results shown use 4s samplewindows, except when noted; • and, finally, we made use of cubic Hermitesplines [16] to envelope ˙VI maxima. Resultsuse an inter pivot distance that dynamicallyvaries up to a maximum of 80s depending on theposition of the observed value in the probabilitydensity function of ˙VI (figure 1). Given that the space of all clusterings is substan-tial, corresponding to a lattice of over 4 . × points (Bell number B ), the amount of uniqueclusterings we can observe is just a small fraction ofthis space, gated by the total of samples collected(average 58283, σ = 1336). Assuming a randomdistribution, the probability of observing the sameclustering, that is the same sets of simplices, is forall purposes nil when considering the space size.Obviously the real distribution is not random andis heavily condition by its prior state. But, whenexcluding consecutive observations, a significantlevel of clustering re-appearances still emerges (av-erage 6 . σ = 0 . ˙VI and the standard deviation forthe whole match is consistent across matches, with atotal average of 0 .
597 bps, σ = 0 . > . ms − ) 1.4% of the time [7], 10Hz is a samplingfrequency that often generates no clustering changesin consecutive samples. In fact, in almost 80% ofthe network observations clusterings do not change.The standard deviation per match has an average ˙VI of 1.30 bps, with a maximum of 1.37 and aminimum of 1.25 bps across all nine matches. A fullreport for all matches can be found in table 3.The dispersion of ˙VI as measured by the coeffi-cient of variation of all match observations averages218%, reflection of the high activity level of thesoccer game.We found no correlation between the time or-dered sets of VI observations between the matcheswe have analysed. However, a similar ˙VI averageand dispersion is observed across matches. Theprobability density functions for all nine matches,which can be seen in figure 1, are strikingly sim-ilar. There is a clear consistency of dynamics asmeasured by ˙VI , in which matches exhibit similarprobabilities of finding given levels of dynamics. Anexplanation is player’s regulation of exertion duringthe match to manage fatigue, particularly at thehigh intensity professional matches are played [28].In 8 out of the 9 matches we examined, we ob-serve a descending slope when the time ordered ˙VI set is linearly regressed as seen in figure 2( p = 0 . H : normal null average distribu-tion, single tailed). It is not a very pronouncedslope. Two interpretations for this observationare increased fatigue as the matches progresses onone hand, and adjusted tactics as a result of in-creased acquaintance with competitor behavior onthe other. Similar observations have been previ-ously reported [23]. However, it is important tonote that the same team plays in every match. Alarger sample of matches, from a wider population,may offer more consistency to this pattern.At a sampling rate of 10Hz, VI f is the majorcontributor to the total VI . Typically VI f VI c ≈ VI f VI c ≈ . Figure 1.
Probability Density Function ( f ( ˙ VI )) and Cumulative Distribution Function ( F ( ˙ VI )) for allnine matches measured on a 4s moving average window. Games color coded. There is a consistency ofpatterns that likely mirrors energy expenditure and management throughout the game [17].and clusterings take time to evolve, but the for-mer has a much more restricted space. The num-ber of possible formations is given by the integerpartition function, which is P = 1575 reducingto 320 when considering that formations with iso-lated players are not allowed, while the space ofclusterings, as referred above, is given by the Bellnumber B ≈ . × , that only reduces to ≈ . × when excluding clusterings with sin-gleton clusters. In practice we observed an averageof 11070 , σ = 678 unique clusterings, but only anaverage of 193 , σ = 29 of full formations per match(i.e. with 22 players on the pitch, red cards impactthese results as it reduces the number of players,preventing clusterings from reappearing).Although VI f far outweighs VI c in its contri-bution to the information distance, the differencein maximum scores is much less dramatic, which points to less frequent contributions but equallyimpactful at certain moments of game play.This can be seen when comparing the envelopesplines for the same match with and without a mov-ing average (4 s ) at 80 s pivot separation (see figure3). This trend can also be seen on the average ofthe coefficient of variation for ˙ VI c and ˙ VI f , respec-tively 547% and 227%. The impact of the samplingrate is sizable and further exploration of the signif-icance of VI f and VI c in the context of a soccermatch warrants a deeper analysis of the interactionof the sampling rate, the game dynamics and theresulting VI .To validate VI as an indicator of game dynam-ics, we searched for correlations between known mo-ments of intensive player repositioning and peaksin the information distance. To identify those mo-ments in our datasets we made use of publicly avail-Page 6 of 18he Soccer Game, bit by bit Figure 2.
Linear Regression of ˙VI for all nine matches. Games color coded. Only one match (purpleline) has an average positive gradient. (a)
4s moving average window (b)
No rolling average
Figure 3.
On a moving average with sample window of 4 s , ˙ VI f has a ≈ VI than ˙ VI c when sampled at 10Hz (3a). However, when looking at individual sample maxima,that difference almost disappears (3b). If we equate ˙ VI to energy expenditure, we can interpret this isdue to energy management by the individual players, being judicious about their marking and unmarkingefforts. Page 7 of 18he Soccer Game, bit by bitable match commentary, as visual information wasnot available to us for most matches. The time Figure 4.
Conditional probability of having a peak( ¨VI = 0) if a corner is taken P ( peak | corner ), withpeaks taken from the cubic hermit spline using theinter-pivot distance to control the number of peaksobtained. Analysis performed with ≈ .. P ( peak ∩ corner ), but P ( peak ) as well.accuracy of these commentaries is restricted to aresolution of 60 seconds, leading to a potential er-ror of ±
300 observations, discounting other timingcommentary errors. This mismatch between com-mentary and sampling resolution was addressed asdescribed ahead.We collected timed tags for goals, redcards, cor-ners and substitutions among others. Out of these,only corners are intuitively associated with quick player re-positioning, which justifies their selectionfor analysis. It should be noted that there is nospecial reason to select corners except for the ob-servation that if VI , as used in this article, is agood measure for game dynamics, then we shouldexpect peaks when corners are taken, and their timecorrelation useful for validation of the hypothesisthat VI is a good proxy for playing dynamics. Inthe 9 matches, we observed an average of 10 cornersper match σ = 2 . VI as a measureof game dynamics, knowing that corners should rateas moments of highly changeable player positioning,we computed the conditional probability P ( peak | corner ) of observing a peak every time a corneris taken. As match commentary resolution is 1minute, P ( peak ∩ corner ) was measured at the realpeak ± s . This is contrasted to the probability offinding a random peak under the same conditions,see figure 4.This analysis is done per match, as this proba-bility is dependent on the number of peaks observedin the match and their time distribution. At ± s overlaps can occur, as successive corners are notinfrequent. Peaks were collected from the Hermitesplines, with inter pivot distance adjusted to gen-erate ≈
20 peaks. With one minute resolution thisstill covers, assuming no peak overlaps, a little over20% of the whole match, which is confirmed bythe results obtained for the probability of finding apeak at random. Using total VI , we were able torecognize 72.4% of all corners. These results shoreup the compelling observations recovered from thematch ˙ VI graphs. VI , as used in this study, is clearly a proxyfor game dynamics, understood as a rapid pace ofinter-players relative displacement, i.e. without afixed frame of reference. This is notably obviousduring set pieces. Corners and free kicks invariablygenerate a spike in VI , especially supported by VI f ,which could indicate the execution of set routines.Conversely, other events, like substitutions or send-offs, generate pauses that are captured by a dropin VI .Examples can be seen in figures 5, where VI is plotted for a whole match, with vertical barsindicating the type and time of events.To analyse player contribution to the overall VI , we apply equation 4. We consider the playerindividual ˙VI , and his overall activity compared tothe average ˙VI per player. This may be useful toassess his activity during the match (figure 6).Page 8 of 18he Soccer Game, bit by bit (a) Match 1, 0-0 (b)
Match 2, 2-2
Figure 5.
Plots for two matches where orange and purple points are, respectively, observations of ˙ VI f and ˙ VI c at each sample transition, and the colored lines the respective peak envelope. ˙ VI f and ˙ VI c seemto be heavily correlated with match events, such as corners, where a high level of player repositioning isexpected, and player substitutions, usually associated with a trough in ˙ VI . It is also visible at minute 92in 5a that the match virtually ”stopped” during the send-off of two players from opposing teams.Page 9 of 18he Soccer Game, bit by bit Figure 6. ˙VI for a single player, in a single match, with maxima envelope. His total ˙VI is comparedagainst the match average for the whole match on the red bar on right hand side of this plot. In thiscase, a center forward player is represented, showing a lower than average ˙VI , which may be expected,because a forward is typically less active than the other players during his team defensive sub-phases ofthe match.We also introduce the concept of a simplextransition, a tuple of simplices ( c ti , c t + δj ) such that( c ti ∩ c t + δj (cid:54) = ∅ ) ∧ ( c ti (cid:54) = c t + δj ), that, at successiveobservations, involves always the same players.We visualize the type of transition, color codedto denote the number of home and visiting playersinvolved. Each simplex transition plot is scaled byoverall VI contribution for that set of transitions,and details when those transitions occurred (seefigure 8). Plotted in reference to a single player,the major simplex transitions he was involved inbuild a full view of the player activity during thematch. This is depicted in figure 7, where thevisual representation of this view is detailed. Anaggregation of all simplex transition charts providesa full view of a complete match. In this section we discuss the relevance, principles,relationships and generalizations that can be derivedfrom the results presented above. We cover eightmajor findings informed by expertise about thesoccer game. ˙VI is highly variable throughout a match. Evenwith a 4 second moving average sample we foundan average ˙VI coefficient of variation of ≈ ˙VI time series across matches, as everymatch is different.Although these findings confirm empirical expec-tations from a typical soccer match, it is compound-ing evidence that ˙VI reflects the game dynamics. When comparing different matches, we found con-sistent ˙VI averages, with a coefficient of variationof the averages of ≈ ˙VI measurements is highly con-sistent across matches as seen on figure 1. We didnot find matches where ˙VI is consistently high orconsistently low. All matches come from the offi-cial English premier league games, usually playedat a similar competitive level, so these results arenot surprising, if ˙VI is indeed a proxy for gamedynamics. We observe a general decreasing trend in ˙VI as thematches progress. When linearly regressed eight outof nine games exhibit this trend. Player fatigue andinter-team tactical adjustments may be a determin-ing factor, although the evolving match score andsignificance in the context of each teams generalendeavors, may play a role as well.
There is evidence that the peaks and troughs ob-served in the values of ˙VI correlate with knownevents, such as corners, free kicks or substitutionsthat similarly affect the game dynamics.
Sampling at 10Hz generate ≈
80% of null ˙VI mea-surements. A lower sampling rate may producesimilar outcomes, resulting in a more efficient datacapture and computing process. However, not allPage 10 of 18he Soccer Game, bit by bit
Figure 7.
This chart shows the top ten simplex transitions player 22 of match 1 (figure 5a) was involvedin, as well as their formation. His contribution to the match VI resulting from participating in thesesimplex transitions, is proportionally encoded in the area of the circle: larger circle signifies highercontributions. Each formation is coded in color and shade, with green and blue representing, respectively,home and visitor players, and the number of shades the number of participating players in the simplex.Each tick signals a transition and the match moment when it occurred, with a full match taking a fullcircle. The lower and upper semicircles describe, respectively, the formation of the prior (source) andimmediately subsequent (destination) simplices, where the player was involved. Finally, simplices areidentified by the participating players’ numbers, with home players first, followed by visitors. Player 22is a visiting forward, and as seen in the picture, is frequently observed alone (the single shade of blue inthe semi circles) in a simplex with opposing back player(s), a typical pattern. Transition from formation3 −
22 to 3 , −
22, when home player 12 joins the simplex, has the highest accumulated VI contributionfrom player 22. It occurs throughout the match but with an emphasis in the first half of the first 45 min.Player 22 is supported by a teammate in only two transitions out of the 10 represented. Page 11 of 18he Soccer Game, bit by bit Figure 8.
This chart uses the same symbolic elements as figure 7 but operates at a different level.Each circle represents the overall contribution to the match VI of a whole transition and not just theplayer’s contribution. Here we represent a match top ten transitions. The encoded information in thisand in figure 7 can be useful to study and train high frequency transitions that contribute significantlyto playing dynamics.events develop in the same time scale, and furtheranalysis would be required to fine tune the samplingrate to the specific analysis sought. We found that clusterings reappear throughout thematches with a probability (0 . , σ = 0 . . − ). This can be interpreted as player dynamicplacement on the pitch according to a game plandesign. ˙VI components We found that, at 10Hz, average ˙ VI f is the maindriver of total ˙VI , meaning that when clusteringschange, players end up in the clusters that frequentlyminimize the information distance. However if weinspect the maxima of these two components, wefind that player repositioning within the clustering,i.e. ˙ VI c , sometimes contribute as much as ˙ VI f tototal ˙VI . An hypothesis to justify this observationis that players are judicious with their energy ex-penditure, while individual initiative can heavilyimpact game dynamics. The proposed way of measuring the soccer gameenables a multi-layer decomposition of its dynamicsfrom macro level (a full match) to meso (clustersof players, transitions and teams), to micro (indi-vidual players), as exemplified by the informationpresented, respectively, in figures 5, 8, and 6. Thisenriches the information that can be extracted, help-ful to evaluate the dynamics generated by individualplayers, but also cluster changes experienced duringa clustering transition, which can be helpful to un-derstand which sets of players are more prevalent,how they change and how they impact the overall ˙VI . The presented results endorse the status of ˙VI asa measure for game dynamics. The fact that itcaptures with accuracy and precision well knownmoments of players jostling for position, such aswhen corners are taken, supports this interpretation.With error free and detailed metadata, a moreaccurate analysis would be possible, especially withconcurrent visualization and representation. ThePage 12 of 18he Soccer Game, bit by bitpresent work is based on prior data, captured andclustered independently, that abstract the reality ofa soccer match. Based on the promise shown here bythe variation of information as an analysis tool, theproposed methods could be valuable to evaluate dif-ferent approaches to data capture, such as samplingrates, as well as different clustering methods andgame representations, such as overlapping, distanceweighted networks, non-inertial frames of referencethat accommodate ancillary factors, centroid basedclustering, among many others.Although the soccer game was the subject mat-ter of this article, we believe the principles andapproaches used extend to other socio-biologicalsystems with structural competing interactions, ofwhich those found in competitive team sports arean example.This is left for future research.
Appendix
To illustrate how ˙ VI is computed, consider thetwo moments in a fictional match represented infigure 9. The corresponding confusion matrix, whichdescribes the transition of nodes between simpliceswhen going from moment t to t+0.9s during thematch, is given in table 1.Null matrix elements, as well as unchanged sim-plices (simplices 1, 2 and 9), do not contribute toinformational distance. The contribution of the oth-ers is computed according to equation 4. The resultis shown in table 2, where the contribution fromeach simplex transition can be seen.The end result is VI = 0 . VI = . . =0 . Declarations
Funding
This project was partly supported by Funda¸c˜aopara a Ciˆencia e Tecnologia through project UID/Multi/ 04466/ 2019. R. J. Lopes was partly sup-ported by the Funda¸c˜ao para a Ciˆencia e Tecnolo-gia, under Grant UID/50008/2020 to Instituto deTelecomunica¸c˜oes. D. Ara´ujo was partly funded byFunda¸c˜ao para a Ciˆencia e Tecnologia, grant numberUIDB/00447/2020 attributed to CIPER – CentroInterdisciplinar para o Estudo da Performance Hu-mana (unit 447). Page 13 of 18he Soccer Game, bit by bit (a)
Clustering at time t (b)
Clustering at time t+0.9s
Figure 9.
Clustering for two moments of a fictional match separated by 900ms. Cluster 1 (goal andgoalkeeper of the red team) and Cluster 9 (goal and goalkeeper of the yellow team), are not visible. Theclustering process ensures that a node and its closest neighbor are nodes of the same simplex. Homeplayers are numbered in red circles, visitors in yellow. Blue hexagons identify the simplices. White linesare only used to identify simplex membership. Formation for (a) is { , , } and for (b) is { , , , } ,which correspond to the row and column sums of the matrix in table 1. Page 14 of 18he Soccer Game, bit by bit Simplex 1 2 10 11 12 13 14 15 9
Table 1.
Confusion matrix going from t to t+0.9s
Simplex 1 2 10 11 12 13 14 15 9
Table 2.
Computing VI Page 15 of 18he Soccer Game, bit by bitMatch 1 2 3 4 5 6 7 8 9Result 0-0 2-1 2-2 1-0 3-0 1-0 0-1 2-1 1-0Avg 0.544 0.591 0.631 0.665 0.622 0.573 0.568 0.599 0.581 σ VI t a -4.6E-4 -6.0E-4 -2.9E-4 -9.9E-4 1.4E-4 -1.2E-3 -8.7E-4 -1.3E-3 -4.7E-4˙ VI h Avg 0.277 0.290 0.329 0.330 0.314 0.284 0.301 0.302 0.292 σ σ VI v a -4.0E-4 -1.8E-4 -5.3E-4 -6.2E-4 2.4E-4 -6.2E-4 -1.3E-3 -6.6E-4 -1.8E-4˙ VI c Avg 0.083 0.102 0.102 0.112 0.105 0.095 0.095 0.104 0.097 σ σ VI f a -4.6E-4 -3.5E-4 -1.2E-4 -6.5E-4 2.6E-4 -1.1E-3 -5.4E-4 -1.1E-3 -4.5E-4 Table 3.
Average (avg), standard deviation ( σ ), and linear regression slope (a) for ˙ VI results (Total,Home, Visitor, Compositional and Formation) for the nine matches used in this article Page 16 of 18he Soccer Game, bit by bit References
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