New Indicator for Centrality Measurements in Passing-network Analysis

of Soccer

Masatoshi Kanbata, Ryohei Orihara, Yuichi Sei, Yasuyuki Tahara and Akihiko Ohsuga

Graduate School of Informatics and Engineering, The University of Electro-Communications, Tokyo, Japan

Keywords:

Big Data, Data Mining, Graph Mining, Graph Network, Soccer.

Abstract:

A number of ﬁelds including business, science, and sports, make use of data analytics. The evaluation of

players and teams affect how tactics, training, and scouting are conducted in soccer teams. Data such as the

number of shots and goals in match results are often used to evaluate players and teams. However, this is not

enough to fully understand the potential of the players and teams. In this paper, we describe a new analysis

method using passing-distribution data from soccer games. To evaluate the performance of players and teams,

we applied graph mining. We also used an index called centrality, which evaluates individual contributions

with an organization. In this research, we propose a new centrality model to improve existing conventional

models. In the calculating the centrality of a given player pair, we consider not only the shortest sequence of

passing but also longer ones. In this research, we veriﬁed the signiﬁcance of these indicators by applying the

data of UEFA EURO 2008, 2012, and 2016. As a result, we found our method to be more consistent with

game results than conventional methods.

1 INTRODUCTION

In recent years, data analysis has been widely used,

and it has enriched our lives. Among its uses in sports,

data analysis is actively performed during games.

This analysis has the inﬂuence on team management,

such as the evaluation of teams and players, team tac-

tics, training of players, and scouting of new play-

ers. Another beneﬁt of data analysis is its ability to

provide an alternative viewpoint for sports audiences.

For example, in baseball, Saber Metrics conducts an

objective analysis from a statistical viewpoint, which

has an inﬂuence on players’ evaluations and tactics

(Beneventano et al., 2012).

However, conducting an objective analysis in soc-

cer is more complicated. Team play is important

in soccer, and ball possession in a soccer game fre-

quently switches between the offensive and the de-

fensive side. Furthermore, players and teams are of-

ten evaluated by data such as the number of assists

and goals in a game’s results, and the players with

these statistics in the game tend to be highly evalu-

ated. For this reason, we examine not only the players

who scored goals and assists but also the players in-

volved in goals and assists. In addition, by evaluating

players and teams through a more appropriate analy-

sis method, this research can contribute to the team’s

performances using a logical analysis of tactics, eval-

uation of player’s characteristics and abilities.

In this research, we focus on passes that are con-

sidered to lead to goals and assists in soccer, and we

evaluate the players based on them. In particular, by

applying the graph theory, we obtain the sequence of

the passes that ended with the shots. We apply the

concept of centrality in the graph theory to propose

a new evaluation model that improves current con-

ventional models. At the moment, there is no ob-

jective method to show if a criterion of an individual

player’s evaluation is sound. Therefore, we evaluate

the soundness of our model by calculating the accu-

racy of team evaluations based on the model. The

accuracy of the team’s evaluation can be determined

by comparing it to actual game results.

Section 2 introduces conventional research appli-

cations of the graph theory to soccer and the conven-

tional method of centrality. In Section 3, we will ex-

plain a new centrality model that will improve the

conventional method of centrality. In Section 4, we

explain the details of the data and the experimental

method used to verify the usefulness of the new cen-

trality model. We also present the results of our exper-

iment in this section. Section 5 discusses the results.

616

Kanbata, M., Orihara, R., Sei, Y., Tahara, Y. and Ohsuga, A.

New Indicator for Centrality Measurements in Passing-network Analysis of Soccer.

DOI: 10.5220/0007377506160623

In Proceedings of the 11th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2019), pages 616-623

ISBN: 978-989-758-350-6

2 RELATED WORK

Analysis related to sports have been widely studied,

among them, the application of the graph theory to

passes in a soccer game has been studied. When con-

ducting this research, data called passing distribution,

which records the number of passes as well as the

passer and receiver of the pass in a game, are often

publicly available.

In soccer, a pass that moves the ball to the oppo-

nent’s goal is considered to be an important offensive

move. Therefore, the analysis to create a graph net-

work based on passing distribution has been done by

other researchers. (Goncalves et al., 2017) conducted

a tactical analysis that considers the player’s position-

ings and the passes between players using passing dis-

tribution. They evaluated the performance of teams

participating in international youth games

and ar-

gued that players and tactical features are different

when comparing the two. (Cotta et al., 2013) ana-

lyzed the pass tactics of the Spanish national soccer

team of the 2010 FIFA World Cup. They investigated

the games from the quarter-ﬁnals to the ﬁnal, taking

the transition of the number of passes and player’s po-

sitioning into account.

In addition, (Duch et al., 2010) measured central-

ity as a method to evaluate the players and the teams

in games by creating a graph network from passing

distribution. Furthermore, they modiﬁed between-

ness centrality used for the centrality measurement

and proposed ﬂow centrality. This is a centrality mea-

surement using data on only the sequences of passes

leading to a shot. They measured ﬂow centrality from

the passing-distribution data of EURO 2008. They

also evaluated each team by simply averaging play-

ers’ evaluations. From the results, they argued that

the team evaluation based on their method reﬂects the

game’s results better than one based on betweenness

centrality.

Indicators such as betweenness centrality and ﬂow

centrality are used for centrality measurements. Flow

centrality is a method that measures centrality using

data on only the sequences of passes leading to a shot

and highly evaluates athletes who participated in the

scoring opportunity. However, if it is a small num-

ber of sequences in some games, ﬂow centrality may

yields inaccurate measurements.

In this research, we propose a new model to alle-

viate this problem by improving ﬂow centrality. We

also evaluate teams based on the model to show the

soundness of the model. Let us emphasize that pass-

ing distribution only records the number of passes

between players and cannot reproduce sequences of

The teams have players under 15 or 17 years of age.

Figure 1: Graph Network Example.

passes with three or more players. However, realiz-

ing the tendency of the sequence from a large number

of passes is possible. Therefore, we will apply and

verify this data in our research.

2.1 Betweenness Centrality

Centrality is an index of the degree of inﬂuence that

any node gives to other nodes in a graph network

(Tsugawa and Ohsaki, 2014). This index makes it

possible to estimate an important individual in an or-

ganization. Because soccer places more emphasis on

organizational team play, applying the index would be

appropriate.

Creating a graph network using nodes, edges, and

edge weights is a feasible endeavor. Figure 1 shows

an actual graph network example. When it is applied

to a soccer match, the node is the player, the edge is

the direction of the pass, and the weight of the edge is

the number of passes.

When this concept is applied to the graph theory, it

is possible to measure the centrality of the player node

by calculating how much the player node involved in

the sequence of a speciﬁc pass from the graph was

created by passing distribution. In soccer, there is a

consensus that the more passes a player is involved

in, the more contribution to a match she/he makes.

We accept this idea and conduct our study based on

it.

Betweenness centrality is calculated by counting

the number of times a node appears in one or more

shortest sequences of passes between nodes. The

weights of the sequence are the minimum value of the

edge weights. In the example of Figure 1, the shortest

sequence from node 7 to node 1 is 7 → 3 → 2 → 1,

7 → 4 → 2 → 1, 7 → 5 → 2 → 1, and 7 → 6 → 2 → 1.

The edge weights of 7 → 3 → 2 → 1, 7 → 5 → 2 → 1,

and 7 → 6 → 2 → 1 are one. In addition, 7 → 4 →

2 → 1 has the edge weights of three, and the number

of passes of the shortest sequence is three. Therefore,

the number of passes in the shortest sequence from

New Indicator for Centrality Measurements in Passing-network Analysis of Soccer

617

Figure 2: Example of Betweenness Centrality’s Graph Net-

work.

node 7 to node 1 is six.

Let σ

be the number of passes in the shortest se-

quence from node s to node t. Let σ

(v) be the num-

ber of passes through node v in the shortest sequence

from node s to node t. Betweenness centrality repre-

senting the centrality of the nodes is deﬁned as fol-

lows.

betweenness(v) =

∑

s,t∈V

s̸=v̸=t

(v)

(1)

Where, V = {1,2, ..., 11 } denotes a set of player

nodes (Barthelemy, 2004). For example, the shortest

sequence from node s to node t in Figure 2 is s →

u → t, s → v → t, s → x → t. In this case, to calculate

(v)/σ

, the number of edge weights of s → u → t

is one, the number of edge weights of s → v → t is

three, and the number of edge weights of s → x → t is

one. Therefore, we can write equation (2) as follows:

(v)

= 3/5

betweenness(v) =

(v)

= 3/5 + 0 +0 + 0 + 0 = 3/5

(2)

2.2 Flow Centrality

Betweenness centrality is a method to measure cen-

trality from the shortest sequence between nodes, but

when you applied it to soccer, the scoring opportu-

nity to measure centrality between the player nodes is

unclear.

Here (Duch et al., 2010) improved betweenness

centrality. They only treat the process of the short-

est sequence leading to the shots. Speciﬁcally, they

newly create the shot node in the graph network and

divide it into a shot on goal and a shot wide, and the

direction of the player’s shot is the edge, and the num-

ber of shots is the weight of the edge (Duch et al.,

Figure 3: Changes in Graph Network from Betweenness

Centrality to Flow Centrality.

2010). Figure 3 exempliﬁes this. For this reason, they

evaluate the players by considering the opportunity to

score directly. They deﬁne ﬂow centrality as follows.

f low(v) =

∑

s∈V

t∈U

s̸=v

(v)

(3)

Where, U = {12, 13} is the set of shot nodes and

the shot wide node is 12 and the shot on goal node is

13. In Figure 3, to calculate f low(u), we can write

equation (4) as follows:

f low(u) =

s shot

(u)

s shot

t shot

(u)

t shot

v shot

(u)

v shot

x shot

(u)

x shot

= 1/5 + 0 + 0 + 0 = 1/5

(4)

(Duch et al., 2010), who believed that a player’s

degree of inﬂuence in the game will have the largest

evaluation in the sequence directly connected to the

goal, arguing that ﬂow centrality can evaluate players

more effectively than betweenness centrality.

3 PROPOSED METHOD

The shortest sequence in ﬂow centrality is determined

by ﬁxing node t as shot nodes. However, in a soccer

game, a team may adopt defensive tactics or maybe

unilaterally attacked during a game. As a result, the

number of shots would be relatively small in these

cases. In applying ﬂow centrality to such games, the

number of the shortest sequence to be calculated de-

creases, and the players involved in the score in the

measurement would be limited.

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

618

Figure 4: Flow Centrality Problem.

In the example of Figure 4, when ﬂow centrality is

applied, the shortest sequence is s → t → shot. In this

case, σ

s shot

(t)/σ

s shot

= 1 and σ

s shot

(u)/σ

s shot

(v)/σ

s shot

= σ

s shot

(x)/σ

s shot

= 0. Therefore,

the values of nodes u, v, and x, which are involved

in the opportunity of the score, are not evaluated. To

evaluate each player’s contribution to the score, we

need to consider not only node t but also nodes u, v,

and x.

Therefore, in this research, in order to consider

the players involved in the passes leading to the shot,

we increase the number of sequences to be calculated

by searching for a sequence whose sequence length is

l+1, where l is the length of the shortest sequence.

Let σ

′

be the number of the sequence with a

length is l+1 from node s to node t. Let σ

′

(v) be

the number of sequences through node v in the num-

ber of the sequence whose length is l+1 from node s

to node t. f low

is deﬁned as follows.

f low

(v) =

∑

s∈V

t∈U

s̸=v

′

(v)

′

(5)

Figure 5 shows an example of a graph network

created from sequences whose length is l+1.

From this graph network example, sequences with

a length of 1+1 include is s → u → t → shot, s →

v → t → shot, and s → x → t → shot. Therefore, it

is possible to evaluate nodes u, v and x. Furthermore,

we can write equations (6) and (7) as follows:

Figure 5: Flow+’s Graph Network Example.

′

s shot

(t)

′

s shot

= 1

′

s shot

(u)

′

s shot

′

s shot

(x)

′

s shot

= 1/5

′

s shot

(v)

′

s shot

(6)

f low

(t) =

s shot

(t)

s shot

u shot

(t)

u shot

v shot

(t)

v shot

x shot

(t)

x shot

= 1 + 1 + 1 + 1 = 4

(7)

4 EVALUATION EXPERIMENT

4.1 Data Sets

We conducted this experiment using the passing-

distribution data of UEFA EURO 2008, 2012, and

2016. Table 1 shows the details of the data.

“Matches” represents the number of matches in the

tournament, “Teams” the number of teams, and

“Evaluated Matches” the number of games excluding

draws. “Passes attempted” indicates the number of

passes in all games in the tournament, “Passes com-

pleted” represents the number of successful passes of

all games, and “Ave. Passes” indicates the average

number of successful passes. The passing-distribution

data were prepared as public data from FIFA and

UEFA and it can be obtained directly from the web-

sites (FIFA, 2018) (UEFA, 2018).

New Indicator for Centrality Measurements in Passing-network Analysis of Soccer

619

Table 1: Data Details.

Data Sets 2008 2012 2016

Teams 16 16 24

Matches 31 31 51

Evaluated Matches 26 24 42

Passes attempted 23756 27517 44490

Passes completed 17679 19893 37947

Ave. Passes 339 414 744

4.2 Experiment

We examined whether the proposed model is an ap-

propriate measure for the players and teams in the

match. P

betweenness

, P

f low

, and P

f low

are arrays of

betweenness, f low, and f low

of players in the ini-

tial lineup of team A, where the arrays are sorted in

descending order. We deﬁned the team’s evaluations

in matches in equations (8), (9), and (10). The players

are represented as n ∈ {1, 2, ...,11}.

betweenness

(n) =

∑

i=1

betweenness

(i)

betweenness

(1) > P

betweenness

(2) > ...) (8)

f low

(n) =

∑

i=1

f low

(i)

f low

(1) > P

f low

(2) > ...) (9)

f low

(n) =

∑

i=1

f low

(i)

f low

(1) > P

f low

(2) > ...) (10)

(Duch et al., 2010) used team evaluations as the

average of the top n players’ values from the equa-

tions (8), (9) and (10). They calculated this estimate

from the team evaluation and by using the bootstrap

method. As a result, they showed the usefulness of

this method by creating the sample distribution of an

estimate and performing a hypothesis test.

In this research, we set n = 3, 11 based on (Duch

et al., 2010) and veriﬁed whether the values of P

(n)

and P

(n) are directly linked to the outcome of a

match of A versus B. We calculate the accuracy of

our method as the ratio of agreement between the val-

ues of P

(n) and P

(n) and a match’s outcome agree

(Duch et al., 2010). We did not conduct the evaluation

in the case of a draw.

4.3 Result

The results of our experiments on the data from

EURO 2008, 2012, and 2016 will be described below.

Table 2: Accuracy of Match Results.

EURO 2008 betweenness f low f low

P(11) 0.35 0.42 0.46

P(3) 0.35 0.50 0.65

EURO 2012 betweenness f low f low

P(11) 0.42 0.46 0.46

P(3) 0.38 0.42 0.42

EURO 2016 betweenness f low f low

P(11) 0.59 0.35 0.49

P(3) 0.41 0.35 0.41

Table 2 shows the accuracy of betweenness central-

ity, ﬂow centrality, and ﬂow+ centrality as a ratio of

the level of agreement between the experiment’s ar-

ray values and match outcomes. Furthermore, Table

3, 4, and Table 5 depict P

f low

(3) and P

f low

(3) as well

as P

f low

(3) and P

f low

(3). The country’s name is

expressed in FIFA code, and when the match results

and values do not match, the values are followed by

the letter F (FIFA, 2010).

5 DISCUSSION

Table 2 shows that f low

of P shows higher values

than betweenness and f low. Because f low

calcu-

lates sequences whose length is l+1, more sequences

leading to the shot are calculated than the conven-

tional model. From this, the player involved in the

sequence that ended with a shot was evaluated. There-

fore, this is considered to be more accurate than the

conventional method. In this research, we veriﬁed

f low

’s soundness with sequences with a length of

l+1. From now on, it is necessary to estimate the ap-

propriate length of the sequence. It is also necessary

to verify a new index in which both the sequences

with a length of l+1 and the sequences with a length

of l are applied.

Furthermore, in EURO 2012 and EURO 2016, the

accuracy was low overall. The value of f low

in P

did not exceed f low in EURO 2012 and betweenness

in EURO 2016. From this result, in order to verify the

cause of mismatch of f low

against the outcomes,

we examined the distribution of the value of f low

As an example of a mismatch of P, Figure 6 shows

the distribution of f low

for the match between Italy

and the Republic of Ireland in EURO 2012. Further-

more, Figure 7 depicts the distribution of f low

for

the match between Croatia and Spain.

Figure 6 shows an example where Italy won but P

in f low

is higher for the Republic of Ireland than for

Italy. The variance of the value of f low

for Italy is

4.68, whereas the Republic of Ireland is 15.1, which

ICAART 2019 - 11th International Conference on Agents and Artiﬁcial Intelligence

620

Table 3: Team Evaluation of Each League Game in EURO

2016.

Gr.A P

f low

(3) P

f low

(3)

FRA-ROM 3.92-5.82,F 7.05-8.92,F

ALB-SUI 6.33-4.55,F 8.60-7.06,F

FRA-ALB 4.52-6.54,F 7.55-8.70,F

ROM-ALB 5.28-3.62,F 6.84-6.67,F

Gr.B P

f low

(3) P

f low

(3)

WAL-SVK 5.11-6.33,F 8.46-8.69,F

RUS-SVK 6.4-4.44 8.42-7.32,F

ENG-WAL 4.24-7,F 7.84-9.98,F

RUS-WAL 4.03-4.38 6.77-6.51,F

Gr.C P

f low

(3) P

f low

(3)

POL-NIR 4.24-4.33,F 7.53-4.91

GER-UKR 3.79-6.44,F 6.04-8.18,F

UKR-NIR 3.45-4.33 5.62-6.64

UKR-POL 3.33-4.72 6.18-7.60

NIR-GER 4.79-3.2,F 5.28-5.77

Gr.D P

f low

(3) P

f low

(3)

TUR-CRO 4.67-2.27,F 8.22-5.09,F

ESP-CZE 3.40-5.81,F 5.12-8.30,F

ESP-TUR 5-2.67 7.09-4.31

CZE-TUR 4.07-5.61 6.93-7.75

CRO-ESP 3.72-4.61,F 5.91-6.99,F

Gr.E P

f low

(3) P

f low

(3)

BEL-ITA 4.34-6 7.06-8.45

ITA-SWE 6.22-3 7.42-4.59

BEL-IRL 4.01-6.83 5.94-8.53

ITA-IRL 2.67-4.18 4.39-7.63

SWE-BEL 4.89-4.11,F 8.33-6.60,F

Gr.F P

f low

(3) P

f low

(3)

AUT-HUN 3.94-2.12 6.63-5.96

ISL-AUT 3.83-4.39,F 7.12-6.42

shows that there are variations in the value of f low

for the Republic of Ireland.

Although the relative evaluation of each player

does not change depending on f low and f low

, we

can observe a phenomenon in which particular play-

ers get extremely high evaluations with f low

, result-

ing in a high team evaluation.

In addition, the Croatia versus Spain game in Fig-

ure 7 shows that Spain won, but it is the mismatch of

f low

against the outcomes. In this game, the overall

evaluation of the Spanish team is low.

From this, we found that there are mainly two pat-

terns in the trend of f low

in cases of inconsistency.

From this point on, we analyzed these two patterns

further and examined the current problem and solu-

tion of f low

The reason why the value of a speciﬁc player, such

as Keith Andrews in the Republic of Ireland in Figure

6, becomes so large is that the number of shots in the

Table 4: Team Evaluation of Each League Game in EURO

2012.

Gr.A P

f low

(3) P

f low

(3)

RUS-CZE 3.75-2.72 7.33-5.41

GRE-CZE 5.42-5.33,F 7.07-8.49

CZE-POL 3.3-2.64 5.55-5.11

GRE-RUS 5.67-1.67 7.93-4.65

Gr.B P

f low

(3) P

f low

(3)

NED-DEN 2.33-1.71,F 5.49-3.32,F

GER-POR 5.29-4.36 7.59-6.22

DEN-POR 5.25-4.46,F 8.16-7.20,F

NED-GER 3.82-4.2 7.34-5.07,F

DEN-GER 4.22-4.10,F 6.71-5.81,F

POR-NED 3.60-2.88 6.97-5.67

Gr.C P

f low

(3) P

f low

(3)

IRE-CRO 4.34-3.84,F 5.49-3.32,F

ESP-IRE 3.11-6.43,F 5.12-6.43,F

CRO-ESP 6.33-4.52,F 8.76-6.51,F

ITA-IRE 2.54-6.17,F 5.94-8.36,F

Gr.D P

f low

(3) P

f low

(3)

UKR-SWE 3.56-4.21,F 6.32-6.87,F

UKR-FRA 5.12-3.67,F 7.99-6.52,F

SWE-ENG 3.34-2.75,F 6.96-5.54,F

ENG-UKR 4.12-2.86 7.12-6.27

SWE-FRA 4.44-2.44 7.12-6.1

Figure 6: Flow+ of Italy and the Republic of Ireland.

game is extremely small and the tactics used to spec-

ify a particular player as the destination of passes are

carried out. In the future, when the value of the index

of a speciﬁc player becomes large, it would be nec-

essary to discount it. Furthermore, Table 6 shows the

breakdown of the number of shots in a EURO 2008

New Indicator for Centrality Measurements in Passing-network Analysis of Soccer

621

Table 5: Team Evaluation of Each League Game in EURO

2008.

Gr.A P

f low

(3) P

f low

(3)

SUI-CZE 4.37-7.83 6.64-8.71

POR-TUR 4.66-3 7.32-4.55

CZE-POR 4.86-4.89 7.4-7.66

SUI-TUR 3.19-4.67 6.88-7.09

SUI-POR 5-5.31,F 8.17-7.94

TUR-CZE 3.9-4.05,F 6.6-6.49

Gr.B P

f low

(3) P

f low

(3)

AUT-CRO 4.83-5.12 7.23-7.7

GER-POL 5.67-3.87 8.86-6.98

CRO-GER 3.23-4.22,F 6.38-6.87,F

POL-CRO 4.07-3.51,F 6.45-6.87

AUT-GER 4.72-2.71,F 6.92-5.24,F

Gr.C P

f low

(3) P

f low

(3)

NED-ITA 3.27-3.33,F 6.04-5.19

NED-FRA 4.58-3.26 7.24-6.24

NED-ROU 3.68-5,F 6.69-7.18,F

FRA-ITA 5.33-3.57,F 8.33-5.69,F

Gr.D P

f low

(3) P

f low

(3)

ESP-RUS 3.57-4.61,F 6.79-7.13,F

GRE-SWE 3.72-3.12,F 6.07-6.25

SWE-ESP 4.33-4.5 5.91-7.26

GRE-RUS 4.97-4.43,F 6.89-7.53

GRE-ESP 4.58-4.73 6.26-8.29

RUS-SWE 2.86-3.2,F 5.95-6.34,F

Figure 7: Flow+ of Croatia and Spain.

Table 6: Breakdown of the Number of Shots in a EURO

2008 Game.

Winning Team Losing Team

Total Attempts 337 287

On-Target 175 121

Off-Target 162 166

Ave. On-Target 6.73 4.64

Table 7: Breakdown of the Number of Shots in a EURO

2012 Game.

Winning Team Losing Team

Total Attempts 347 292

On-Target 195 135

Off-Target 152 157

Ave. On-Target 8.13 5.63

game, and Table 7 shows the breakdown of the num-

ber of shots in a EURO 2012 game. The number of

shots is “Total Attempts”, the number of shots on goal

is “On-Target”, the number of the shots wide is “Off-

Target”, and the number of the average shots on goal

is “Ave. On-Target”.

From Table 6 and Table 7, it can be presumed that

the winning teams are superior in the number of shots,

the number of shots on goal, and the number of the

average shots on goal. From this assumption, it can

be inferred that the number of shots is larger in the

winning team than the defeated team, and the shot ac-

curacy tends to be higher in the winning teams. In

this research, we calculated f low

by treating team

A and team B equally. In the future, changing the

edge weight by shots out of the frame or shots in the

frame is a necessary step.

Furthermore, we analyzed the cause of the ten-

dency of values like Spain in Figure 7. f low

an index that quantiﬁes how much is involved in the

sequence of passes that leads to a scoring opportu-

nity. Figure 8 shows a scatterplot of the number of

shots and the number of passes of the winning team

of EURO 2012, and Figure 9 shows a scatterplot of

the defeated team’s number of shots and number of

passes in EURO 2012.

In Figure 8, Spain’s the number of successful

passes is over four hundred and the number of shots

increases accordingly. In fact, the correlation coefﬁ-

cient between the number of shots and the number of

passes of the winning team is 0.45, whereas it is 0.63

for the defeated team. From this, we can say there

is a correlation. Based on this, it is thought that the

more the number of successful passes increases, the

more the number of scoring opportunities increases

as well. In Spain, the number of successful passes

is much larger than that of other teams, therefore the

value of f low

will most likely not become too large

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622

Figure 8: Scatterplot of the Number of Shots and the Num-

ber of Passes of EURO 2012’s Winning Teams.

Figure 9: Scatterplot of the Number of Shots and the Num-

ber of Passes of EURO 2012’s Defeated Teams.

even if the number of shots increases. This can be im-

proved by limiting the number of successful passes to

a certain range.

6 CONCLUSIONS

In this research, we analyzed the contribution of a soc-

cer player in the matches by utilizing graph theory’s

centrality concept from passing-distribution data. As

a result, we found that the proposed model sometimes

has better accuracy than the conventional model. We

were able to verify the soundness of the proposed

model. By improving f low

, new indicators may

emerge in future studies. In addition, an objective

method is necessary to show whether the individual

player’s evaluation is sound.

ACKNOWLEDGMENTS

This work was supported by JSPS KAKENHI Grant

Numbers JP16K00419, JP16K12411, JP17H04705,

JP18H03229, JP18H03340, JP18K19835. We ac-

knowledge the professor Shinichi Honiden of the

Waseda University for providing opportunities to

study and the members of our laboratory.

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