Six Degrees of Zlatan Ibrahimovic

Zlatan Ibrahimovic, Ibracadabra, the self proclaimed god. Or simply Zlatan.  An incredible football player, well known for his shenanigans outside of the football ground. The Internet has created quite a cult around Zlatan and I thought to contribute by conducting the “Six degrees of Zlatan” experiment.

The study is based on the Six Degrees of Kevin Bacon which itself is kind of an adoption of the Erdős number in math. Readers familiar with the concepts can skip the following paragraph and go directly to the calculation of the Zlatan number.

What is the Erdős or Bacon number?

Paul Erdős wrote an incredible amount of scientific papers (around 1500) with around 500 collaborators. Due to his prolific output, the Erdős Number was created. An Erdős number describes a person’s degree of separation from Erdős himself, based on their collaboration with him, or with collaborators of Erdős. Erdős is the only one with an Erdős number of 0 (for being himself), while his immediate collaborators have an Erdős number of 1, their collaborators have Erdős number at most 2.

The concept of the Bacon number is basically the same. Similar to Erdős, Kevin Bacon has quite a prolific screen career such that many actors can be linked by co-occurrences to Kevin Bacon.
Kevin Bacon is the only one with a Bacon number of 0. Actors occurring in a movie with Bacon have a Bacon number of 1 and so on.

Both the distribution of Erdős and Bacon numbers are examples of the small-world phenomenon:
Everyone is so widely connected in modern times, such that we are connected to every individual in the world by no more than 6 links of mutual friends. Apparently, the same rules apply to the mathematicians and actor collaboration networks. Most actors are at most at distance 5 from Kevin Bacon and the median of Erdős numbers among mathematicians is around 5.

The numbers can also be aggregated to the Erdős-Bacon number.  To obtain an Erdős-Bacon number, you either have to be an acting mathematician or an actor who wrote a scientific paper.
According to Wikipedia the smallest existing sum of both numbers is 4.

The Erdős-Bacon-Sabbath number seems to be the craziest extension. Natalie Portman, for example, has an Erdős-Bacon-Sabbath number of 10.

The Zlatan Number

Zlatan doesn’t do auditions science. But I do. So I put together a dataset of football players and the clubs they played for dating back to the beginning of 2000 and for some leagues to the mid 90s. The data contains a great variety of countries and were all scraped from here[1]. I chose to only look at clubs and neglected national teams.

The dataset contains 101581 unique soccer players with almost 5 million links between them. A link between players indicates, that they were playing at the same time at the same club. This induced graph is used to determine the distances from each player to Zlatan via shortest paths of co-occurrences in clubs.

Zlatan, of course, is the only one with a Zlatan number of 0. No one is like Zlatan! A total of 337 players had the privilege of being in a team together with Zlatan and thus have a Zlatan number of 1. An example for a Zlatan number of 2 is shown below.

The maximal Zlatan number achieved in the dataset is 5, obtained by 14 players of PFC Chernomorets Burgas Sofia (Season 2006/2007) . The mean Zlatan number is  2.85 and the median is 3, such that we should actually rename the study to “Three degrees of Zlatan Ibrahimovic”.

If you are interested in specific Zlatan numbers, you can check out all numbers of players here.

 

Most central football player

Although we all know that Zlatan is the most awesome player there is, I checked the data for the most central player. That is, the player with the lowest aggregated distance to all others. The five most central players in this sense are:

  1. Robbie Keane
  2. Mido
  3. Duško Tošić
  4. Florent Sinama Pongolle
  5. Roy Caroll

So Robbie Keane is the closest to have played with any player in the dataset. His mean distance to all players is 2.68.

By the way, Zlatan is only the 1028th most central player!

To wrap up, we see that the football “collaboration” network seems to be a further example for the pervasive small-world phenomenon. No player is more than 5 steps away from Zlatan.

The Bacon-Zlatan Number

As for the combined Erdős-Bacon number we can of course also check for football players who appeared in movies to create a combined Bacon-Zlatan number. Surprisingly many football players played in movies and with the two documentaries about the German national team, we have a quite large number of players with a combined Bacon-Zlatan number. The lowest Bacon-Zlatan numbers (Shown as Bacon Number+Zlatan number) are
  1. David Beckham 2+1=3
  2. Eric Cantona 2+2=4
  3. Zinedine Zidane 2+2=4
  4. Lionel Messi 3+1=4
  5. David Luiz 3+1=4
David Beckham is the only player with a combined number of three and it is highly unlikely to find someone with that number again. Until of course there is a documentary about the life of Zlatan which is inevitable.The highest I could find is a combined number of 6 for some former German players.

  1. Jürgen Klinsmann 3+3=6
  2. Andreas Köpke 3+3=6
  3. Matthias Sammer 3+3=6
  4. Michael Ballack 3+3=6
  5. Oliver Neuville 3+3=6
Note that it might well be that I am missing some non-German players that appeared in a movie. A notable player missing is Vinnie Jones with a Bacon Number of 2. He unfortunately ended his football career a bit to early to end up in the dataset.

The Erdős-Zlatan Number

Could there be an Erdős-Zlatan number? I somehow doubt that there is a professional football player out there who ever wrote a scientific math paper. I, for one was a passionate Sunday league player once and I happen to have an Erdős number of 4. A few years back, former professional player Sean Dundee started playing in exactly that Sunday league, so I went through his team and found someone I played with. Sean Dundee has a Zlatan number of 3 which gives me a Zlatan number of 5. My Erdős-Zlatan number is thus 4+5=9. Until proven otherwise, I have the lowest combined Erdős-Zlatan number!

 

You can check out Bacon numbers at the Oracle of Bacon and Erdős numbers can be calculated here.

All Zlatan numbers can be found here.[1] “Source of Material is http://www.footballsquads.com. Material: © FootballSquads.com, 1999 – 2016, All Rights Reserved”

Posted in: Network Analysis, Soccer Analytics

Written by Dmathlete

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