by Ben Blatt and Arjun Modi
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Treating two actors that were in the same film as ‘connected’, the game Six Degrees of Kevin Bacon seeks to connect actors to Kevin Bacon in six links or less. The number of people needed to connect to mathematician Paul Erdős, when co-authors are viewed as ‘connected’, is a mathematician’s Erdős number. Applying this same type of thinking into the world of sports, I sought to find shortest link, using teammates as connections, between every player in the history of the NBA.
In the interest of full disclosure, I am not the first one to apply this to sports or even basketball. However, the only NBA version I could find includes players from only 1996-2001. For my analysis, I decided to include all players from the entire 62 year of the NBA as well as any player that played on a BAA or ABA team as long as they were on a franchise that was eventually incorporated into the NBA.
In order to find the shortest link between two players I used a shortest-path algorithm. It’s the same sort of algorithm that Google Maps would use to determine the shortest route. In order to use the algorithm, I first had to create an adjacency matrix. An adjacency matrix is just a matrix that represents if two vertices of a graph, or in this case two players in the NBA, are connected. For instance, the entry in the matrix representing a connection between Paul Pierce and Kevin Garnett would be a ‘1’ since they have played on the same team while the entry in the matrix representing a connection between Paul Pierce and Kobe Bryant would be a ‘0’ since they have never played on the same team.
The specific algorithm I used, Dijkstra’s algorithm, works by finding the distances between a starting node (NBA player) and all other nodes (NBA players) that can be connected with one edge (NBA team). The algorithm then finds the distances between any node found to be one edge away from the starting node to any node that is one additional edge away. This is then updated in the matrix so that it has distance lengths for any node within two edges of the original node. This continues until all the distances are found. There are many publicly available codes for this algorithm to choose from and I ended up choosing one for MATLAB which I only had to modify slightly to complete this project.
The number six often turns out to be the maximum number of connections to link all entities, but is this accurate for the NBA? Not if you want to include the entire history of the NBA. It turns out that the maximum number of degrees needed to connect any two players who have ever played in the NBA is eight. For example, to link current Los Angeles Clipper Blake Griffin to Charlie Parsley of the 1949-1950 Philadelphia Warriors takes eight connections. The reason some connections are long is simply the number of players needed to cover the years separating current NBA players to players that played back in the 1950s or 1960s. If only current NBA players are considered, the maximum number to link all players is four with Byron Mullens to Xavier Henry being an example. While four and eight are the maximum, if any two players were chosen at random the expected number of degrees separating them is only 3.37. Below is a chart that shows the distribution of number of degrees separating any two players.
We can also look at individual players to determine how close they are on average to all other players. The player with the shortest average separation between him and all other players, ‘The Center of the NBA Universe’, was James “Buddha” Edwards. By playing nineteen seasons for eight different teams in the middle of the NBA’s history, Edwards was able to build enough connections that he is now on average just 2.46 degrees from any other player. If only the current NBA is considered, ‘The Center of the Universe’ is Eddie House who can be connected with just 1.83 degrees on average to other current NBA players. There is also hope for Shaq, who is the current player with the shortest average link to the historical NBA. One day, he will most likely claim the Center of the NBA Universe title, in mathematical – if not gravitational – terms.
We can also determine the player with the largest average separation between him and his colleagues. Let’s call him ‘The Outcast of the NBA Universe’. For the entire history of the NBA, Bill Roberts of the 1949-1950 St.Louis Bombers takes top honors with 5.63 degrees of separation on average between him and all other players. Logically many rookies are at the top of the list for ‘The Outcast of the Current NBA Universe’, with Memphis Grizzlies’ Greivis Vasquez and Xavier Henry tied for first with 2.72 degrees on average.
Below is a chart showing the top and bottom five players in terms of average links for historical NBA and the current NBA. For the current NBA, any player that is retired was excluded from being a link.
If you would like to play around with some of these connections yourself, you can check out the ‘6 Degrees of Separation’ website developed from the results of this project. I would like to give a huge thanks to HSAC member Arjun Modi, who developed this very fun website for everyone to enjoy.
Ben Blatt can be contacted at firstname.lastname@example.org.