Positive/Negative Relationships, Strong/Weak Ties, and Rivalry in the English Premier League
Before we begin our discussion of sports rivalries and graph theory, we should first review two important concepts within graph theory: the strength of ties and positive and negative relationships. First, let’s define and review the strength of ties. Within a certain person’s network, there can be people they are close with (strong tie) and people that they are merely acquaintances with (weak tie). Although there is no one size fits all classification for strong and weak ties, we will use a number range for our purposes. For ease of understanding, let’s classify the strongest tie as being a 10 and the weakest tie as being a 1. The numbers in between represent varying strengths of the tie.
Next, let’s discuss positive and negative relationships. Every connection between two nodes in a graph can be annotated with a positive or negative sign. A positive sign indicates a friendly relationship whereas a negative sign indicates a more hostile relationship. One concept involving positive and negative relationships is the Structural Balance Property. This property suggests that there are only two types of balanced triangles (three nodes from a graph): triangles with three positive edges or triangles with only 1 positive edge. Configurations with 3 negative edges or 2 positive and 1 negative edges are considered unbalanced.
Let us expand both of these ideas by combining the two into one. If we number each positive and negative edge in a graph, we can obtain a more detailed and informative graph. Here is an example:
Note that we have essentially extended our range to -10 to 10. A positive number suggests that there is a friendly relationship between the two nodes whereas a negative number suggests a hostile relationship. Now, how does this concept apply to sports rivalries?
In order to explain this topic, let’s consider the English Premier League, the top division of football (or soccer) in England. There are 20 teams in the EPL, consisting of teams from all around England. Each team has its own unique fan base, stadium, club structure, and club beliefs. All 20 teams play 38 games each every season in hopes of finishing first and winning the Premier League. Within this twenty team structure, there are also several intense rivalries. Aside from the general competition for the trophy, what causes these rivalries? Well, many sports rivalries are based on location, past competition between the two teams, and previous success (Kilduff et al., 2010). The Premier League provides several fitting examples of these rivalries. One example is the North London Derby (NLD) which is a rivalry between Arsenal and Tottenham Hotspur, two teams that are located in North London. Their proximity has caused the NLD to be one of the fiercest rivalries in the Premier League. Let’s try to put this rivalry into a simple graph by adding a third team, Chelsea. Chelsea is also a team from London, but they are not as close in proximity to Arsenal and Tottenham. They are also rivals, but the rivalry is not as intense. Below is a graph representation of this scenario:
This graphical representation shows the varied rivalries between the three teams. Although they are all rivals, Tottenham and Arsenal have the more intense rivalry. According to our prior discussion of the Structural Balance Property, this graph is unbalanced. The three negative edges suggest the possibility that one of the teams will eventually ally themselves with each other against the third team. However, this is not the reality. The three teams are still embroiled in a serious rivalry against each other despite having played against each other for decades. Why is this the case? Why does the graph not revert back to one of the “balanced” configurations? Well, the rivalry itself appears to make a big difference. Rivalry can actually benefit football clubs tremendously, so much so that their antagonistic state becomes natural (or even balanced). There are several reasons behind this assertion. First, rivalry has been shown to improve team performance and motivation both in the short term (in a head to head match-up ) and in the long term (in the context of a tournament or league) (Pike, et al., 2010). This boost in performance is already enough for many football clubs and players to want to maintain their rivalries. It becomes a way to increase interest and ticket sales as well as a way to motivate players. Pike, et al. (2010) also suggests that rivalries can motivate players and teams for up to a year after their most recent head to head game. This boosts performance even against nonrival opposition. Teams want to match or best the successes of their rivals, so they have more motivation to win as many games as possible to keep ahead or level with them. In short, the graph does not revert to the “balanced” configuration because it benefits the teams involved to remain in their current state.
In our case, the three clubs all benefit from their rivalry to different extents. For example, the Arsenal vs Tottenham rivalry might motivate Arsenal and Tottenham players more than their rivalries with Chelsea. This suggests both the sign and the strength of the tie matters when determining the effect of rivalry on our graph structure.
To conclude, we have examined the combination between two types of graph labelling, positive/negative and weak/strong ties. We have also explored the example of English Premier League rivalries using this type of graph labelling and determined why teams within rivalries remain in a traditionally unbalanced configuration. Further investigation into rivalries could expand into game theory or more global examples (i.e. rivalries between nations), but these topics are not within the scope of this blog post.
References:
KILDUFF, G., ELFENBEIN, H., & STAW, B. (2010). THE PSYCHOLOGY OF RIVALRY: A RELATIONALLY DEPENDENT ANALYSIS OF COMPETITION. The Academy of Management Journal, 53(5), 943-969. Retrieved September 12, 2021, from http://www.jstor.org/stable/20788803
Pike, Brian & Kilduff, Gavin & Galinsky, Adam. (2018). The Long Shadow of Rivalry: Rivalry Motivates Performance Today and Tomorrow. Psychological Science. 29. 095679761774479. 10.1177/0956797617744796.