Social Networks in Legally Blonde
The film “Legally Blonde” celebrated its 20-year anniversary over the summer. Multiple articles covered highlights of the movie, things we may not have known, and some dove deep into the characters of the film. In the article attached, a lot of interesting conversation is had about the film, but one point it brings up, in particular, is the dynamic between characters in a love triangle that unfolds as a part of the plot. Upon thinking about this more deeply, I was reminded of our discussion in class about graph theory, and specifically about balanced networks.
For those who may be unfamiliar with the movie, the protagonist Elle Woods goes to Harvard Law School to try to win back her ex, Warner. When she arrives, it is revealed that he’s engaged to another girl, Vivian. However, at the end of the movie, Vivian actually sides with Elle and breaks up with Warner. The girls are now friends and both dislike Warner- a classic “the enemy of my enemy is my friend” situation. While this love triangle may appear trivial on a surface level, I thought back to what we learned in class about balanced networks. A balanced network is either one in which everyone is friends with each other (+, +, +) or exactly two groups dislike each other (+, -, -). In the scenario between Elle, Vivian, and Warner, there is a balanced graph in the beginning, with Warner and Vivian being friends (+) since they are dating, Vivian and Elle being enemies (-), as Vivian bullies Elle in class, and Elle and Warner being enemies (-), since Warner dumps Elle. However, this turns into a balanced graph with different relationships once Elle and Vivian join forces and both mutually dislike Warner. It ends up with Warner and Vivian being enemies (-), since Vivian breaks up with Warner, Vivian and Elle being friends (+), since the two girls are now on the same side, and Elle and Warner still being enemies (-) – they never get back together and mutually dislike each other. These graphs follow the Structural Balance Property (SBP) that we learned in class, which states that a complete graph has the SBP if all of its triangles are balanced.
We can examine the characters’ relationships even further, by applying the Balance Theorem to the situation. Similar to examples we did in class and on homeworks, we can add in another node to these graphs, such as the character Emmett, and the network will still be balanced. The Balance Theorem states that if a labeled complete graph has the structural balance property, then either all pairs of nodes are friends or the nodes can be split into two groups, X and Y, such that every pair of nodes in X likes each other, every pair of nodes in Y likes each other, and everyone in X is the enemy of everyone in Y. Emmett is Elle’s new love interest, who supports her and would certainly be in her group. Let group X be Elle’s group with her, Vivian, her friend Paulette, her parents, etc. Emmett would join group X, and everyone in group X would like each other. Let group Y be Warner’s group with him, his brother, his friends, etc. Everyone in group Y would like each other, and the two groups, X and Y, would mutually dislike each other. This would be a balanced network.
Article: https://www.npr.org/transcripts/987337889