GRAPH THEORY IN FICTIONAL UNIVERSES (FROM MARVEL TO CHARMED)
Marvel is starting to use graph theory, networks specifically, to organize all the characters from its countless comics into one big network for the delight of fans. The network is meant to showcase the interactions between all Marvel characters across their many comics, in their many disguises. The utilization of graph theory in the Marvel universe has got Peter Olson, the Marvel AP of Web and Application Development and the person responsible for the new network, thinking about all other people who could be organized in just such a way, including historical figures and other tv and movie characters. Being as how I am a Charmed tv series fan, I decided to look at the networks formed by the Halliwell sisters and their many friends, acquaintances, and enemies.
At the beginning of the first episode we already see a structurally imbalanced network with two sisters being connected by a positive tie with the third sister but by a negative tie with each other. Because the network is structurally imbalanced and it is in Piper’s (the third sister) best interest that her sisters be connected by a positive tie, we soon see a change into a structurally balanced triangle with all three sisters being connected by positive ties.
In fact, it is not only positive but also very strong (statistically made so by the amount of time they spent together and their familial relationship). The three sisters are greatly embedded in each other’s networks and have high neighborhood overlap because of their common “neighbors”. What makes their relationship even stronger is the presence of a common enemy, or enemies, – the bad wizards and witches and demons. These common enemies, who refuse to follow a balanced relationship structure and remain generally enemies among themselves, also help the three Halliwell sisters form a positive connection to the elders and whitelighters. Charmed also has the epitome of a structurally unbalanced relationship, which showcases the enormous likelihood that these relationships have of going wrong. It is the romantic relationship between Phoebe (the youngest sister) and Cole Turner (the demon Balthazar). Despite Phoebe having positive ties to both of her sisters and Cole, her sisters are connected to Cole with negative ties, forming a structurally imbalanced network.
In fact, Piper and Prue’s relationship with Cole is so scant that the ties between them and him might as well not exist, which would make the edge between Phoebe and Cole a bridge (since her close-knit network consists of good witches and wizards while his consists of demons). It is true that for this to occur the Strong Triadic Closure property would have to be violated as Phoebe is connected to Piper, Prue, and Cole with strong ties but Piper and Prue would not be connected by any ties to Cole.
However, even in this scenario the edge does not last long as a bridge because the Strong Triadic Closure property proves too strong. About halfway through the show Cole joins the good side for Phoebe and forms strong, positive ties with Phoebe’s sisters and all her other allies. His strong, positive ties to the demons and other creatures of the bad side then become weak, negative ties.
The networks I have shown here are only a few of those that can be seen on Charmed. Throughout the show there are many examples of networks and their versatility, that is how quickly they change over time and the number of snapshots in time that would be needed to show their development is endless. It is, after all, an imitation of real life.