Problems always arise when people live in close proximity to one another. Even the closest of friends often find fault with each other’s habits and routines if they’re around one another enough. In my own apartment, this is certainly the case, though the number of conflicts has increased substantially this semester. I attribute this conflict explosion to the one of our roommates, Raymond, going abroad to Australia. There are five of us that currently reside in Apartment 319: Casey, Alex, Ryan, Brian, and Dave (some names were changed for identity preservation). Dave moved in this semester to fill Raymond’s spot while he is abroad. In speaking to Ray on the phone last week, I blamed him for the increased level of conflict in our apartment. He responded by saying “It’s impossible. Dave gets along with more people in the apartment than I do, things have to be easier than before.” While it may be true that Dave meshes better with more roommates, I believe that his presence actually causes more conflict.

In delving further into the logic behind structural balance, I came across Frank Harary’s 1953 paper “Of the Notion of Balance of a Signed Graph.” It is from this paper that the idea of structural balance originally arose. Harary claims that everyone in a social network either must be friends, or the network is split up into two factions. I believe that our apartment satisfied the latter of these two conditions with Raymond present, and that the balance has been disrupted by the roommate switch. Perhaps graph theory can shed some light on why this is the case, and prove why my insight is correct.

The graph to the right is a depiction of our apartment at the present. An edge with a (+) means that two roommates have little or no conflict. Conversely, a (-) indicates that two roommates argue frequently about apartment-related things. Given Harary’s rules, it is clear to see that the (+) edge between Ryan and Dave causes a structural imbalance, and is perhaps the root of the increased conflict in Apartment 319. According to the rules set forth by Frank Harary in 1953, this edge causes two (+) (+) (-) triangles, and hence puts latent stress on Dave and the network as a whole. It’s seems strange that a good relationship between two roommates causes more conflict than a bad one, but perhaps an example of a recent conflict can elucidate why this is the case.

Casey and Alex prefer the apartment temperature to be 72 degrees, and turn the thermostat up to 72 whenever Brian and Ryan aren’t around. Brian and Ryan, however, change it back to 68 whenever they get the chance. Neither group addresses the problem with the other, and the game continues indefinitely without conflict. However, Dave sees that two sets of his friends are not getting along, and takes in upon himself to correct the problem. His addressing the problem causes the two groups to argue, and perpetual conflict ensues. If Dave simply chose one group (Alex and Casey, or Ryan and Brian), then the two groups could get along by rationalizing that the members of the other group are just jerks.

When Dave graduates, and Raymond (node Y) returns in the fall, I would expect the level of conflict to diminish. Raymond doesn’t get along with Ryan in the way that Dave does. The graph below illustrates this new scenario. Here, the graph can clearly be divided in two groups, with Alex/Casey/Raymond in one group and Brian/Ryan in the other. Since there are two distinct groups, and each group gets along with everyone within it and argues with everyone in the other group, the graph achieves structural balance, just as Harary would predict. It is worth noting though, that it is still early in the semester, and relationships may change. There is pressure on Dave to choose one group or the other, and so in time, he may sever the positive relationship with Ryan, since he does not argue with either of the members of the other group. However, this hasn’t happened yet. The rules for structural balance seem to make logical sense, but it is interesting to see that they hold true under empirical observation.