Graph Theory: Opposing Political Parties in 1 Household
Within American politics there is constant turmoil between the Democratic Party and the Republican Party. The two political parties have very different ideals and ways of approaching crucial issues within the US. This causes tension and constant disagreement between the two. With such hostility between the parties in government, it is inferred that the followers of each party do not associate or agree, let alone like each other, as a result. This can be very difficult when looking at a family that does not have a homogenized political view.
We can look at how this case relates to class and Graph theory. More specifically Structural Balance theory / The Balance Theorem and how this case satisfies and/or violates them.
Looking at one family, say that the mother (node M) and son (node S) are Republicans and the father (node F) is a Democrat. Let’s also add 2 more nodes, node C, that is a democratic friend of the father, and node R, that is a Republican friend of the son. Nodes M,R, and S have positive tie between them and the and nodes F and C also have a positive tie between them. But, there are positive ties between nodes F and C, and nodes F and M even thought they are in different parties because they are family.
The Structural Balance Property (SBP) states that a triangle is balanced when either all 3 edges are positive or there is exactly one positive edge; never two.
When looking at just the family. It is a balanced triangle because all the ties are positive (M-S, S-F, M-F). So in that case the SBP is satisfied. On the other case, when looking at a different triangle, the triangle between nodes F, C and S for example. Assuming that everyone of a different party has negative ties except the family. Nodes F and C and nodes F and S would have positive ties but nodes C and S would have a negative tie, meaning that the triangle would have 1 negative and 2 positive ties. SBP states that because of that, the triangle is not balanced.
For a larger graph to balanced, all of its triangles should be balanced. Furthermore, the full graph is not balanced because not all of the triangles are balanced as seen above.
The Balance Theorem: If a labeled complete graph is balanced, then either all pairs of nodes are friends, or else the nodes can be divided into two groups, X and Y, such that each pair of people in X likes each other, each pair of people in Y likes each other, and everyone in X is the enemy of everyone in Y.
Turning back to the example. Let’s look at the 2 parties as 2 separate nodes of mutual friend (similar to the example in the textbook). This family case violates the Balance Theorem. When looking at the democrats and the republicans as a whole node with the followers as smaller nodes within, if we said earlier that the 2 groups do not like each other, meaning that there is a negative tie between the 2, this should translate to the followers. But in the case of the family the mother and the son would have a positive tie because of their similar political party, but the tie between the son and the father, and the mother and the father would be negative because they separate parties. This would be the case that satisfies the Balance theorem, but it would cause a very hostile family life and marriage in general.
This reasoning could be used to defend and further enhance the understanding of the ideas seen in the article Marriage Between Democrats and Republicans are Extremely Rare by Wendy Wang (https://ifstudies.org/blog/marriages-between-democrats-and-republicans-are-extremely-rare). The article talks about the decline in of politically-mixed marriages and how most of them end abruptly or are very unhealthy for a long time. There should be that negative tie between people of opposite political parties, but since it is a relationship, there is not. But there is also that aspect of just looking at the family and seeing that the family triangle itself is balanced. The article also touches on some “loop wholes”. It also brings in the independent party and gives statistics of the relationship dynamic between all parties.