Structural Balance Theorem and Rival Schools
We can observe two rival schools, Cornell and Harvard. We can also easily observe that students at Cornell are all on good terms with each other, and likewise for Harvard students. Thus, we can group students in Cornell into one group, and students in Harvard into another group. As these schools are rivals, we can observe the relationships between Cornell students and Harvard students as enemies.
At a low level, we can observe the Structural Balance Property, which states: for every set of three nodes, if we consider the three edges connecting them, either all three of these edges are labeled +, or exactly one is labeled +, otherwise is imbalanced. We can observe four possible set of three nodes: (Harvard, Harvard, Harvard), (Cornell, Cornell, Cornell), (Harvard, Harvard, Cornell), and (Cornell, Cornell, Harvard). The set of (Harvard, Harvard, Harvard) nodes are all connected by positive edges. The set of (Cornell, Cornell, Cornell) nodes are all connected by positive edges. The set of (Harvard, Harvard, Cornell) is connected with two negative edges from Cornell to each Harvard node, and a positive edge between the Harvard nodes. The set of (Cornell, Cornell, Harvard) is connected with two negative edges from Harvard to each Cornell node, and a positive edge between the Cornell nodes. Thus, we see that these sets all satisfy the Structural Balance Theorem.
Furthermore, we can see that this property implies the global property of the Balance Theorem.
We can apply the Balance Theorem to this situation. We know the Balance Theorem to be: 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 every pair of nodes in X like each other, every pair of nodes in Y like each other and everyone in X is the enemy of everyone in Y. In our scenario, we see the complete graph as all students in Cornell and Harvard connected by edges. Nodes can be connected into pairs of (Cornell, Harvard), (Harvard, Harvard), or (Cornell, Cornell). We can see a (Cornell, Harvard) pair is connected by a negative edge. We can see (Harvard, Harvard) and (Cornell, Cornell) pairs is connected by a positive edge. We can easily observe all students from Cornell represented by nodes and connected by positive edges, and all students from Harvard represented by node and connected by positive edges. Additionally, we observe that the nodes of students from Harvard are all connected to the nodes of students from Cornell by negative edges. Thus, we can observe the Balance Theorem.