Overview of Graph-Theoretic Formalization of Balance Theory

This document summarizes some key ideas in balance theory and details how those ideas are represented within the Cartwright-Harary graph-theoretic formalization of balance theory. At the end of the document are some "problems" that deal with complex structures. We may work through some of these problems in class, giving concrete examples of situations that might manifest the kinds of relations and dynamics that the theory addresses.

Signed Line: Connection between two nodes with a valence, such that a positive valence indicates positive sentiment between the two points, and a negative valence indicates negative setiment between the two points:

	OR
(POSITIVE SENTIMENT)		(NEGATIVE SENTIMENT)

Cycle: A non-intersecting path that begins and ends at the same point:

(Figure 1)

Cycles in Figure 1: (prqp), (qrpq), (rpqr)

Sign of a Cycle/Path: Product of the valences of each line in the cycle/path.

Sign of Cycles in Figure 1:

• (prqp) = (pr) x (rq) x (qp) = (-) x (-) x (+) = (+)
• (qrpq) = (qr) x (rp) x (pq) = (-) x (-) x (+) = (+)
• (rpqr) = (rp) x (pq) x (qr) = (-) x (+) x (-) = (+)

Imbalance: When one or more cycles in a structure are negative.

Thus, Figure 1 is balanced, since all three cycles are positive. Note, however, that balance and a preponderance of positive sentiment are not the same. As this figure demonstrates, balance can be achieved in a structure characterized by a preponderance of negative sentiment. This figure is captured by the assertion, "My enemy is my friend's enemy."

II. Some Balance Theory Assumptions

Assumption 1: If a structure is balanced, then the structure is unlikely to change.

Assumption 2: If a structure is unbalanced, then the structure is likely to change.

[How does change occur?]

Assumption 3: Incomplete structures (i.e., ones in which one or more nodes are not connected) tend toward completion so as to achieve balance.

III. Derivations of Graph-Theoretic Formalization of Balance Theory (Based on Logic of Graph Theory)

• Degree of Balance: Proportion of positive cycles to negative cycles.
• Structure Theorem: Every balanced stucture can be broken down into two mutually exculsive subsets (one of which may be empty, in which case the theorem is trivially true), such that each positive tie joins two points of the same subset, and each negative tie joins two points from different subsets.

In Figure 1 (which is balanced), the subsets are: {PQ} and {R}

In complex structures, the Structure Theorem can identify the formation of cliques.

• Fixed-Point Theorem: If there is a point , p, in a digraph, which is connected to all other points, and if all cycles are positive with respect to that point, then the structure is balanced.

In Figure 1, the Fixed-Point Theorem tells us that we need only look at the cycles through one of the points in order to ascertain if the strucure is balanced. This is because, in Figure 1, all of the points are connected to all other points. For instance, according to the Fixed-Point Theorem, we could use all cycles through p, and if all of those cycles are positive (which they are), then we can be confident that the structure is balanced, without assessing the cycles through q and r.

The Fixed-Point Theorem is useful for assessing whether complex structures are balanced, and thus, the likely dynamics of such structures (e.g., whether they are likely to change).

IV. Analyzing More Complex Structures

(Figure 2)		(Figure 3)

A. How many cycles are there in each of the above figures?

B. Is Figure 2 balanced?

C. Is Figure 3 balanced?

D. If either figure is balanced, what are the likely cliques?

E. If either figure is unbalanced, what kind of dynamics are likely to lead to balance?