Balance Theory with Incomplete Awareness

Abstract

In this paper, we reformulate balance theory by allowing actors to possess incomplete awareness of the evaluations held by other actors, and by adopting balance closure (modified to allow incomplete awareness) as an equilibrium concept. Our treatment highlights psychological mechanisms, maintains a clear distinction between actors and objects, emphasizes the effects of self-awareness and self-evaluations of actors, and permits actors to hold ambivalent (simultaneously positive and negative) evaluations. Our analysis extends previous results linking the imbalance of a signed graph to ambivalence in its balance closure and reveals that an actor's “indirect awareness” of imbalance is necessary but not sufficient for that actor's ambivalence in the balance closure.

Keywords:

• ambivalence
• balance closure
• signed graph

I am grateful to John Martin for helpful comments.

Notes

¹See Doreian (2004) for further discussion.

²Adopting notation developed in the next section, we may offer two (more formal) definitions of the transitive closure. First, the transitive closure R_∗ is the minimal set such that R_∗ ⊇ R ∪ R_∗R_∗. Second, proceeding recursively, let R _t+1 = R _t  ∪ R _t R _t . Setting R₀ equal to R, the transitive closure R_∗ is given by R_∞.

³Formal definitions will be developed below.

⁴While Harary et al. (1965) may be credited with this result, we adopt the term “balance closure” from Batagelj (1994). Both sources compute the balance closure using a semiring, while we employ Boolean algebra more familiar to social network analysts.

⁵Going beyond the present analysis, it would seem natural to allow actor i to be aware of some – not necessarily all – of actor j's evaluations. However, we leave this extension for future research.

⁶Consider a semicycle originating at node i. Choosing (arbitrarily) some node j along the semicycle, there is a semiwalk of type ω₁ from node i to node j, and a semiwalk of type ω₂ from node j to node i. Thus, the semicycle is a semiwalk of type ω₁ω₂ from node i to node i. If ω₁ and ω₂ always have the same sign, then ω₁ω₂ must be positive.

⁷To see this, note that the sets of evaluations must weakly expand at each step, in the sense that P _t+1 ⊇ P _t and N _t+1 ⊇ N _t for all t. If either of these containment relations is strict (so that P _t+1 ⊃ P _t or N _t+1 ⊃ N _t ), then the recursion continues. If both containment relations are equalities (so that P _t+1 = P _t and N _t+1 = N _t ), then the recursion stops. Because S is finite, the recursion must stop after a finite number of steps.

⁸For each mechanism, there are four cases that might be illustrated: positive-positive, negative-negative, negative-positive, and positive-negative. Thus, if each case of each mechanism was labeled as a distinct submechanism, there would be 16 submechanisms. While Davis' (1967) analysis of clusterability presumes that the negative-negative submechanisms are weaker than the others, the present analysis assumes that all 16 submechanisms are operating.

⁹Batagelj (1994) offers similar examples. But to our knowledge, Proposition 4 has not been stated explicitly in the existing literature.

¹⁰As an exercise, the reader might use equation (2) to enumerate all of the compound relations comprising P₂. To give a fragment of the answer, P₂ ⊇ (P₁P₁′ ∩ A) ⊇ [[P ∪ (PP′ ∩ A) ∪ (NN′ ∩ A) ∪ (P ∩ A)P ∪ (N ∩ A)N] [P′] ∩ A] = [PP′ ∩ A] ∪ [(PP′ ∩ A)P′ ∩ A] ∪ [(NN′ ∩ A)P′ ∩ A] ∪ [(P ∩ A)] PP′ ∩ A] ∪ [(N ∩ A)NP′ ∩ A]. Note that it is possible to obtain each of the preceding relations by step 2 of the procedure specified in Definition 4.

¹¹Formally, we would need to establish a mapping from the nodes in the graph in Figure 3 to the nodes of the graph of (P, N, A). It is important to realize that this mapping need not be one-to-one. That is, two (or more) distinct nodes in the graph in Figure 2 may be mapped into the same node in the graph of (P, N, A). For instance, if nodes 4 and 6 in the graph in Figure 2 both map into some node k in the graph of (P, N, A), the positive edge 4P6 reflects actor k's positive self-evaluation, and actor 1 is both aware of k's evaluations and holds a positive evaluation of k.

¹²Beyond inspection of Figure 3, it is also possible to establish actor 1's indirect awareness of these evaluations algebraically. It can be shown that (F ∩ A)(G ∩ A) ⊆ (F ∩ A)G ∩ A² and that F(G ∩ A′) ∩ A ⊆ (F ∩ A²)G ∩ A for any relations F and G on S. Using these results, we obtain φ ⊆ ((((PP′ ∩ A)N′ ∩ A²)N′ ∩ A)PN′ ∩ A²)P. This condition makes it easier to detect actor 1's indirect awareness of the evaluations comprising the semiwalk of type ω_φ from 1 to 8. Reconciling this condition with Figure 3, note that 1PP′3 and 1A3, that 1(PP′ ∩ A)N′4 and 1A²4, etc.

¹³It may again be instructive to return to Heider (1946). Although he did not interpret the “unit” relation as awareness, he did suggest that actors might resolve an imbalanced situation by breaking the relevant unit relation without altering their evaluations.

¹⁴Interpreting the recursive computation of balance closure in Definition 2 as real-time dynamics, we could easily add another process (at the end of each step t) by which evaluations are randomly removed. Thus, evaluations would be added deterministically (through balance-theoretic mechanisms) and then removed stochastically (by chance) at each step.