Heider balance, asymmetric ties, and gender segregation

Abstract

To remove a cognitive dissonance in interpersonal relations, people tend to divide our acquaintances into friendly and hostile parts, both groups internally friendly and mutually hostile. This process is modeled as an evolution towards the Heider balance. A set of differential equations have been proposed and validated (Kulakowski et al, IJMPC 16 (2005) 707) to model the Heider dynamics of this social and psychological process. Here we generalize the model by including the initial asymmetry of the interprersonal relations and the direct reciprocity effect which removes this asymmetry. Our model is applied to the data on enmity and friendship in 37 school classes and 4 groups of teachers in México. For each class, a stable balanced partition is obtained into two groups. The gender structure of the groups reveals stronger gender segregation in younger classes, i.e. of age below 12 years, a fact consistent with other general empirical results.

Gain and loss of esteem, direct reciprocity and Heider balance

“…It is easy to notice that close to the stationary state, where (|x i j | ≈ 1), the equation reduces to the case where γ = 0. Yet, as we have found already in [18], the action of the mechanism of direct reciprocity is to reduce the asymmetry of the relations before the system is trapped in a jammed state. If the coefficient α is too small, the process is too slow and the Heider balance is not attained.…”

“…Recall that in the case of symmetric relations, jammed states are not generic; this is so both for discrete [13] and continuous [28] dynamics. Basically however, experimentally collected data on relations are not symmetric, as discussed in [31]; a recent example has been reported in [18]. Exceptions from this rule can be induced by an experimental method, as when the intensity of communication in pairs is measured [32].…”

“…In parallel, a set of deterministic differential equations has been proposed as a model of RCD [14,15,16]. This approach has been generalized to include asymmetric relations [17] as well as the mechanism of direct reciprocity [18], which was supposed to remove the asymmetry.…”

“…Now let us consider jammed states in asymetric matrix of relations, where the condition x ji = x i j does not hold. In [18], a series of such states has been identified by a numerical check. Here we show that all of them can be captured in the following scheme.…”

“…Accordingly, the coefficient 1 − α is a measure of intensity of RCD. The former mechanism has been introduced in [18]; for initially asymmetric relations, its appearance is a necessary condition for the Heider balance. Varying α, we control the mutual rate of the two processes.…”