Gain and loss of esteem, direct reciprocity and Heider balance

Abstract

The effect of gain and loss of esteem is introduced into the equations of time evolution of social relations, hostile or friendly, in a group of actors. The equations allow for asymmetric relations. We prove that in the presence of this asymmetry, the majority of stable solutions are jammed states, i.e. the Heider balance is not attained there. A phase diagram is constructed with three phases: the jammed phase, the balanced phase with two mutually hostile groups, and the phase of so-called paradise, where all relations are friendly.

The Heider balance and the looking-glass self: modelling dynamics of social relations

“…Stochastic algorithms designed to locate energy minima have been applied to social networks 19,20 . Besides global minima, equivalent to the structural balance, unbalanced stationary states have been identified 19 , hereafter called ‘jammed’ or ‘glassy’ states 19,21–23 .…”

“…On the other hand, the concept of balanced triads has found a psychological support in terms of cognitive dissonance 24 . While most of works have been concentrated on the case of symmetric relations, where negative and positive relations towards each other are always reciprocated, some papers have dealt also with the asymmetric case 21,25 . Although asymmetric relations are established in social modeling 26,27 , their consequences for the structural balance are not fully explored.…”

“…Instead of stochastic algorithms 19,20 , we intend to apply a set of differential equations 28,29 which is supposed to drive the system to a stationary state most close to the initial state. In most cases, jammed states are obtained 21 ; this is the main difference with respect to algorithms of simulated annealing, a standard method formulated as to reach a global minimum of energy 30 . Although some jammed states are possible also for symmetric case 19 , for asymmetric relations they are generic 21 .…”

“…In most cases, jammed states are obtained 21 ; this is the main difference with respect to algorithms of simulated annealing, a standard method formulated as to reach a global minimum of energy 30 . Although some jammed states are possible also for symmetric case 19 , for asymmetric relations they are generic 21 .…”

Glassy States of Aging Social Networks

“…In order to introduce the influence of memory on social networks, we have followed the Heider balance theory [1] that has provided a fundamental platform for tackling many sociological problems ranging from social systems [23][24][25][26] to political ones [27], and e.g. online networks [28,32] (see [33] and references therein).…”