A small chance of paradise —Equivalence of balanced states

AGH University of Science and Technology, Faculty of Physics and Applied Computer Science al. Mickiewicza 30, 30-059 Kraków, Poland

Received: 11 April 2017
Accepted: 24 July 2017

Abstract

A social network is modeled by a complete graph of N nodes, with interpersonal relations represented by links. In the framework of the Heider balance theory, we prove numerically that the probability of each balanced state is the same. This means in particular, that the probability of the paradise state, where all relations are positive, is . The proof is performed within two models. In the first, relations are changing continuously in time, and the proof is performed only for N = 3 with the methods of nonlinear dynamics. The second model is the Constrained Triad Dynamics, as introduced by Antal, Krapivsky and Redner in 2005. In the latter case, the proof makes use of the symmetries of the network of system states and it is completed for .