Topologically induced changes in thermal properties of structurally balanced systems

The dynamics of social relations and the possibility of reaching the state of structural balance (Heider balance) are discussed for various topologies of the networks of interacting actors under the influence of the temperature modeling the social noise level. For that purpose, two main types of lattices are considered. The first is created by removing some links from a regular triangular lattice to produce a diluted triangular lattice or adding more links to create an enhanced triangular lattice. The second is the classical random graph. In both those cases, the full range of possible graph densities is discussed, limited by the extreme cases of networks which consist of a small number of separated triads and fully connected networks. The performed simulations indicate that for random lattices a transition between the balanced and imbalanced states is always possible and occurs at the critical temperature which depends on the graph density. On the other hand, it is shown that this transition and existence of the balanced state are not possible for networks based on the diluted triangular lattices if the average node degree is too close to the value characterizing the regular triangular lattice. The critical temperatures are size-independent only for the diluted triangular lattices and depend on the size of the system for the enhanced triangular lattices and random graphs.