  Balance theory proposed by Heider for the first time modeled triplet
interaction in a signed network, stating that relationships between two people,
friendship or enmity, is dependent on a third person. The Hamiltonian of this
model has an implicit assumption that all triads are independent, meaning that
state of each triad, being balanced or imbalanced, is ineffective to others.
This independence forces the network to have completely balanced final states.
However, there exists evidence indicating that real networks are partially
balanced raising the question of what is the mechanism preventing the system to
be perfectly balanced. Our suggestion is to consider a quartic interaction
which dissolves the triad's independence. We use mean field method to study
thermal behavior of such systems where the temperature is a parameter that
allows the stochastic behavior of agents. We show that under a certain
temperature, the symmetry between balanced and imbalanced triads will
spontaneously break and we have a discrete phase transition. As consequence
stability arises where either similar balanced or imbalanced triads dominate,
hence the system obtains two new imbalanced stable states. In this model, the
critical temperature depends on the second power of the number of nodes, which
was a linear dependence in thermal balance theory. Our simulations are in good
agreement with the results obtained by the mean field method.
