Second- to first-order phase transition: Coevolutionary versus structural balance

In social networks, the balance theory has been studied by considering either the triple interactions between the links (structural balance) or the triple interaction of nodes and links (coevolutionary balance). In the structural balance theory, the links are not independent from each other, implying a global effect of this term and it leads to a discontinuous phase transition in the system's balanced states as a function of temperature. However, in the coevolutionary balance the links only connect two local nodes and a continuous phase transition emerges. In this paper, we consider a combination of both to understand which of these types of interactions will identify the stability of the network. We are interested to see how adjusting the robustness of each term versus the other might affect the system to reach a balanced state. We use statistical mechanics methods and the mean-field theory and also the Monte Carlo numerical simulations to investigate the behavior of the order parameters and the total energy of the system. We find the phase diagram of the system which demonstrates the competition of these two terms at different ratios against each other and different temperatures. The system shows a tricritical point above which the phase transition switches from continuous to discrete. Also the superiority of the local perspective is observed at low temperatures and the global view will be the dominant term in determining the stability of the system at higher temperatures.