Reply to "Comment on 'Phase transition in a network model of social balance with Glauber dynamics' "

Recently, we introduced a stochastic social balance model with Glauber dynamics which takes into account the role of randomness in the individual's behavior [Phys. Rev. E 100, 022303 (2019)2470-004510.1103/PhysRevE.100.022303]. One important finding of our study was a phase transition from a balance state to an imbalance state as the randomness crosses a critical value, which was shown to vanish in the thermodynamic limit. In a similar study [Malarz and Kułakowski, Phys. Rev. E 103, 066301 (2021)10.1103/PhysRevE.103.066301], it was shown that the critical randomness tends to infinity as the system size diverges. This led the authors to question the appropriateness of the results in our Monte Carlo simulations, when compared with the non-normalized form, used in their work. The normalized form of energy in our model is, in fact, a common choice when one deals with systems comprising long-range interactions. Here, we show how their probabilistic definition leads to vanishing possibility of forming negative bonds, thus leading to a frozen ordered (paradise) state for any amount of finite randomness (temperature) for large enough system size. On the other hand, in the same large system size limit, our model is unstable to thermal randomness due to global, long-range effect of changing a bond's sign. We also address the rule of different updating mechanisms (synchronous vs sequential) in the two models. We finally discuss the distinction between the balanced states reached by each model and provide arguments for social relevance of our model.