The lack of signed random networks in standard balance studies has prompted us to extend the Hamiltonian of the standard balance model. Random networks with tunable parameters are suitable for better understanding the behavior of standard balance as an underlying dynamics. Moreover, the standard balance model in its original form does not allow preserving tensed triads in the network.

Therefore, the thermal behavior of the balance model has been investigated on a fully connected signed network recently. It has been shown that the model undergoes an abrupt phase transition with temperature. Considering these two issues, we examine the thermal behavior of the structural balance model defined on Erdös-Rényi random networks within the range of their connected regime.

We provide a mean-field solution for the model. We observe a first-order phase transition with temperature for a wide range of connection probabilities. We detect two transition temperatures, $T_{\mathrm{cold}}$ and $T_{\mathrm{hot}}$, characterizing a hysteresis loop. We find that with decreasing the connection probability, both $T_{\mathrm{cold}}$ and $T_{\mathrm{hot}}$ decrease. However, the slope of decreasing $T_{\mathrm{hot}}$ with decreasing connection probability is larger than the slope of decreasing $T_{\mathrm{cold}}$.

Hence, the hysteresis region gets narrower until it disappears in a certain connection probability. We provide a phase diagram in the temperature-tie density plane to accurately observe the metastable or coexistence region behavior. Then we justify our mean-field results with a series of Monte Carlo simulations.

R. Masoumi, F. Oloomi, S. Sajjadi, A. H. Shirazi, G. R. Jafari, Modified Heider balance on Erdös-Rényi networks, Physical Review E 106 (2022) 034309