Mean-field approximation for structural balance dynamics in heat-bath

A critical temperature for a complete signed graph of $N$ agents where time-dependent links polarisation tends towards the Heider (structural) balance is found analytically using the heat-bath approach and the mean-field approximation as $T^c=(N-2)/a^c$, where $a^c\approx 1.71649$. The result is in perfect agreement with numerical simulations starting from the paradise state where all links are positively polarized as well as with the estimation of this temperature received earlier with much more sophisticated methods. While heating the system one observes a discontinuous and irreversible phase transition at $T^c$ from a nearly balanced state state when the mean link polarisation is about $x_c=0.796388$ to a disordered and unbalanced state where the polarization vanishes. When initial conditions for links polarisation are random then in low temperatures a balanced bipolar state of two mutually hostile cliques exists that decays towards the disorder and there is discontinuous phase transition at a temperature $T^d$ that is lower than $T^c$. The system phase diagram corresponds to the so-called fold catastrophy when a stable solution of mean-field equation collides with a separatrix and as result a hysteresis-like loop is observed. When the system crosses the tipping point $(T^c,x^c)$ then a return to the paradise state is not possible. Cooling the system from a higher temperature leads to a phase with two hostile groups.Comment: 8 pages, 6 figures, 1 tabl