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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 polarization 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. When heating the system, one observes a discontinuous and irreversible phase transition at $T^c$ from a nearly balanced state when the mean link polarization is about $x_c=0.796388$ to a disordered and unbalanced state where the polarization vanishes. When the initial conditions for links polarization are random, then at low temperatures a balanced bipolar state of two mutually hostile cliques exists that decays towards the disorder and there is a discontinuous phase transition at a temperature $T^d$ that is lower than $T^c$. The system phase diagram corresponds to the so-called fold catastrophe when a stable solution of the mean-field equation collides with a separatrix, and as a result a hysteresis-like loop is observed.