  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.
