  The dynamics of social relations and the possibility of reaching the state of
structural balance (Heider balance) are discussed for various networks of
interacting actors under the influence of the temperature modeling the social
noise level. For that purpose, two main types of lattices are considered. The
first is created by removing some links from a regular triangular lattice to
produce a diluted triangular lattice, and the second by adding more links to
create an enhanced triangular lattice. In both those cases, the full range of
possible graph densities is discussed, limited by the extreme cases of networks
which consist of a small number of separated triads and fully connected
networks. It is shown that the existence of the balanced state is not possible
if the average node degree is too close to the value characterizing the regular
triangular lattice. Otherwise, both balanced (or partially balanced) and
imbalanced states are possible, depending on the temperature. However, only for
graphs which are dense enough a phase transition of the first kind is observed,
while less enhanced networks (and all diluted) indicate a smooth cross-over
between the two states. The cross-over temperatures are size-independent only
for the diluted triangular lattices and depend on the size of the system for
the enhanced triangular lattices, as is the case also for the critical
temperatures of the phase transition observed in denser enhanced lattices.
