  We consider triad dynamics as it was recently considered by Antal \emph{et
al.} [T. Antal, P. L. Krapivsky, and S. Redner, Phys. Rev. E {\bf 72}, 036121
(2005)] as an approach to social balance. Here we generalize the topology from
all-to-all to a regular one of a two-dimensional triangular lattice. The
driving force in this dynamics is the reduction of frustrated triads in order
to reach a balanced state. The dynamics is parameterized by a so-called
propensity parameter $p$ that determines the tendency of negative links to
become positive. As a function of $p$ we find a phase transition between
different kind of absorbing states. The phases differ by the existence of an
infinitely connected (percolated) cluster of negative links that forms whenever
$p\leq p_c$. Moreover, for $p\leq p_c$, the time to reach the absorbing state
grows powerlike with the system size $L$, while it increases logarithmically
with $L$ for $p > p_c$. From a finite-size scaling analysis we numerically
determine the critical exponents $\beta$ and $\nu$ together with $\gamma$,
$\tau$, $\sigma$. The exponents satisfy the hyperscaling relations. We also
determine the fractal dimension $d_f$ that fulfills a hyperscaling relation as
well. The transition of triad dynamics between different absorbing states
belongs to a universality class with new critical exponents. We generalize the
triad dynamics to four-cycle dynamics on a square lattice. In this case, again
there is a transition between different absorbing states, going along with the
formation of an infinite cluster of negative links, but the usual scaling and
hyperscaling relations are violated.

Statistical Mechanics