Universality class of triad dynamics on a triangular lattice

Radicchi, F, Vilone, D, Meyer-Ortmanns, H
Phys. Rev. E 75,  021118 (2007)
Times cited: 5

Abstract

We consider triad dynamics as it was recently considered by Antal
[Phys. Rev. E 72, 036121 (2005)] as an approach to social balance. Here
we generalize the topology from all-to-all to the 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 parametrized 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 kinds of
absorbing states. The phases differ by the existence of an infinitely
connected (percolated) cluster of negative links that forms whenever p
<= p©. Moreover, for p <= p©, the time to reach the absorbing state
grows powerlike with the system size L, while it increases
logarithmically with L for p > p©. From a finite-size scaling
analysis we numerically determine the static critical exponents beta
and nu(perpendicular to) together with gamma, tau, sigma, and the
dynamical critical exponents nu(parallel to) and delta. The exponents
satisfy the hyperscaling relations. We also determine the fractal
dimension d(f) that satisfies a hyperscaling relation as well. The
transition of triad dynamics between different absorbing states belongs
to a universality class with different 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.