Universality class of triad dynamics on a triangular lattice

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<or=p(c). Moreover, for p<or=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 static critical exponents beta and nu(perpendicular) together with gamma, tau, sigma, and the dynamical critical exponents nu(parallel) 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.