Universality class of triad dynamics on a triangular lattice

Abstract

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 $β$ and $ν$ together with $γ$, $τ$, $σ$. 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.

9 pages, 19 figures